J . Phys. Chem. 1986, 90, 2273-2280
2273
Parameter Space Mapping of First-Order Linear Ordinary Differential Equations Lynn M. Hubbard, Carl Wulfman,+ and Herschel Rabitz* Department of Chemistry, Princeton University, Princeton, New Jersey 08544 (Received: May 16, 1985)
A Lie group formalism for global parameter space mapping of ordinary differential equations, described in the preceding paper, is developed further here. The need for such mapping arises in a variety of physical contexts. The procedure is demonstrated on, but not restricted to, the system of coupled equations x = cx. The Lie group generators are obtained exactly from both the time-independent and time-dependent generating equations. The transformationsobtained from these generators leave the system of differential equations invariant. The time-independent transformations map any solution of the linear system x = cx into any other solution with the same frequency or time constant. The time-dependenttransformationsinterconvert solutions with different frequencies and/or time constants. Any solution of x = c x can be mapped into any solution of x = c'x. Thus the behavior of x ( t ) can be examined as a function of changes in any of the system parameters, or in the initial conditions, x(0). As an example, one of the time-dependentmappings demonstrates the continuous transformation of oscillator solutions into nonoscillatory ones, and vice versa.
I. Introduction
Many problems of chemical interest involve nonlinear equations, and the Lie group approach discussed here is fully capable of The response of a dynamical system over a wide region of addressing these systems. At this early stage of development, and parameter space is of interest in many different contexts. For for simplicity and ease in demonstrating the use of this formalism example, the stability of a classical mechanical or kinetic system for parameter space mapping, the application herein is to a system with respect to variations in the space of initial conditions is an of coupled linear equations. No generality is lost, however, because important problem currently receiving attention in the l i t e r a t ~ r e . ~ . ~ the interpretation of the formalism as applied to linear systems In a similar context, the practical implementation of classical will not change in the application to nonlinear systems. S-matrix theory entails exploring wide regions of initial coordinate The analysis of nonlinear ordinary differential equations often space in order that the final momenta or action be q ~ a n t i z e d . ~ begins with their reexpression as a set of first-order ordinary Chemical kinetic modelling (e.g., atmospheric, combustion, and differential equations. The critical points of these are then deair pollution chemistry) involves solving a large number of coupled termined and the equations are linearized in the region of each differential equations with uncertainties or unknowns in many of critical point. The linearized equations are then used to determine the reaction rate constants and initial concentration^.^-^ In a the phase portraits of the system in the critical regions.1° If one similar fashion, there continues to be a desire to understand how is to be able to map a variety of nonlinear equations into linear molecular dynamical behavior would be influenced over a wide equations it is clearly necessary to have mappings which can range of parameter values in the intermolecular potential. In some interconvert solutions of linear equations with different phase problems the system parameters may actually be controlled in portraits. With these observations in mind we demonstrate here the laboratory and the optimum choice of these parameter values that operators can be obtained that will transform any given phase for fundamental or practical applications would require exploring portrait in the plane into any other phase portrait in the plane. a full domain of control parameter space. These types of questions Section I1 describes the general procedure for obtaining the fall into the general realm of sensitivity analysis.' The practical generator from the determining equations for the system of difimplementation of conventional sensitivity analysis, however, is ferential equations x = cx. In section 111, the general analysis limited to study of behavior in the local vicinity of a chosen of section I1 for an arbitrary number, N , of differential equations operating point in parameter space? This paper and ref 1 present is specialized to the case of N = 2, and an analytic form for the a formalism which uses Lie group theory to obtain a tractable generator is obtained. The solution to the system of differential method for performing global parameter space mapping, thus equations with N = 2 is known analytically, and it serves as an addressing the types of questions mentioned above. The formalism aid in understanding and demonstrating the usefulness of the finite may be applied to any set of ordinary differential equation. transformations obtained from the Lie generators. We can thus The mapping procedure demonstrated here, and shown schedemonstrate exactly what the generators do to a system which matically in Figure 1, moves the solution of the system of difis already well understood. Also, the procedure for obtaining the ferential equations through parameter space to any chosen point generators for any arbitrary linear system of equations will not in this hyperdimensional space. This is accomplished by the use entail different operations than is involved here. The fact that of the theory of Lie groups, which provides a framework for the illustrated system has both oscillatory and exponential solutions obtaining a first-order differential operator, U, called the group also provides an opportunity for showing how the Lie group opgenerator or Lie deri~ative.