Chaos and fluctuations in nonlinear dissipative systems - American

Chaos and Fluctuations In Nonlinear Dissipative Systems. Raymond Kapral,* Mark Schell, and Simon Fraser. Department of Chemistry, University of Toront...
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J. Phys. Chem. 1982,86,2205-2217

Substitute x = (2[)1/2 and e, - 1/2 = n

H,” - 2rHn’ + 2nH, = 0

or (A‘10)

This is the Hermite equation and has Hermite polynomials as solution. Transforming back, we get Ib,) = e-xa/4vzH,(X/21/2V)

E , = -nb

H,(

$)e-x2/4(

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-:a) +

= ( n + ;)H,( $)e-x2/4 (A.14)

or

(A.ll)

(A.15)

(A.12)

To solve the eigenvalue problem (A.3), we essentially follow the same steps up to eq A.lO). We get

Performing the inverse transformation f r, we get H , , ( x / ~ ~ / ~=) F- n b H , ( ~ / 2 ~ / ~ ) (A.16) -+

so

(b,l = H,(X/21/2V) E,’= -nb

(A.17) (A.18)

Chaos and Fluctuatlons In Nonlinear Dlsslpatlve Systems Raymond Kaprai,’ Mark Schell, and Simon Fraser Depertment of chemistry, University of Toronto, Toronto, Ontario M5S 1A 1, Canada (Received: Ju& 20, 198 1)

The dynamics of many condensed-phasesystems can be described by forced, damped, nonlinear oscillator models. The deterministic motion of such oscillators is complex. In addition to a variety of periodic states, there are attracting sets on which the motion is chaotic (strange attractors). The origin and character of the chaotic states, which arise either by an infinite sequence of subharmonicbifurcations or abruptly by other mechanisms, are discussed. The types of chaotic behavior are illustrated with results for the Morse, Duffing, and cosine oscillators. Fluctuations play an important role in the study of rate processes in these systems and the effects of noise on the dynamics is also considered. Some specific noise-induced effects which are examined include transitions between chaotic and periodic solutions, and barrier crossing dynamics.

Introduction The operation of a large number of electrical and mechanical devices can be described by forced, dissipative, nonlinear oscillator equations of motion. As a consequence, certain aspects of the rather rich variety of dynamical behavior have been extensively studied and several excellent texts which describe these results exist.’ These studies focused primarily on the periodic solutions of the equations as well as the approximation techniques for analysis of such nonlinear oscillations. More recent investigations have been stimulated by the observations2that these systems can exhibit aperiodic states with many of the same features as the chaotic flows of nonlinear ordinary differential equations in three or more variables, which model some features of turbulence in fluid dynamics or macroscopic chemical rate processes? Also there has been work on the possible application of these nonlinear oscillator equations to microscopic condensed-phase relaxation processes.@

Many of the suggested applications are to solid-state phenomena such as pinned charge density waves: superionic conductivity: noise phenomena in Josephson junctions: and dislocation dynamics in crystal^.^ All of these examples have the feature that for reasonable system parameter values the chaotic states might be attained by pumping with available radiation or mechanical sources. These applications are also of interest from a chemical point of view since they often involve barrier crossing dynamics; indeed problems such as superionic conductivity have traditionally been discussed7 in terms of either transition-state theory or Kramers models.8 On a more speculative level, such forced oscillator equations may serve as crude models for molecular relaxation processes. Here the oscillator coordinate corresponds to a molecular degree of freedom, the driving term to a coupled periodic radiation field, and the viscous damping to the dissipative contribution of thermal fluctuations in the surrounding fluid or the remainder of the molecule. The nature of the coupling to the radiation field and the validity of such a simple

(1) See, for example, J. Stoker, “Nonlinear Vibrations”, Interscience, New York, 1950; N. Minorski, “NonlinearOscillations”,van Nostrand, Princeton, NJ, 1962; N. N. Bogoliubov and Y. A. Mitropolsky, “Asymptotic Methods in the Theory of Nonlinear Oscillations”,Gordon and Breach, New York, 1961. (2) P. Holmea, Appl. Math. Modelling, 1,362 (1977);Y. Ueda, J. Stat. Phys., 20,181 (1979); Ann.N.Y. Acad. Sci., 357, 422 (1980). (3)0.Gurel and 0. E. Rbsler, Ed., “Bifurcation Theory and Applicatione in Scientific Disciplines”, New York Academy of Science, New York, 1979; ‘Nonlinear Dynamics”, R. H. G. Hellman, Ed. New York Academy of Science, New York, 1980.

(4) B. A. Huberman and J. P. Crutchfield, Phys. Reu. Lett., 43,1743 (1979). (5) B.A. Huberman, J. P. CrutcEeld, and N. H. Packard, Appl. Phys. Lett., 37, 750 (1980). (6) C. Herring and B. A. Huberman, Appl. Phys. Lett., 36,975 (1980). (7) See, for example, “Physicsof Superionic Conductors”,M. B. Salamon, Ed., Springer-Verlag, New York, 1979. (8) H. A. Kramers, Physica, 7,284 (1940);S.Glasstone, K.J. Laidler, and H. Eyring, T h e Theory of Rate Processes”,McGraw-Hill,New York, 1941.

0022-365418212088-2205$01.25/0

0 1982 American Chemical Soclety

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classical model clearly should be considered before any serious study was undertaken. In all of the above-mentioned microscopic applications fluctuations enter in an important way. In a Langevin description of the dynamics a statistical specification of the force exerted by the bath on the oscillator is adopted; the damping is related to the random force correlations by a fluctuation-dissipation theorem. Hence the question of the response of the nonlinear oscillator to such fluctuations must be investigated. In this article we consider the chaotic dynamics of these systems with a view toward microscopic applications. Section I1 surveys some of the features of the periodic and chaotic states of the deterministic equations of motion. We focus in particular on the possible mechanisms which give rise to chaos and on the properties of the flow on the strange attractors, which are responsible for the complex dynamics. The extent to which the chaotic motions can be analyzed in terms of one-dimensional mappings of the unit interval onto itself is examined. Other features, such as, bistability and symmetry breaking are also briefly discussed. The effects are illustrated by considering three oscillator potential models: the Morse potential, which can model some aspects of molecular dynamics, the Duffing oscillator, which locally can describe barrier crossing dynamics, and the cosine potential, which can be used to model a number of systems such as superionic conductors. A Langevin equation description of the dynamics is presented in Section 111. Here, the effects of fluctuations on both the chaotic and periodic states are examined. When fluctuations are present escape from local potential minima is possible as well as transitions between bistable states. The dynamics associated with these processes is studied. The last section contains a discussion of the results with an aim to their application to condensed-phase rate processes.

11. Deterministic Dynamics of Driven Oscillators The structure of flow which is governed by the deterministic equations of motion controls to a large extent the character of the response of the system to fluctuations. Hence, we begin with an examination of how chaos can arise and be analyzed in a deterministic, dissipative system. The oscillator equations of motion we consider have the general form fj = -yq + m-1 F ( q ) + p cos ut (2.1) where F(q) = -dV(q)/dq is the (nonlinear) restoring force, y is f/m, with f the friction coefficient and m the mass, and p cos ut is the external driving field. Higher-order differential equations like eq 2.1, even with explicit time dependence, can be converted into a system of coupled first-order equations by defining new variables. A vector in the phase space of such variables uniquely defines the direction of the flow (velocity field) since the system is first order.g In this picture the time dependence is implicit (autonomous system) and our geometrical descriptions will always refer to motion in such a space or on maps derived from it. Due to the presence of viscous damping (TO), the phase space flow corresponding to these equations is contracting (negative divergence) and thus the volume of a parcel of (phase) fluid, followed along a trajectory, decreases steadily with time. The divergence is in fact -y and the flow is globally contracting; an initial volume V, of phase fluid (9) See, for instance, G. Birkhoff and G . C. Rota, “Ordinary Differential Equations”, Wiley, New York, 1969.

Kapral et al.

shrinks as V(t) = exp(-yt)V@Hence, the flow approaches a measure zero set of points in the phase space (an attractor). The existence of such attractors distinguishes dissipative system from conservative systems where phase space volumes are preserved. The character of the attracting set depends, of course, on the nature of the oscillator equations. Consider, for example, the simple case of a viscously damped oscillator with a harmonic restoring force, F(q) = -mu&; in this case all solutions are attracted to the origin (q = 0, q = u = 01, which is a stable fixed point of the motion. However, in a two-dimensional phase space when nonlinearities are present, more complex limit cycle attractors may also exit. A standard example which illustrates this type of behavior is the van der Pol oscillator’

4 = -y(cq2

- 1)q - w:q

Here, the nonlinear damping, which is positive for small q and negative for large q, in contrast to simple viscous damping, is responsible for the appearance of the (onedimensional) limit cycle attractor. Such single secondorder equations are equivalent to a coupled pair of firstorder equations, i.e., plane autonomous systems, and for such systems of equations these two types of attractors exhaust the list of possible attracting sets (PoincarBBendixson theorem).’O However, in higher dimensional systems a new class of attractors is found. These strange attractors” arise from flows which, although contracting in some directions, have the property that neighboring trajectories exponentially separate in other directions. This leads to the possibility of a flow with statistical properties in a deterministic, dissipative system.12 The driven, dissipative oscillator with one degree of freedom (eq 2.1) is a system which can exhibit such complex behavior since this second-order equation of motion may be written as a coupled system of three fiist-order, ordinary differential equations (a three-dimensional autonomous system):

Q =u

U

= -YU

+ m-l F(q) + p

COS

6

b = w (2.3)

We call the phase space of this system r (q,u,OI. Next, we give a qualitative discussion of the nature of the attractors (periodic and chaotic) for this model dynamical system. Attractors and Basins. We first note that 0 increases monotonely as the system evolves. Thus driven systems have no fixed points and 0 extends to infinity over an infinite epoch. (In contrast, “intrinsically” autonomous systems may ultimately be confined to a bounded region of phase space, and may possess fixed points.) Given the contracting nature of a dissipative flow, there is typically a measurable set of points in the phase space which eventually reach the attractor under the phase flow; this set of points is the basin of the attractor. In general, even for the simple driven, dissipative flows considered here, there may be more than one, indeed infinitely many attractors, each with its associated basin. Different basins are separated by boundaries and the geometry of basins, boundaries, and attractors may be very complicated. For driven flows, attractors and their basins must be extended in the timelike variable 6, and if the potential in which the particle moves is periodic, they can be configurationally unbounded as well. The cosine potential, V ( q ) = a cos (10) H. Haken, ‘Synergetics”, Springer-Verlag, New York, 1978. (11) D. Ruelle and F. Takens, Commun. Math. Phys., 20,167 (1971). (12) Numeroua relevant articles can be found in ref 3. See also, R. Shaw, 2.Naturforsch. A, 36,M (1981); R. G. Helleman in “Fundamental Problems in Statistical Mechanics”, Vol. 5, E. G . D. Cohen, Ed., NorthHolland, Amsterdam, 1980.

Chaos in Nonlinear Dissipative Systems

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The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 2207

r

b +2lT

r

Figure 2. Picture of helical flow in for a bifurcated attractor with one twist between (periodic) surfaces Zooand Ze+2s. Velocity v and position q are the coordinates in the planes Z. 11’ is the unstable orbit that has given rise to the winding segments 22‘ and 33’, where n’ is the image of n under the Poincar6 map. Since 2‘ = 3 and 2 = 3’ the winding segments can be joined in two distinct ways to form subharmonic orbits. Locally relaxation takes place across the ribbon from the neighborhood of 11‘ to the outer subharmonic segments. The double loop P is the projection of the segments 23‘ and 32‘ onto &0+2,.

sections of the flow at separation A8 = 27r we can construct this Poincar6 map the Poincar6 map of the surface (zoo); is a difference equation in the q,u plane with general form x,+1 = fb,) Figure 1. Projections of the strange attractors on the q , v plane for (a) cosine oscillator, w = 0.567, p = 0.118, y = 0.4; (b) Morse oscillator, w = 0.54, p = 0.42, y = 0.4; and (c) Duffing oscillator, w = 0.546, p = 0.1 175, y = 0.4. The calculations were carried out with eq 2.3 in dimensionless form; times were scaled by w0-l while distances were scaled by qO,where q 0 is the distance between the successive maxima in the cosine potential, the distance between the two maxima in the quartic potential, and 9 0 = p-’ for the Morse potential. Also, for the Morse potential, De = mw,2/2B2. Hence, three dimensionless parameters enter the equations c?, = w/wo, 7 = y/wo, and p = p / ~ : 9 0 . I n reporting the values of these parameters above and in the rest of the paper we dispense with the overbar. Equation 2.3 was integrated in double precision on a SEL 75 machine by using a predictor-corrector method with a variable time step.

(27rq/q0), affords an example; apparently statistical hopping of the particle from one well to another is associated with a strange attractor periodically extended in configuration space.5 Potentials like the Duffjig, V(q)= ‘ / g ” q 2 + bq4 ( b < 0), which are unbounded from below, provide models for the study of barrier crossing dynamics. The basins and attractors for such potentials must be confined to a region surrounding the potential well as the particle rapidly escapes to infinity outside the well. The bounded phase flow in these systems is restricted to a cylindrical region; the vector field points everywhere into the surface of the cylinder and the attractor is contained in the cylinder. When it is not possible to fiid such a surface, escape from the potential well may occur. The Morse potential, V(q)= D,[exp(2&) - 2 exp(-Pq)], provides an instance of a function that goes smoothly to infinity so that attractors that are elongated in configuration space are possible for this system. Examples of chaotic attractors with these characteristics are shown in Figure 1;these representations of attractors are projections in the q,u plane and are of course extended in the 8 variable. Since the focus of this article is primarily on such chaotic states and their influence on relaxation processes, we shall now give a brief description of how such states may arise and discuss the character of the chaotic flow. Chaotic Flows and Poincarg Maps. For driven-dissipative motion the simplest response is periodic entrainment; the asymptotic trajectory is a helix in r, wound periodically about the 8 axis. From successive, plane cross

(2.4)

where x, = (q(8=80+27rn),~(8=8,+2~n)) and f is some (vector-valued) nonlinear function. For periodic entrainment the helix in I’ intersects &,, at a single fixed point, so that x = f(x); f and therefore x depend parametrically on eo. Locally, in this simplest example, Zoopossess no other fixed points and the periodic helix is an attractor in I’ corresponding to the set of all attracting fixed points x(8,) on the surfaces Z . Imagine moving Z so that 80 increases from 0 to 27.eh% ! trajectory traced out by ~ ( 8 , ) on this movable surface Z is a limit cycle and is the projection of the periodic helix onto the q,u plane (Figure 2). As the parameters of the differential equation are changed a number of different types of bifurcation may occur and throughout this section one should imagine the parameters of the appropriate equation of motion being slowly adjusted to produce the progression of dynamical behavior described. An ubiquitous bifurcation mechanism involves the generation of period doubling subharmonics in I?. More precisely, at the bifurcation parameter value, the originally stable helix becomes marginally stable and then gives rise to two neighboring stable orbits of twice the period of the driving term. Each such subharmonic is a helix with double period and one can be generated from the other by translation along 0 through a distance 27r. The appearance of subharmonics implies that bifurcation takes place (locally) on a manifold with an odd number of twists per period13 (see Figure 2). (With periodic boundary conditions on 8 this manifold is a Mobius strip, embedded in a toroidal phase space.) Correspondingly on any surface Zoothe original fixed point becomes unstable, spawning two stable fixed points in its neighborhood. The diagram of this process has a characteristic “pitchfork” configuration as a function of the system parameters. The two stable fixed points on any Zoocorre(13) At each subsequent subharmonic bifurcation it should be clear that for the orbit to double its period an odd number of twists (usually one) is introduced per period of the subharmonic, not per period of the driving term. The fact that the manifold is twisted can be seen from the alternate (as opposed to monotone) character of the convergence to the periodic orbit near bifurcation;such alternate convergence is only possible if the manifold has a twist. Also see, J. Mallet-Paret and J. A. Yorke, Ann. N.Y. Acad. Sci., 357, 300 (1980).