~ The operator exp(aU), with a being the group parameter, acts upon the variables in the original set ( I ) C. Wulfman and H. Rabitz, J . Phys. Chem., preceding paper in this of differential equations and is required to leave these equations issue. (2) M. Tabor, Adu. Chem. Phys., 46, 73 (1981). invariant. This results in a set of equations determining U, the (3) (a) S. Rice, Adv. Chem. Phys., 47, 117 (1981). (b) P. Brummer, Adu. solution to which leads to a set of finite invxiance transformations, Chem. Phys., 47, 201 (1981). (Le., the desired mappings), for the system variables and param(4) W. H. Miller, Adu. Chem. Phys., 25, 69 (1974) eters. The formalism in this paper and ref 1 uses this framework (5) See, for example, B. G. Heikes and A. M. Thompson, J. Geophys. Res., C: Oceans A m " , 88, 10883 (1983). with the constraint that complete control is allowed over the form (6) See, for example, J. G. Calvert and W. R. Stockwell, Can. J. Chem., of the part of the generator which acts upon the system parameters 61, 983 (1983). in the set of differential equations. This control allows the various (7) (a) P. M. Frank, "Introduction to System Sensitivity Theory", Acasystem parameters to be changed at will. The effect of these demic, New York, 1974. (b) R. Tomovic and M. Vukobratonic, "General Sensitivity Theory", Academic, New York, 1974. changes on the solutions to the original differential equations can (8) H. Rabitz, M. Kramer, and D. Dacol, Annu. Reu. Phys. Chem., subthus be easily monitored. mitted for publication. 'Present address: Department of Physics, University of the Pacific, Stockton, CA 9521 1.
0022-3654/86/2090-2273$01.50/0
(9) A. Cohen, "An Introduction to the Lie Theory of One-Parameter Groups", reprinted, Stechert, New York, 1931. (10) N. Minorsky, "Non-linear Oscillations", Krieger, New York, 1974.
0 1986 American Chemical Society
2274
The Journal of Physical Chemistry, Vol. 90, No. 10, 19616 tt
X
Figure 1. The hyperdimensional space of x, c, and c are collapsed into three dimensions for pictorial purposes. We seek a mapping which will take the trajectory x(t), at a given value of the system parameters c and transform it to the new trajectory % ( t )at a different and specified value of E.
erators can map between the two types of solutions. In section 111 both the time-independent and time-dependent solutions to the determining equations, and the resulting generators, will be discussed. It is shown that the time-dependent generator allows for either the initial conditions or the parameters or both to be varied. A subsection is included which discusses the use of the theory for mapping through the space of initial conditions. The conclusion in section IV will describe anticipated future uses of the Lie group formalism presented here and in ref 1. The use of Lie groups to study the dependencies of solutions to differential equations on the system parameters is not entirely a new idea, and ref 1 lists many of the relevant works in this area. The present research aims to further develop the basic concepts and bring the method to practical fruition. We have also recently become aware of the work by Dragt et al., who have used Lie algebraic techniques to study trajectories near a given trajectory for a general Hamiltonian system, with particular application to accelerator design, charged particle beams, and light 11. General Analysis of the Determining Equations
A systematic way of obtaining finite transformations that will take the solution at one point in the hyperdimensional space of system parameters and transform it to the solution at a different chosen point in parameter space is outlined below. Further details may be found in ref 1. These trmsformations are accomplished by the action on the system variables of an operator of a oneparameter Lie group
T(a) = exp(aU)
(2.1)
where the parameter a is real and the generator U is a first-order differential operator. The finite transformation operator T(a) is a propagator of the system of differential equations in the hyperdimensional space of all system variables, Le., parameters as well as state variables, coordinates, and/or time. To ensure that T(a) transforms solutions of the differential equations into solutions one demands that the operator leave invariant the dynamical equations. In the present case, once T(a) is known, the finite transformations of the system variables x, t , and the system parameters c are obtained by x = exp(nU)x (2.2a) E = exp(aU)c
(2.2b)
7 = exp(aU)t
(2.2c)
where the barred symbols represent the transformed values. It is important to keep in mind the distinction between the Lie group parameter a and the system parameters, c. It will become clearer in the example given in section I11 that the general form of the generator, U, is composed of a linear combination of generators, Ui.Each of these Vicorresponds to a one-parameter Lie group (11) A. J. Dragt and E. Forest, J . Math. Phys., 24, 2734 (1983), and references therein. (12) A. J. Dragt, J . Opt. SOC.Am., 72, 372 (1982), and references therein.
Hubbard et al. dependent on a different parameter a,. In fact, we will see in the example discussed below that for a given number of independent parameters {c] in the original set of differential equations it is possible to find sufficient independent one-parameter Lie groups to change each of the parameters cIJindependently. Each of these independent Lie groups, with its corresponding a,, will give a mapping of the solutions through the space of one of the original system parameters, cIJ. In what follows we will use the generic symbol a as the Lie parameter, unless distinction is necessary between parameters of different one-parameter groups. The generator U is constructed from the solution to a set of determining equations. These equations are derived from the action of U on the original set of differential equations, with the requirement that U leaves these equations invariant. The following treatment is a derivation of the general form of the determining equations for any system of linear differential equations given by XI
=
(2.3a)
CIJXJ
where i, j = 1, ..., N , and x(0) = x o . The standard repeated index summation convention is used here and in all that follows. In vector notation eq 2.3a may be written as x = cx
(2.3b)
The generator U is by definition a first-order differential operator. It includes terms which differentiate with respect to all the dependent and independent variables in the system of differential equations, treating all the variables as independent. (The action of the generator U is analogous to the action of the rotation operator x(a/ay) - y(a/dx) on the equation x2 + y 2 = 1 defining the unit circle. The variables x,y are considered as independent variables subject to a side condition to be applied after the action of the generator. This treatment of variables is basic to all of Lie's work.) The generator for the system of eq 2.3 is given by
U = &(x,c,t)-
a
axj
a
a
+ $ij(c)-acij + s(x,c,t)z
(2.4a)
It is convenient to define the vector c which contains the elements of c in the order (C~,.C,~,...,C~~). The derivation which follows is for s(x,c,t) = 0, so that eq 2.4a becomes
a
U = &(x,c,t)-axi
a + *{j(C)-at,
(2.4b)
No generality is lost, however, if &(x,c,t) in eq 2.4b is redefined as &(x,c,t) E $i(x,c,t) = &(x,c,t) - ~ T ( x , c , ~ ) ( 2 . 4 ~ ) so that
u = '$(x,c,t)- a
8x1
+ $ij(c)- a
at,,
(2.4d)
It is not hard to show that the transformations obtained by using the generator in eq 2.4d will map onto the same solution (or trajectory) as that obtained from eq 2.4a. The only difference is that eq 2.4d results in a vertical mapping in time, Le., 7 = t , whereas eq 2.4a results in the nontrivial finite transformation i = exp(aU)t. We are at liberty to choose any form for the $,, coefficients which is convenient for a given problem. For example, the choice of = 1 results in simple translations in parameter space; i.e., eq 2.2b becomes E , = cy + a The corresponding transformations of the system variables x will be dependent on x, c, and t. The choice of the $, determines how the transformations move through parameter space (e.g., linearly, exponentially, etc, in the group parameter a). This gives the ability to tune the solution as a function of a predetermined and usually simple motion through parameter space. The coefficients in eq 2.4 are restricted to having dependence only on the c,, -a*l, =at
W I ,
' = 0 ax
The Journal of Physical Chemistry, Vol. 90, No. 10, 1986 2275
Parameter Space Mapping of Differential Equations and the ci, are required not to depend on the xi or t. This ensures that the transformed variables cij are independent of x,t as well as the transformed variables x,T. The requirement that U be the generator of an invariance transformation of (2.3) is (cf. ref 1) (2.6a) 0(X - cx) = 0 The extended generator, 0, includes a term differentiating with respect to the xi, treating them as independent variables,
a +u 0= &(X,x,c,t)-aki
(2.6b)
with (2.6~)
The general time-dependent solution is given by