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Flgure 3. A subharmonic sequence for the Morse oscillator which leads to chaos. In this sequence y and p were fixed at 0.4 and 0.42, respectively, while w was decreased: (a) period two, w = 0.8; (b) period four, w = 0.65; (c) period eight, w = 0.615; (d) a (possibly) eight-banded strange attractor, w = 0.6. The figures show the projections of the orbits on the q ,v plane (ordinate is v and abscissa is

4 1.

spond to distinct intersections of the two subharmonic trajectories in 8. In projection on the q,u plane the subharmonics appear as double loops (which would be traced on a surface 2: moving through two periods of the driving term along e). As the differential equations’ parameters are further changed a cascade of these pitchfork bifurcations occurs, leaving unstable hyperbolic fixed points on ZB, at each bifurcation vertex. Since the constant negative divergence of the flow in I’ implies a global contraction of area for the Poincar6 mapping on 8,unstable nodes are inconsistent with this global c~ntraction.’~Correspondingly in r we observe a rapid proliferation of helices, doubling their period of at each bifurcation. (A series of such subharmonics for the Morse oscillator is shown in Figure 3.) Locally each (stable) binary vertex in the tree of this cascade looks more and more like a scaled down version of ita predecessor, the bifurcation points converging geometrically in parameter space. Cascade implies an exponentially increasing number of attracting fixed points (-29, which in the limit ( n a)becomes a Cantor set; any deterministic trajectory on this set must be aperiodic. As these fixed points multiply in 8 the set immediately attracted by any one of them shrinks, i.e., for large n a (discrete) trajectory must approach an attracting fixed point very closely indeed before its distance from the fixed point decreases geometrically under f. This complicated transient behavior is associated with successive generations of hyperbolic points born during the cascade. At the cascade’s limit (in parameter space) no finite attracting set remains and the motion of the system is governed entirely by the confiiation of these hyperbolic points and their associated manifolds. (A Poincarg surface of section of the chaotic Morse oscillator flow in Figure I b is shown in Figure 4.) One of the most successful parts of recent bifurcation theory has been the description of the scaling behavior of this cascade of attractors into chaos. We give a s u m m a r y of some of these results in the next subsection. It is evident from this descriptive account of the subharmonic cascade that the nature of the chaotic flow can be examined by considering the motion in the vicinity of the hyperbolic fixed points in Zoo,A quantitative study

-

(14) For driven-diesipativesystem the corresponding maps have the stronger property of being uniformly contracting, i.e., having constant Jacobian, J,over the q,u, plane; they are Cremona maps. In fact, the contraction factor for the area is the same as that for the volume discussed earlier, J =

Flgwe 4. PolncarO maps of the chaotic flow in Figure lb. The figure shows maps obtained from four surfaces of section corresponding to (1) 0, = 0, (2)0, = d 2 , (3)Bo = a,and (4)Bo = 3d2. Since the attractor has two bands each PoincarO map consists of two disjoint segments.

of these fixed points entails, of course, a precise specification of the Poincar6 map appropriate for the particular oscillator flow under investigation. Detailed studies of model two-dimensional maps can be found in a number of sources.1s One of the most widely studied dissipative maps in two variables is that due to H6non:16 yn+1= bx,

xn+1= 1 + yn -

(2.5)

This map was constructed” to mimic certain features of the Poincar6 map of the Lorenz equations,18 which model convective instabilities in fluid systems. Recently Holmeslg has constructed and studied the map yn+l = X ,

xn+1

= -by,

+ xn(d - x n 2 )

(d I O , b

> 0)

(2.6)

which captures much of the dynamics of the Poincar6 map associated with a driven Duffing oscillator. Rather than review the results obtained for these specific maps or discuss the detailed properties of the Poincar6 maps of the oscillator models studied here, we give below a qualitative discussion of certain general features which are common to all such maps. As mentioned earlier, at or beyond the first subharmonic limit, in the absence of finite attractors, the asymptotic motion of the system is governed entirely by hyperbolic points. As the equations’ parameters are changed attractors may appear suddenly out of chaos by a “tangent” mechanism (discussed later); such attractors and their harmonics are embedded in the sequence of chaotic “hyperbolic” motions, which we now examine. Each hyperbolic point, h say, is associated with a stable (W,) and unstable (W,) manifold (Figure 5). The manifolds are invariant since they transform into themselves under the (15) I. Gumowski and C. Mira, “Recurrence and Discrete Dynamic Systems”, Springer-Verlag,New York, 1980; G. Iooas, ‘Bifurcations of Maps and Applications”,North-Holland, Amsterdam, 1979. (16) M. HBnon, Commun. Math. Phys., 50,69 (1976). (17) M. HBnon and Y. Pomeau, Lect. Notes Math., No. 565, 29-68 (1976). (18) E.N. Lorenz, J . Atmos. Sci., 20, 130 (1963). (19) P. Holmes, Phil. Trans. R. Soc. London, 292,419 (1979).

The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 2209

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I

I

J.

2

h I

m e 5. (a) Schematic representation of the homoclinic intersections of the stable (W,) and unstable (W,) manifolds emanating from the hyperbolic point h . (b) Diagrams of the deformation of a neighborhood N containing h under powers of the PoincarB map. Stage 0 to 1 represents stretching and incipient folding of N , in 1 to 2 the production of a complete fold and in 2 to 3 repetition of this folding process. Note that we make successhrely 0 , 1 , 2 , and 4 intersections with the vertical line. For dissipative maps the area of N decreases rapidly, thus the transformation is like a contracting baker’s transformation.

Poincar6 map, W,,, = f (W,,,). Locally these invariant manifolds can be found by a linearized analysis of the Poincar6 map, eq 2.4, about the fixed point and correspond to the eigenvectors of the linearized analysis. The unstable manifold (that associated with the eigenvalue IA21 > 1)may then be continued throughout Z from the iterates of a small segment along the unstable eigenvector under the map f. Similarly, the stable manifold (that associated with the eigenvalue 1x1 < 1 may be generated from a small segment on the stable eigenvector under the time-reversed mapping f-’. These manifolds of h in Zooare folded lines of infinite length, in general, which may be confined to a finite region or wander unboundedly. For an isolated hyperbolic point W, and W, may either not intersect, just touch, or intersect transversally. It is easy to see20 that one transverse intersection (homoclinic point) implies an infinite number since the image of any point x E W, n W, also belongs to this intersection, as W,,, are invariant manifolds. But as we iterate x ( # h) under f forward in time we must eventually approach h since x E W,. Conversely all preimages of x belong to W, and therefore must have been arbitrarily close to h in the infinite past. So no point in W, n W, (but h) is fixed or periodic and thus W, and W , must intersect in an infinite set of homoclinic points. Since the associated fixed point, h, is hyperbolic the mapping f leads to exponential separation of almost all points on W,(in the future) and W , (in the past) near h, except for those points in contained W , n W , that are locally connected to h. Consequently, near h the folds of W, (respectively W,) are infinitely compressed transverse (20) See, for example, D. R. J. Chillingworth,‘Differential Topology with a View to Applications”, Pitman Publishing, London, 1976.

to the direction of the stable (respectively unstable) eigenvector of h, and infinitely extended parallel to the unstable (respectively stable) eigenvector of h, to form an arbitrarily fine mesh21 (Figure 5a). (A similar picture in terms of surfaces applies in I’.) In spite of the fact that f is an areally contracting map almost all iterates o f f separate exponentially near h. For contracting maps and flows the above description of homoclinic intersections provides an explanation of the ultimately attracting nature of the W , belonging to any particular hyperbolic point (h)fixed under f say (in the absence of other complicating features).22 Consider the successive images under f of a small neighborhood N containing h. Every f,(N) in the set is connected as h is fixed by assumption. After many iterates n the portion of f n ( N ) associated with the homoclinic points emanating from h along W, returns to h along W,. The “feedback” of material to h occurs infinitely often as points returning close to h along W , are re-emitted along W,. Note that at every iteration the area (measure) of the set f,(N) decreases by a constant factor (the Jacobian of the Cremona map f). This global contraction of area combined with the return mechanism associated with the homoclinic points implies that, for large n, returning segments of f,(N) near h lie closer and closer to W,. Certainly W, C fm(N)since by definition W, is invariant under f and can be generated from a small but measurable segment of W, near h. It has therefore been suggested23that, in the absence of other complications, the closure of W , is a strange attractor. This sketch of the feedback mechanism for generating chaotic orbits also suggests the way in which the strange attractor associated with the closure of W, has a structure that is locally the product of a manifold and a Cantor set. This geometrical form underlies the components of the banded chaos appearing beyond the subharmonic limit where points at this limit have been replaced by lines in Z (or surfaces in I’) as typical elements of the Cantor set. Globally the transformation of N under some sufficiently high power of f produces a folded image, or Smale “horseshoe”.” Under further iterations off the “free” end of the horseshoe returns ultimately to h; the middle segments are mapped away from h along W , and the remote curved extremity returned to h along the homoclinic orbits C W,. This transformation is like a continuous contracting baker’s transformation25(see Figure 5b). Repetition of this mapping, which corresponds to some fixed power o f f , doubles both the length of the image of the horseshoe and the numbers of intersections made by this image on a cut transverse to W,; globally successive kneading transformations ultimately produce a fractal curve of (exponentially) infinite length and locally the intersection of this curve with a transverse to W , appears like a Cantor decimation of the original intersection.22 So far we have considered the (local) behavior of a hyperbolic point that is isolated and fixed under a particular power off. During the subharmonic cascade of the attractor an infinite (binary) tree of hyperbolic points was produced, each point being fixed under an increasing power off and previous generations of points remaining as degenerate fixed points at each stage. For any given (21) In a single iteration the mea decreases by a factor J = IX1X21 < 1. This implies that the expansion is weaker than the contraction. (22) For a detailed discussion of some of these points for the HBnon map see, C. Simb, J. Stat. Phys., 21,465 (1979). (23) S. E. Newhouse,Ann. N.Y. Acad. Sci., 357,292 (1980);ref 19 and 22. (24) S.Smale, Bull. Am. Math. SOC.,73, 747 (1967). (25) A contracting bakeri transformation model has been constructed by J. L. Ibaiiez and Y. Pomeau, J . Non-Equilib. Thermodyn., 3, 135 (1978).

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power off such fixed points persist as the parameters are suitably changed. On the "chaotic" side of the subharmonic cascade we observe that initially separate manifold bundles of pairs of hyperbolic points expand and merge as the parameters are changed. (The transverse intersections of the stable and unstable manifolds of different hyperbolic points result in heteroclinic points.) Very near the cascade limit the manifolds look like tiny, thin, swelling islands on Z (supporting banded chaotic trajectories), coalescing very rapidly in the parameter space. (Each island represents a strange attractor component that is a minute replica of the structure previously discussed.) The islands' configuration reflects the distribution of their associated hyperbolic points fixed under a very high power off. Correspondinglyfarther from the cascade limit larger and larger islands join across initially greater gaps in a mirror tree of band fusions, and decreasing power o f f identify the hyperbolic point associated with a particular band. Finally, a singlebanded attractor emerges from this pairwise fusion, the corresponding hyperbolic point being the continuation, in parameter space, of the original fixed point attractor. The band fusion process obeys complementary scaling laws to the original attractor cascade, as might be anticipated from its dependence on the distribution of hyperbolic points. (We again remark that other structures may appear in this fusion sequence because of the tangent mechanism for creating attractors.) Next Amplitude Maps. As a consquence of the negative divergence of the driven-dissipative phase flow the folded structures described above collapse, globally, in a thin "arachnoid" layer;%the gross folding of this layer onto itself under the flow in r allows a great simplification in the description of the associated chaos. (The Smale horseshoe map is a model of this folding.) Trajectories rapidly approach the strange attractor along the stable manifolds, and once "inside" the attractor the trajectory behavior is determined by the foliation of the webbed layer by the hyperbolic manifolds. The folding character of the chaotic flow of the Morse oscillator is clearly evident in the surfaces of section in Figure 4;the flow on the two-banded strange attractor which was shown in Figure l b has one fold every two periods of the driving term. The highly compressed nature of the flow is also demonstrated by the linelike character of the trajectory intersection on the surface of section in Figure 4. The dynamics on the attractor correspond to a diffeomorphism; however, the simple global folding of the attractor layers to form a very thin ribbon suggests that the "visible" features corresponding to a contracting diffeomorphism could be successfully modeled by an endomorphism of the lines2' The endomorphism can be constructed from the surface of section in the following way: we lets denote the distance along the surface of section with the origin of s at one end. The next amplitude map consists of the plot of s, the distance from the origin (scaled to lie in [0,1])of the nth intersection of the flow with Zoovs .s,,+~.~*If this method (26) For reasonable parameters of the autonomous differential equations most studied, i.e., the Lorenz and Raasler systems, the compression of the hyperbolic manifolds onto themselves is so fine that layers are indistinguishable in the c r w section on the scale of the 'diameter" of the attractor. In HBnon's map the contraction parameter b had to be adjusted so that the layen, of W,could be resolved to display the geometrical structure. For the driven oscillators studied here typical values of y do not produce as strong a contraction aa in the Lorenz or Rbsler systems. (27) Pointlike Cantor seta can arise for these endomorphisms at the limit of the subharmonic bifurcation sequence of a periodic attractor in the same way that they arkie from contracting maps in the plane. However, the endomorphism cannot imitate the 'transverse" Cantor layer of a chaotic banded strange attractor since these layers are fused (trivially) under the coarse graining of the model endomorphism.

Sn Figure 6. A next amplitude map for the Morse oscillator, which was constructed from Poincar6 map 3 in Figure 4. The solid line is the quadratic one-dlmensionalmap, eq 2.7, with intercept, a - 1, chosen to match Morse oscillator data: a = 0.173, or X = 3.581 (see text). The map deviates m e w h a t from the quadratic form. I n addition, a small but detectable thickness is evident on the lower right portion of the map.

is used on the surface of section 3 in Figure 4 the resulting map is roughly quadratic. A truly quadratic map can be parameterized in terms of the s = 0 intercept, which we call a - 1,as sn+1 = Qa(sn) = -(a 1 + ~ c Y ' / ~ ) s , 2(a ~ + C Y ' / ~ ) S ,- CY 1 (2.7)

+

+

+

The comparison of this form with the next amplitude map derived from the flow is shown in Figure 6. The map may also be written in the standard form sn+1

= k n ( 1 - sn)

Q(sn;X)

(2.8)

with X related to the intercept by X = 1 + (5 + CY'/^)'/^. Since, for this parameter range, the underlying dynamics is adequately represented by a quadratic one-dimensional map, the well-known properties concerning the cascade into chaos are expected to be applicable to these oscillator flows. Like the flows from which it arose, this quadratic map exhibits a sequence of subharmonic bifurcations with periods 7, = 2" as X is increased. (The relevant range of X is [0,4].)In addition, the metric properties associated with this cascade have been established recently.31 Let A, denote the value of X at which the subharmonic with period 2" first became stable. As X is varied the "windows" corresponding to stability of a given subharmonic shrink until an accumulation point, A,, is reached where the motion is truly aperiodic.30 One of the interesting properties of such cascades is that the sizes of the windows decrease asymptotically as

(28) The form of the next amplitude map which is obtained by this which is selected. In the course procedure will depend on the surface Zb, of motion on the attractor of r the flow may be deformed (stretched) nonuniformly in the transverse direction. However, all such maps are conjugate, Le., if F&) and F&) are maps obtained from Zoand Zv and h is a continuous one-to-one map, which takes points from Z8to Zf,then F&) = h(Fo(h-l(s))).See, for example, ref 20. (29) If the surface of section is rectilinear then either qn or u, can be used directly to construct the map. (30) For a review of some of the properties of this map see, R. M. May, Nature (London),261,459 (1976).