where the x(0,c) are the initial conditions of X(t,c). Section IIIc includes a discussion of the relationship between the initial conditions for the X(t,c), and the initial trajectory to which parameter space mapping is applied. The linear expansion of &(x,c,t) in terms of the coefficients ordinary differential equations instead x..(t,c) in eq 2.9 leads to of the original N partial differential equations. We may rewrite the N vector equations in (2.10a) equivalently in terms of N matrix equations thus leaving the determining equations for xij(t,c) dimensionally equivalent to the original dynamical system. Equation 2.7 can be written in terms of the matrix X(t,c), defined in eq 2.8, and the matrix J,(c) as
(for 7 = 0). Operating with 0 as shown in eq 2.6a gives the determining equations for U as equations for 5
at
- + x.(cx) at
- ct = J,x
The homogeneous and particular solutions are, respectively (2.7)
where 6, x, and x are column vectors of length N. The matrices c and J, are square of dimension N a n d the matrix x is given by
. :1
axN
laxi
X =
.
x*(t,c) = exp(-ct)x(O,c) exp(ct) and xP(t,c) = exp(-ct)F(t,c) exp(ct) where
I
.El
F(t,c) =
1‘ 0
dt’exp(ct’)$(c) exp(-ct’)
with
Equation 2.7 is a partial differential equation in x and t for the coefficients ((x,c,t). The coefficients 5 depend on the choice for $ through the source term on the right-hand side of eq 2.7. One approach to solving eq 2.7 is by expanding 5 in a power series in the fixed function x. Collecting coefficients and making use of the linear independent of the powers of x, a series of equations corresponding to these powers are obtained. Only those equations resulting from the linear terms in x include the J, term on the right-hand side of eq 2.7. These are the (inhomogeneous) equations of immediate interest because we are seeking information about the generator with nonzero J, coefficients. The higher order terms in x give homogeneous equations, Le., with J, = 0. For now, consider only the linear terms in the expansion of [ ti(x,c,t) = Xij(c,t)xj
(2.9)
where the expansion coefficients xi,(c,t) (equivalent to the matrix in eq 2.8) are to be determined. Inserting eq 2.9 into eq 2.7 leads to the following determining equations (2.10a) for k = 1, ..., p,1 = 1, ..., p,which are written in vector notation as
+ J,
a x / a t = KX
(2.10b)
Notice that there are equations in (2.10a) and N equations in (2.7). The elements of the matrix Kare functions of the system parameters, c, and are given by K V. .J.,J., = c..,6 I l J .,J. - cJ., J.6.., r l (2.1 OC) where ij counts the rows and iy‘counts the columns. The vectors x and J, are defined by J,
=
x
= (x119 x12t
($119
(2.13)
$12,
...1
$”)
...9
X”)
F(0,c) = 0 The general solution is given by x(t,c) = exp(-ct)[ x(0,c)
+ j‘ dt’exp(ct’) 0
i(c)eP’] exp(ct) (2.14)
which is equivalent to eq 2.12. Equation 2.14 expresses the coefficients X(t,c) in terms of the propagator, exp(ct), of the original dynamical system. The Green’s function method of sensitivity analysis yields expressions for linear sensitivity coefficients which are very similar in form to eq 2.14.13 A related observation is that x(t,c) is a sensitivity of the generator coefficient, &(x,c,t), with respect to xj, Le., a5i(x,c,t)/axj = Xij(t,c)
(2.15)
Further discussion on the connection between sensitivity analysis and the Lie group formalism for parameter space mapping can be found in ref 1. We will use the form of the determining equation given in eq 2.12 in all that follows, to keep the notation here consistent with that in ref 1. 111. The Case of Two Coupled Equations
In this section we look in detail at the solutions to both the time-independent and time-dependent determining equations for a system of two coupled equations (Le., N = 2 in eq 2.10 and 2.1 1). This simple case provides an illustrative example of how the Lie group generators can be used to study the behavior of the system variables as a function of the parameters. Section IIIC will discuss mapping through the space of initial conditions. a. Time-Independent Transformations. The matrix K i n eq 2.10b with N = 2 is explicitly 0
1
(2.10d)
in J,, and similarly so that, e.g., $2 in eq 2.10b corresponds to for x. The time-independent solutions of eq 2.10b obey the equation Kx(c,t) = -J/(c) (2.1 1)
-e12
As the rank of K is 2 and its order is 4. K has no inverse. Solutions (13) J. Hwang, E. P. Dougherty, S. Rabitz, and H. Rabitz, J. Chem. Phys., 69, 5180 (1978).
2276 The Journal of Physical Chemistry, Vol. 90, No, 10, 1986
Hubbard et al.
TABLE I: Finite Transformations Generated by the Time-Independent Generators JO J,
Jz J3
x1 e x p ( a / 2 )
c,
XI
c,
u2 cosh a
c1 cosh a , -u3 sinh a c, cos a v2 sin a
c2 u2 cos a , --[>I sin a
x2 e x p ( a / 2 ) uo x 2 exp(-a/2) exp(fl/2) 1’0 x I cosh ( a / 2 ) + x 2 sinh ( a / 2 ) x2 cosh ( a / 2 ) + xl sinh ( a / 2 ) ro x I cos ( a / 2 ) + x 2 sin ( a / 2 ) x2 cos ( a / 2 ) , -xI sin ( a / 2 ) I:~
+.
of eq 2.1 1 are therefore not uniquely determined by Nevertheless, all possible forms for x can be expressed in terms of four linearly independent solutions. Because is as yet unspecified, we may treat the components of x as the independent variables to be chosen in any arbitrary (and convenient) form. Four fundmental solutions to the equation Kx = can be obtained by evaluating this equation for the four orthogonal vectors x = (1,0,0,0), (0,l ,O,O), etc. This yields the four linearly independent generators
+
02
+
a + c,2- a - c21- a ac,,
ax2
- c12-
The general time-independent generator is a linear combination of the UIJin eq 3.2, Le., (3.3)
The vector q = (qll,q12,q21.q22) is an arbitrary function of the clJ, and each component of q is set equal to 1 or 0 in what follows. Each U,] in eq 3.2 is a generator for a one-parameter group, and the successive action of each of these generators gives rise ultimately to a four-parameter group exp(al u l l ) exp(a2u12) exp(a3u21) exp(a4U22) where a,, a2, a3, and a4 are the Lie group parameters. There are standard group theoretic methods for investigating and classifying the Lie algebras and Lie groups obtainable from sets of Lie derivatives such as those in (3.2). In this case we can however use intuitive shortcuts, and define the new generators
u2 =
CI2
+ C2I>
03
= c12 - c21
Expressed in terms of the new variables the Ss are
+
w = (uo f (u12 v22- u
=
(CII
+ c22)
(3.8)
-J3
+
-
-
=
(3.7b)
with limits on a’going from 0 to a, x,, from x , to X,, and similarly for u,.. The coefficients L,(v) are related to the $,](c) in the same way that the u,’s are related to the c,,’s as given in eq 3.5. The resulting transformations for each J, are listed in Table I. The action of the group on the surface ( u , ) , ( u ~-) ( ~u ~ =) ~ c is well-known. When the constant c is positive the surface is an hyperboloid of one sheet, and when the constant is zero the surface is a pair of cones joined a t the apex; the group SO(2,l) acts transitively on these surfaces. When the constant is negative, the surface is an hyperboloid of two sheets and the group acts transitively only on the subsurfaces in the upper half space, u3 > 0, and on the subsurfaces in the lower half-space, u3 < 0. By adjoining the reflection operation u3 -u3 to SO(2,l) one obtains a group (a subgroup of O(2,l)) which acts transitively on the entire surface in this case. We must take care however that this reflection in the u subspace is compensated by discrete operations in the complete v,x,t space so that eq 2.3a are left invariant. Written in terms of the parameters v eq 2.3a are
1
00
~ ~ ) I / ~ ) / ~
These are the commutation relations of the generators of the Lorentz-like group SO(2,l) acting in a 3-space of real variables. The Lie group parameters a, determine how far and in what direction the system variables and parameters move through the parameter subspace. The actions of finite transformations upon each of the system variables and parameters gives new values for these quantities dictated by the choice of a,. The finite transformations can be obtained from eq 2.2 or by integrating the equations