Chaos in Nonlinear Dissipative Systems

The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 2211

Figure 8. (a) Projection of a period three orbit for the Morse oscillator on the q,v plane. The parameter values are y = 0.4, p = 0.48, and w = 0.6375. (b) Chaos near the period three: y = 0.4, p = 0.48, and w = 0.640. Note that the chaotic orbit has a structure which is similar to the period three orbit from which it arose.

for two values of A. The solid line Flgure 7. (a) Plot of Q(3)(s;X) corresponds to a value of X 5 1 4 8 = A, in the chaotic region; The rightmost hill and leftmost two valleys just fail to intersect the bisectrix. The dashed line is for X 2 A, where period three is stable. (b) An enlargement of the region in the central square in Figure 7a. The progression of iterates in the channel is also shown. As A approaches A, from below the number of steps in the channel increases as (A, The large number of iterates in this region (and by in the other resonance spikes) is responsible mapping under Q and d2) for the buildup of density in these regions.

+

with 6 = 4.66920 ... a universal number for all one-dimensional maps with a single, locally quadratic maximum.31 This value of 6 also applies to subharmonic bifurcations from any basic period k to yield cycles with periods k X 2n. A similar sequence of bifurcations occurs in some of the chaotic attractors; the attractors are banded, as they were for the differential equation flow, the bands merge in a sequence which mirrors the periodic cascade.32 The banded character of the Morse attractor, which is clearly evident in Figures 1and 3 was commented upon earlier. The chaotic nature of the attractor can be confirmed by computing the Liapunov number33 l T L(X) = lim - C In IQ(sk;X)I (2.9) t--

Tk=l

which measures the rate at which nearby points separate upon iteration of the map. It is positive for chaotic states and negative for periodic states. Once again, the Liapunov number which characterizes this chaotic cascade behaves with t a universal exponent related to as L(X) = c(X 6 by t = In 2/ln 6 = 0.4498....34 Similar scaling properties hold in the periodic regime when an external noise source is i n ~ l u d e d . ~ ~ , ~ ~ (31) M. J. Feigenbaum, J. Stat. Phys., 19, 25 (1978); 21, 669 (1979). (32) S. Grossmann and S. Thomae, 2. Nuturforsch.A, 23,1353 (1977); E. N. Lorenz, Ann. N.Y. Acud. Sci., 357, 282 (1979); S. J. Chang and J. Wright, Phys. Rev. A, 23, 1419 (1981). (33) V. I. Oseledec, Trans. Moscow Math. Soc., 19, 197 (1968); G. Benettin, L. Galgani, and J.-M. Strelcyn,Phys. Rev. A, 14,2338 (1976). (34) B. A. Huberman and J. Rudnick, Phys. Rev. Lett., 45,154 (1980).

The subharmonic bifurcations discussed above arise through a pitchfork bifurcation mechanism.30 In this case the slope of the 2”th power of the map, Q(2”)(s;X)= Q(Q(2n-1)(s;X);X),is +1 at the birth of the periodic cycle with period 2n and -1 when this cycle becomes unstable, giving rise to a cycle with period 2n+1. However, stable periodic cycles need not arise from the bifurcation of an existing fixed point; new cycles may be born by a tangent bifurcation mechanism when the hills or valleys corresponding to some power of the map intersect the b i s e ~ t r i x .When ~~ this mechanism is operative, chaotic states can arise directly from the bifurcation of a periodic orbit. A number of interesting phenomena accompany this process. The system behaves in an intermittent fashion; long periods in which the motion is predictable are interspersed with chaotic b u r ~ t s . ~The ~ 3 origin of this intermittent behavior can be understood by considering the quadratic map near period three. Figure 7a shows the third power of the map; when Q(3)(s)intersects the bisectrix three new stable (and three unstable) fixed points are born. A portion of this figure is expanded in Figure 7b and shows the regular character of the iterates in the vicinity of the period three fixed points. The probability density associated with the map for this value of X has three large spikes in the region of the period three fixed points; the spikes arise from the fact that most of the iterates are confined to this region. Once an iterate moves outside the region of the density maxima it enters a region (between the spikes) where the motion is chaotic. Eventually it will be captured (in a statistical fashion) by one of the spikes. All of these features have been confirmed by detailed calculation on the one-dimensional map.37 In addition, a resonance model for the tangent bifurcation process, which expresses the probability density in terms of the fixed points of Q(3) in (35) J. P. Crutchfield, J. D. Farmer, and B. A. Huberman, to be published. (36) B. A. Huberman and A. B. Zisook, Phys. Rev. Lett., to be published; B. Schrainman,C. E. Wayne, and P. C. Martin, Phys. Rev. Lett., to be published. (37) S. Fraser and R. Kapral, Phys. Rev. A., 23, 3303 (1981). (38) Y. Pomeau and P. Manneville in “Intrinsic Stochasticity in Plasmas”, G. Laval and D. Gresillon, Ed. Les Editions de Physique, Orsay, 1979; Commun. Math. Phys., 74, 189 (1980).

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The Journal of Physical Chemistry, Vol. 86, No. 12, 1982

I

/

c/' S"

Flgure Q. (a) Next amplitude map corresponding to the chaotic flow in Figure 8b. The density of iterates is concentrated in three regions in accord with the resonance picture of tangent bifurcation. Note, however, that the map has a rather pronounced twevaiued character, whlch indicates that the modeling of the process by an endomorphism has only qualitative rather than quantitative significance. (b) Third iterates of the map in Flgure 9a. The hills and valleys just fail to touch the bisectrix.

the complex plane, has been con~tructed.~'In this model the width of probability spikes is directly related to the imaginary parts of these complex fmed points. An analysis of the Liapunov number near bifurcation in the context of this model leads to the prediction that L (X - Xg)1/2 (1, is the value of A at which period three becomes marginally stable), which has been verified by numerical calculations. This type of bifurcation mechanism appears to be relevant for the period three-to-chaos transition in the Morse oscillator (see Figure 8). First, it is clear that the chaos bears a strong resemblance to the period three orbit from which it arose. The density is strongly peaked near the original period three orbit. (This is most evident in the next amplitude map, which was constructed from the surface of section of the flow (cf. Figure 9a).) The fact that a tangent biofurcation mechanism is operative is clear from Figure 9b where the third iterates of the next amplitude map are plotted. Although the description in terms of next amplitude maps successfully models a good deal of the observed behavior, it is clear that for certain parameter values of the differential flow the maps have a nonnegligible thickness (cf. Figure 9). In these circumstances one must resort to an analysis in terms of the two-dimensional Poincar6 map. Also, for some systems the next amplitude maps possess other extrema which may give rise to bistability and other complex behavior.m This is the situation for the Duffing oscillator where a cubic endomorphism provides a more reasonable model.lg This is illustrated by the next amplitude maps in Figure 10 and will be discussed further in the sequel. Next we give a brief description of a number of other features of the bifurcations in driven oscillator models

-

(39) G.Mayer-Kress and

H.Haken, Phys. Lett. A , 82,

151 (1981).

Kapral et al.