+ ~1 ( U O- u I ) x ~
k2 = 5 ( ~ 2- ~ 3 ) x I
Each of these generators annihilates
(3.7a)
The generator Jo commutes with all the other Ji and has no action in the parameter subspace. The remaining Ji obey the commutation relations
ac,,
‘(9) = qi,(c)ulJ
c3
that fixes the spectrum associated with the system x = cx. The allowed frequencies for the set of two coupled equations discussed here are
(3.2)
a + c21- a
ac,,
ti3 cosh
detlc,, - w6,l = 0
[Ji,J21
u2,= x2- a
+ u2 sinh a a , -vI sinh a
ug cosh a
and (u,)’ + ( u , ) ~- ( u ~ =) (c,, ~ - c22)+ 4cI2c2,,so the transformations they generate leave invariant these functions. It follows that these one-parameter groups also have the important property that they each leave invariant the secular equation
-+
U,I = xiax,
03
+ L ; ~sinh a
- -
--
These are left invariant if, as v 3 -v3, one also lets u2 -v2 and x2 -x2, or additionally if, with vj -v3, one lets v i -u,. Other discrete time-independent invariant transformations can also be found. The action of any operation of the O(2,l) transformation group carries a point p(x,c) in the full space to p ( f , f ) ,with the relation between x,c and %,fbeing given in Table I. The evolution operator of the system is exp(ct) as pointed out below eq 2.14, but this operator may be written in Lie form as e x p ( f w with
The Journal of Physical Chemistry, Vol. 90, No. 10, 1986 2277
Parameter Space Mapping of Differential Equations W = f, a / a x j and f/ = cijx,
v? - v32
The operator acts on initial values x ( 0 ) to give trajectories x ( t ) = exp(tmx(0). The SO(2,l) transformations change Wtofj d/dx w i t h 3 = zip, and shift the initial value from x ( 0 ) to n(0). Given these observations it is evident that the effect of the SO(2,l) actions of Table I on a solution curve is to shift c to f and change the initial value x to the new value %. Thereafter the curves evolve through the action of the evolution operator. Finally, we note that in eq 2.1 1 if x vanishes so does $. Thus there are no time-independent invariance transformations of the equations x = cx that leave invariant the vector x ( 0 ) of initial values while changing the ciJ. These correlations between initial conditions and the c's can be eliminated through the use of the time-dependent generator, and the issue of initial conditions will be discussed in more detail in section IIIc. b. Time-Dependent Transformations. The form of x(t,c),and therefore the generator W , will be determined by our particular choice for the coefficients $. The time-dependent solution to eq 2.10b with the x(0,c) = 0 is
[
u1(v~~2u3)t
VI(VZ
--( v 2
- v 3 ) (w(cosh ( u t ) - 1)
sinh (ut) x , a + ax 1
[
03 + -(cosh
v,
-w+ VI
u2(u2
av,
(at)- 1)
+ s ) t - -VI (cash (ut)- 1) +
2wz
2w2
-
wv3
sinh (ut) - -(cosh 2~
I
OJ,- v,)
(v2
1
+
-V 1
v3)
v2
-(cosh
- 2w3
( u t ) - 1)
1
UI
-(cash (wt) - 1) 2w2
UlV2 +sinh ( w t ) 2~
] a xl [ xZ-
+
+ V I 2 0 3 + sinh ( u t ) + -(cosh VI 2(u, + v3)w3 2wz w2u2
2
2
2
2
2 (3.12b)
All of the e,, parameters have been replaced by the u, parameters, as given in the transformations of eq 3.5. The $ij(c)coefficients must also be appropriately altered so that the generator in the v basis is equivalent to the generator in the c basis, i.e., with the new coefficients called L,