Flgure 10. A series of next amplitude maps for the Duffing oscllator, which show how the chaos evohres as the frequency of the drivtng term decreases. For all figures y = 0.4 and p = 0.1 175: (a) w = 0.5490, (b) w = 0.5478, (c) w = 0.5470, (d) w = 0.5460. (The strange attractor is shown in Figure IC.) Note that (a) and (b) are nearly quadratic with a slngle extremum and small thlckness. The map in (b) quite dosely corresponds toa quadratic map wiih X = 4. As the chaos evolves an additlonal extremum and a pronounced two valuedness develops. I n this circumstance modeHng by an endomorphism ceases to be useful. I

0

I

I 0.20

I

I

I

I 0.40

0.60

I

I

I 0.80

1.00

w

Flgure 11. Plot of the peak modulus of the amplitude vs. w for y = 0.4 and p = 0.1175 for the Duffing oscillator, which exhibits a hysteresis loop invoMng a chaotic state. For a further discussion of such phenomena see ref 4. The thlck line denotes the reglon from period two to chaos.

studied here, which are of relevance for the subsequent discussion on fluctuations. Hysteresis. Bistability and the associated hysteresis phenomena are a common occurrence in driven oscillators and Duffing's method can be used to describe these effects for periodic solutions if the forcing and nonlinearity are not too great.' However, in parameter ranges where perturbation or iteration methods fail bistable loops are found which have chaotic attractors on one of the branche~.~ An example of such a hysteresis loop for the Duffing oscillator is shown in Figure 11. Moving from right to left on the upper branch in the diagram, the system undergoes a sequence of subharmonic bifurcations which culminate in a chaotic attractor shown in Figure IC. At the frequency w = 0.5458 the system jumps to a period-one attractor. The system continues to be entrained as the driving frequency is lowered. On the other hand, if the system starts on the lower branch and the driving frequency is increased the amplitude of the period-one orbit increases continuously until w = 0.674... where the amplitude changes discontinuously and the system jumps to the upper branch. We defer further discussion of such phenomena to section I11 where the effects of fluctuations are considered.

The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 2213

Chaos in Nonllnear Dissipative Systems

bistability is possible, which is consistent with the observed chaotic attractors with broken symmetry. Furthermore, for endomorphismswith two extrema a possible bifurcation mechanism involves the period-one fixed point at the origin becoming unstable and giving rise to two distinct periodone fiied points. (A process analogous to this, a period-two fixed point bifurcating to give two distinct period-two fiied points, has been discussed by May4 for a cubic map.) A process of this type for the map underlies the symmetrybreaking period-one bifurcation of the flow observed for the Duffing oscillator. Not unitil a self-similar strange attrador appears do we observe a (local) next amplitude map with two extrema. This is a consequence of the fact that the chaotic trajectory approaches the maxima in the potential sufficiently closely that two foldings are produced per period (cf. parts c and d of Figure 10).

a

> (3

U W

z

b

W

e Figure 12. Plots of energy vs. 0 = ut for the Duffing osclllator. (a) y = 0.4, p = 0.1175, w = 0.57 (self-slmllar perlodone orb%). (b) y = 0.4, p = 0.1175, w = 0.553 (period-one orbit with broken symmetry).

-

Broken Symmetry. For the Duffing and cosine oscillators, eq 2.3 is invariant to the transformation (q,u,e) (-q, -u,O+T). Hence, either a solution has this symmetry property (self-similar) or has mates. Subharmonic solutions necessarily have mates.l9 In the course of the subharmonic cascade into chaos and in the chaotic region itself a number of broken symmetries occur. As an example we consider a sequence of subharmonic bifurcations which lead to chaos for the Duffiig oscillator (y = 0.4,p = 0.1175, decreasing w). Initially the period-one orbit is self-similar. At a value of w N 0.56 the orbit bifurcates into two complementary period-one orbits which, separately, fail to satisfy the above symmetry property. This bifurcation is associated with the separation of r into two basins, which are related by the above transformation. Broken symmetry is clearly signaled by the behavior of the energy as a function of time; for the self-similar orbit the energy has periodicity T / O (Figure 12a), while for the broken symmetry-period-one orbit the energy periodicity is 27r/w (Figure 12b). Also note the change in slope of the amplitude vs. w curve in Figure 11. A further decrease in the frequency results in a cascade of subharmonic bifurcations in each half of r. Thus at the subharmonic with period ( 2 9 2 ~ / wthere are 2 X 2" distinct orbits with this period which, taken together, satisfy the symmetry condition imposed by the differential equation.41 The chaos which arises from either branch also exhibits broken symmetry. In Figure 10 some next amplitude maps corresponding to this case were shown. While the maps in parts a and b of Figure 10 are roughly quadratic and have a single maximum, an analogous map can be constructed from the chaotic flow generated from a trajectory with q -4, u -u, and Bo shifted by T . The composite map constructed from both of these next amplitude maps has the symmetry of the flow, possesses two extrema, and is roughly cubic in The form of this composite map is closely analogous to antisymmetric endomorphisms. For such maps with two extrema it is easy to see that

- -

(40) R. M. May, Ann. N.Y. Acad. Sci., 316,517 (1978); see also ref 15. (41) The symmetry operations of the differential equation also leave the vedor field of the flow invariant. The set of subharmonic solutions embedded in this vector field must posseea this symmetry aa they can be regarded aa the asymptotic state of a uniform phase fluid density under the action of the flow. (42) Such a composite map corresponds to the intersections of both branches of the flow on surfaces L: separated by r .

111. Fluctuations and Chaotic Dynamics In the previous section we observed that the system was very susceptible in the chaotic region to small changes in initial conditions since trajectories exponentially separate on the strange attractor. For any real system in or near such a region thermal fluctuations will be present and one might anticipate that they would have marked effects because of the large susceptibility of the system. In order to investigate these effects we now consider a Langevin equation43model for the dynamics du(t) m -= -S.u(t) + F ( q ( t ) )+ mp cos w t + f ( t ) (3.1) dt i.e., we append a random force f ( t )to eq 2.1. The random force is taken to be a white-noise Gaussian random process, for which the corresponding fluctuation-dissipation theorem is (f(t) f ) = 2 k ~ T 6c( t )

(3.2)

where the angular brackets denote an average over fluctuations.ql In this phenomenologicalmodel the interaction of the oscillator with the heat bath is roughly taken into account through the specificiation of the statistical properties of the random force. Although the justification of the use of such a Langevin model from the microscopic equations of motion is a difficult task, it yields an adequate representation of the dynamics in many instances. We should also point out that the average of the Langevin equation (3.1) over fluctuations is not the same as the deterministic equation (2.1); the presence of nonlinear terms will lead to fluctuation renormalization when such averaging is carried out. The inclusion of the random force in eq 3.1 leads to the introduction of another parameter u = [idzBT/8P2V(qo/ 2)I1l2which gauges the strength of the fluctuations relative to the driving force. For fixed { and p this parameter is controlled by the temperature of the heat bath. One expects that fluctuations would have the effect of destroying some of the detailed structure described earlier, and indeed this is the case. However, one might also expect that the nature of the response of the system would be related to the underlying structure of the deterministic (43) S. Chandrasekar,Rev. Mod. Phys., 1 5 , l (1943);M. C. Wang and G. E. Uhlenbeck, ibid., 17, 323 (1945). (44) The stochastic trajectories were generated by integrating the deterministic equations for a time interval At and then adding the contribution from the random force, sampled from a Gaussian distribution. The time interval At must be chosen small enough so that spurious memory effects, which arise from holding the random force constant over A t , are of no importance, and yet large enough to allow efficient integration of the equations. We found At = (1/400)(2?r/w)to be satisfactory.

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The Journal of Physlcal Chemistry, Vol. 86, No. 12, 1982

Kapral et al.

a

0

#

b

I

0'

I

L

I

C - ~ A - -

.,/"

Flgwo 13. Surfaces of section for the Duffing oscillator y = 0.4, p = 0.1 175, and w = 0.56 (upper branch) at various noise levels: (a) u = 0.0003,(b) u = 0.0010, (c) u = 0.0089. Eight sectkns are shown at equal intervals of 0 between 0 and 2 ~ For . comparison, in (d) surfaces of section for the deterministic system at y = 0.4, p = 0,1175, and w = 0.547 are shown.

phase space flow. Also, fluctuations will allow the system to explore different regions of phase space and thus transitions between bistable states can occur and symmetries which were broken can be restored. At high enough noise levels escape from potential wells becomes a probable event and a study of rate processes for driven nonlinear oscillators can be undertaken. In this section we examine some of these possible effects of the system response to thermal fluctuation^.^^ Character of the Response to Fluctuations. The system responds to fluctuations differently depending on whether or not it is in a state which leads to chaos. The different types of response are clearly illustrated by the Duffing oscillator for the system states shown in Figure 11. Fkall that along the upper branch a sequence of subharmonic bifurcations occur, which lead to chaos, while the lower branch corresponds to simple entrainment at the external driving frequency (period-one orbit). Also, a period-one solution spans a great deal of the upper branch (down to a frequency w N 0.5518); the subharmonic and chaotic regions constitute a small portion of the hysteresis loop. Consider a period-one orbit which is near to chaos on the upper branch (w = 0.56). Figure 13 shows a series of surfaces of section obtained from the flow for differing noise levels and, for comparison, surfaces of section for the deterministic chaotic flow in the same region. The similarity between the noisy system and the chaotic deterministic system is striking. The major effect of the noise is to spread the iterates (under the Poincar6 map) out over the region covered by the strange attractor. Note that the attractor section remains linelike in the presence of weak noise indicating the strong contraction of the flow in directions transverse to the attrador manifold. In this region the noisy system behaves in much the same way as the deterministic system at scaled parameter These results are in accord with the discussion in the previous section where the development of the homoclinic structure was described in some detail; fluctuations can induce the formation of this structure through map parameter renormalization and also encourage the system to explore (45) Studies of the effecta of noise on the driven Duffing oscillator have been carried out by J. P. Crutchfield and B. A. Huberman, Phys. Lett. A, 77, 407 (1980),using an analog computer with a white noise source.

+.d

Flgwo 14. Surfaces of section for the Duffing oscillator y = 0.4, p = 0.1175, and w = 0.70 for various vaiues of u: a) 0.0032, (b) 0.0010, (c) 0.032, (d) 0.10.

the layers of the unstable manifolds. The noise levels required to effect substantial spreading is quite small; in fact, the u values for the results reported in Figure 13 correspond to bath temperatures which are unrealistically low for representative system parameter values. (Higher noise levels induce a transition to the lower branch. This is discussed in the next subsection.&) Even such small amounts of noise can alter the pattern of bifurcations significantly. In the previous section we described the sequence of subharmonic bifurcations which lead to banded chaos and the mirror sequence of bifurcation in which the bands merge. Fluctuations cause the periodic orbits to become diffuse,spreading primarily along the attractor manifold as discussed above, and produce additional spreading on the chaotic attractors as well. At a given noise level some of the higher order ''fuzzy" periodic orbits will blend and cease to be identifiable as distinct entities. Similarly in the chaotic region the fine banded structure will be blurred by the noise resulting in an appearance of band merging. In this case in the presence of noise there is only a quantitative difference between the what one might call noisy periodic and noisy chaotic states; both are chaotic and differ only in the degree of spreading along the unstable manifold. A continuous increase in u produces a continuous increase in the spread of iterates along the attractor (see Figure 13). A t larger values of the driving frequency (farther from the chaotic region for the deterministic system) the effect of fluctuations is to produce an attractor section which is broader and less linelike in character but yet remains the same general structure as that for the chaotic deterministic system (Figure 14). Note also that for the parameter values in this figure the deterministic system has a period-one solution, which is outside the hysteresis loop. Hence the only other possible effect is escape from the well and much higher noise levels can be applied. On the other hand, for the period-one attractor at w = 0.5476 on the (46) Estimates of the temperaturescorresponding to these noise levels are hundredths of a Kelvin for the superionic conductor Li2Ti3O2!

The Journal of phvsical Chemistry, Vol. 86, No. 12, 1982 2215

Chaos in Nonlinear Dissipative Systems

C tt

d

Flgure 15. Surfaces of section for the Duffing oscillator y = 0.4, p = 0.1 175, and w = 0.5467 (lower branch) for various values of u: (a) 0.01, (b) 0.032, (c) 0.10, (d) 0.158.

lower branch of the hysteresis loop, increasing the noise level simply produces a radial spreading of map iterates (Figure 15). All of these results are consistent with the observation that, in the presence of fluctuations, the dynamics of the oscillator flow near chaos is governed by the manifold structure discussed in section 11. Fluctuations can also produce intermittent behavior in the system. In section I1 we discussed the fact that in the chaotic region near period three the time series for the deterministic system consists of periodic regions interspersed with chaotic bursts. If the parameters characterizing the oscillator flow are such that the deterministic system is in a periodic state which arose by a tangent bifurcation mechanism, then small amounts of noise can cause the system to behave as if it were on the period-three like strange attractor. The Morse oscillator illustrates this effect clearly; Figure 16 compares the surfaces of section of the deterministic chaos near period three with the noisy oscillator at parameter values corresponding to stable period three. We described earlier how the transition to chaos near a tangent bifurcation could be modeled by a quadratic endomorphism. Similarly, some features of noise-induced intermittency can be studied by considering a one-dimensional map with an added noise term. An investigation of this type has been recently carried out by Mayer-Kress and Haken.39 If one imagines varying the system parameters at a fixed noise level then the “bifurcation gap” phenomenon of Crutchfield and Huber~uan%*~ is observed. Consider again, for example, the Morse oscillator as the driving frequency is lowered. Earlier we saw that in a certain parameter range such a frequency variation results in a subharmonic cascade into a banded chaotic attractor whose bands merge as the frequency is reduced further. If the same variation is carried out at a given fixed value of the noise then some of this structure will not be resolvable; fluctuations will cause both the periodic states as well as the chaotic states to broaden; as a result only a certain number of bifurcations will be distinguishable, the remainder being “buried” in the noise fluctuations. Figure 17 illustrates this behavior. A “fuzzy” period-one orbit bifurcates to yield a

.j

% . ‘ , .*I.

..

. ,. .*’ ,

.. ,.-

rj.

- . . ::...-

Flgure 18. Surfaces of section for the Morse oscillator near period three: (a) deterministic chaos y = 0.4, p = 0.48, w = 0.0640; (b) effect of noise, u = 0.0032, on the period-three orbit, y = 0.4, /I = 0.48, w = 0.6375.

‘

a

b

C

d

e

\

; -

--...

3

Figure 17. Surfaces of section for the Morse oscillator at a k e d value of u = 0.01 for various values of the frequency. y = 0.4, p = 0.42: (a) w = 1.05, (b) w = 0.8, (c) w = 0.65, and (d) w = 0.54.

“fuzzy” period-two orbit; a further decrease in frequency causes additional spreading of the Poincar6 map iterates until the two bands merge to form one band. Such phenomena have been studied in great detail by considering noisy one-dimensional maps.3s Just as is the case for the deterministic system, many universal properties can be associated with noisy subharmonic cascade into chaos. Since these have already been described in some detail,%” we instead turn to a consideration of the various types of transition that can be induced by fluctuations. Barrier-Crossing Dynamics. The existence of fluctuations permits a variety of additional dynamic phenomena to take place. As mentioned earlier, these include transitions between bistable states and escape from potential wells. We shall now consider a few specific examples of such effects, primarily for the Duffing oscillator. This model provides a local description of particle motion confined by potential barriers and one can examine the escape probability for variations of both the external field and heat bath temperature. Furthermore, since the Duffing oscillator possesses bistable states one may study the noise-induced transitions involving these states. A

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The Journal of Physical Chemistry, Vol. 86, No. 12, 1982

qualitative description of such effects follows. Consider moving along the upper branch of the hysteresis loop in Figure 11by decreasing w for a fixed value of the noise parameter a. It is found that transition to the lower branch takes place at a frequency Wd, which increases as the noise level increases. Thus, noise induces hopping from the upper to lower branches, in effect making the upper branch of the hysteresis shorter. (Actually, as a is increased beyond a certain value, escape becomes an important process and the behavior changes.) Alternatively, if once considers a fixed frequency and varies a then at a certain value of a the upper branch becomes unstable and irreversible decay to the lower attractor occurs. In this instance the upper state has greater average energy and becomes less stable at higher temperatures. This type of behavior does not persist across the entire width of the hysteresis. As might be expected, at the high-frequency end of the hysteresis transitions from the lower to the upper branch take place. Depending on the noise level, the upward transition is observed to be accompanied by the following processes: (1)The system may remain stable on the upper attractor. (2) Transitions back to the lower branch occur, to be rapidly followed by another upward transition. This rapid hopping between branches implies some sort of equilibration; however, a sufficient number of these hops has not been observed to characterize this relaxation process. Under conditions of facile transitions within the well escape from the upper branch also occurs. Some aspects of the above processes can be understood by considering the sizes and geometries of the basins associated with the upper and lower attractors as a function of frequency. For the deterministic system, as the frequency is lowered the basin associated with the upper attractor shrinks and disappears when transition to the lower branch occurs; correspondingly, as the frequency is raised the basin associated with the lower branch shrinks and disappears when the upward transition occurs. Since fluctuations permit the system to explore the neighboring region of an attracting orbit, the structure of the basin of an attractor is an important freature in the response of the system to noise. This change in basin structure clearly determines the gross features of the noise-induced-transition phenomena. In order to study the processes accompanying an attractor’s loss of stability in the presence of fluctuations in more detail, we imagine the system prepared in a state corresponding to a very low temperature (negligible fluctuations). We then suppose that the system is instantaneously heated to a higher temperature and study the average response. The calculations were carried out by integrating an ensemble of trajectories generated by sampling the distribution governing the low-temperature state. We consider two initial states: (A = 0.4, p = 0.1175, w = 0.7) and (y = 0.4, p = 0.1175, w = 0.5467). The former state is a period-one orbit outside the hysteresis loop and the latter is the chaotic state on the upper branch.47 We first discuss the response of the period-one attractor. In the previous subsection we describe the behavior of the system as the noise level is increased; there is increased spreading of iterates along as well as transverse to the unstable manifold (cf. Figure 14). However, at noise levels (47) The sampling over the initial distribution for these calculations was done in the following way: For the case when the initial state was a period-one orbit, the initial distribution was constructed from 200 points (at equal intervals of B from 0 to 2 r ) on the orbit. The construction of the initial distribution for the chaotic state is somewhat more complicated. Here points for the initial distribution were selected from several chaotic trajectories run over ten periods of the driving field.

Kapral et ai. 1

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Flgwe 18. Plot of logarithm of the average escape time f, vs. u-’ for the Duffing oscillator: y = 0.4, p = 0.1175,and w = 0.7. Results = 0.55; for two values of the cutoff distance q mare shown: -, 9,,, -----, 9,,,= 0.5005. The slopes are 0.049 and 0.048, respectively. The error bars represent fl standard deviation. I

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FI ure 10. Plot of the logarithm of the average transition time f, vs. a- for the Dufflng oscillator: y = 0.4, p = 0.1 175,and w = 0.5467. The cutoff distance was taken to be q m = 0.6. The inset shows the fraction n of the ensemble that escapes.

9

corresponding to a N 0.122 there is a detectable probability for escape from the well. [In these calculations a spatial criterion has been used to define escape. Three values of the “escape distance” qm were selected: qm = 0.5005,0.55, and 0.6.1 In Figure 18 it is seen that a linear relationship exists between the logarithm of the average time for escape from the well, &,and a-2. Since a-2 T’ we have a result typical of an activated process, f;l = A. exp(-E,/ kBT).48 It is perhaps useful to compare these noise-induced transitions for the hysteresis with barrier crossing in a bistable (quartic) or unbounded (cubic) potential. In the first case we eventually observe equilibrium and in the second escape. A suitable quintic potential would embrace both phenomena and so imitate the dynamical effects of noise on the Duffing hysteresis. However, the bistability of the driven Duffmg oscillator is dynamically induced and not an intrinsic property of the potential so that a more precise analogy involves mathematical subtilties. Results for a smaller value of q m (qm = 0.505,just outside the potential maximum) are also given in Figure 18. The fact that fe is shorter in this case indicates that complex dynamics involving multiple crossings of the dividing surface at qm.

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(48) The activation energy is found to be E , = 0.12V(q0/2). We also note that the average energy of the periodic orbit plus the activation energy is slightly less than the barrier height.

Chaos in Nonlinear Dissipative Systems

The Journal of Physical Chemistry, Vol. 86, No. 12, 1982 2217

anism that appears to operate in the course of fluctuation-induced downward transitions.

IV. Conclusion

Figure 20. Duffing oscillator trajectory for the deterministic system:

y = 0.4, p = 0.1175, and w = 0.5458. The trajectory was started at an initial condition corresponding to the upper attractor at w = 0.5459. The vertical lines indicate the potential maxima at q0/2.

In the case of the chaotic state transitions to the lower attractor become measurable at a noise level t~ N 0.001. (Escape from the well for the lower attractor occurs at u N 0.245.) However, as shown in Figure 19 the average transition time ft is not as simple a function of u. One reason for this increased complexity is the presence of a competing process; at higher temperatures escape from the well out of the upper attractor occurs (10% at u N 0.0045). An interesting feature associated with transitions to the lower attractor is that most trajectories first pass outside one of the potential maxima in the course of a downward transition. The behavior of these stochastic trajectories is similar to that of a deterministic trajectory when the upper branch becomes unstable. Such a trajectory is shown in Figure 20; the trajectory leaves the potential well at the left and then reenters finally reaching the lower attractor. This shows that not only has the basin of the lower attractor engulfed that of the upper attractor, but it also extends outside of the potential well for this value of the frequency. A t the proper phase the driving term can force particles back into the potential well and allow capture by the lower attractor. It is this dynamic mech-

In the foregoing discussion we have attempted to provide descriptions of chaos in driven-dissipative oscillators and the effects of fluctuations on the dynamics. Such systems clearly exhibit a rich dynamical structure, which persists in the presence of noise. In these driven systems, in common with many autonomous nonlinear differential equations in three or more variables, the chaotic states can arise out of a subharmonic cascade or by a tangent bifurcation mechanism. As a result of such processes a rich manifold structure, associated with the existence of homoclinic points, develops and is responsible for the complex dynamics on the strange attractor. Although noise has the effect of removing some of the detailed fine structure of the deterministic flow, the gross folding features of the underlying chaotic dynamics are both induced and explored due to the presence of heat bath fluctuations. Consequently, a variety of interesting phenomena, such as oscillatory relaxation to a periodic orbit, complicated transient effects during the course of relaxation to a strange attractor and chaotic barrier crossing dynamics, can be studied. (The effects of fluctuations on macroscopic chemical reactions have been investigated p r e v i o ~ s l y . ~ ~ ) We have only briefly touched upon those aspects of the motion that are of direct chemical interest, i.e., molecular relaxation processes and barrier crossing dynamics. The preliminary results on escape and transitions between bistable states, which were presented in the previous section, although limited in scope, indicate how such processes can be studied. Since a large number of condensed-phase rate processes can be modeled by these equations future experimental and theoretical studies should reveal a rich variety of new phenomena. Acknowledgment. This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. M.S. thanks NSERC for a postgraduate scholarship. (49) See, for example, G. Nicolis and I. Prigogine, "Self-Organization in Nonequilibrium Systems",Wiley, New York, 1977.