a a u = lj(x,c,t)- + G j j ( C ) -
a
a + Li(v)axj avi
= Ej(x,v,t)-
8XJ aci, where the Li are given in terms of the
$ij
by
Ll = $11 - $22 Lo = $1' + $22 LZ = $12 + $21 L3 = $12 - $21 Equation 3.13 can now be written as x(t,c) =
4 J'ds e ( f " ) D ] T 1 @ ( v )
(3.13)
[ 2w2 + Ulu3t
-
vZ
-(cosh 2w2
(ut) - 1)
-U3(uZ
- U3)t
2w2
+
( w t ) - 1)
vIv3 +sinh ( u t )
2~
(3.15d) The integration of eq 3.9 has been done analytically only for U I ,for which the following finite transformations are obtained: n , ( t , a ) = P'/2Xl(t,0) n2(t,a)= eo'/2x2(t,0) uo(n) = vo(0) + a (3.16) The finite transformations for the other three generators have been found by rearranging eq 3.9 and numerically solving, for each generator, the following set of differential equations
(3.14)
where W v ) = '/z[(Lo + L M L 2 + L3)& - L ~ ) , ( L-o LdI. BY choosing Li to be a set of basis vectors, Lo = (l,O,O,O), etc., along the v-coordinate axes in parameter space, eq 3.16 can be solved for the four components of x(t,c). This leads to the following set of four fundamental generators, each of which operates on a different vi:
-
1)
u, =
T" = 0
(wt) -
(3.15~) -u1v3t
0
-
+ wzv3 - (cash (wt) - 1) + 2w2 2(v* + v3)u3 u12v2
VI
[ 2w2 rl
X
202
2(u2 - u3)w3
+
w+
1
a +a (3.15b)
~ 2 ax2
v12u2- w2v3
1I
-
+
sinh ( u t )
The matrix K may be expressed as K = TDTI with D being a diagonal matrix having eigenvalues (O,O,w,-w), where w2 = u12 vZ2- u32 = (cll - czZ)'+ 4c12c21.Notice that this u is related, but not equal to, the w of eq 3.7. The transformation matrices T and T-' are given by -1
+ v3)t
-
(3.15a)
-avi(a> - - L,(v) aa
= 1, f o r i = 1, 2, 3
(3.17)
The equations for each finite transformation are uncoupled from each other. Notice that the coefficients xij in eq 3.17 are parametrically dependent on v and t and directly dependent on a. Therefore these equations may be readily solved by using available integrators. The integration of eq 2.3 is along the time axis of the problem while that of eq 3.17 is along the physical parameter
2218
The Journal of Physical Chemistry, Vol. 90, No. 10, 1986
Hubbard et al.
axis, as controlled by the group parameter a. To be explicit about this in what follows, the independent variables will be noted in the argument of the solution as x(t,a), where a is used to indicate that in general there will be dependence on more than one Lie group parameter. Figures 2-5 show the behavior of x,(t,a)as a function of both t and the Lie group parameter a for each of the four generators given in eq 3.17. The initial conditions for the system parameters, ui at a = 0, have been arbitrarily set equal to uo = u l = u2 = 0, and o3 = 2. These values correspond to a harmonic oscillator with unit frequency and amplitude. The initial conditions for the system variable are xio = 1 and x20 = 0. Equations 3.10, at a = 0, thus reduce to u3
(3.18)
XI(t,O) = +,O) 03
X*(t,O)
= --x,(t,O)
2
Along the t axis at a = 0 each of the four figures shows x,(t,O) which satisfies eq 3.18 for one period in time. The solutions x,(t,O) and x 2 ( t , 0 )to eq 2.3 are the initial trajectories from which the propagation in a begins as eq 3.17 are solved. Each generator acts on a different coefficient uias a is increased. Figure 2 shows the action of exp(aU,), which linearly increases or decreases uo, on x,(t,O). As uo changes from its initial value of 0, eq 3.20 become VO(a) u3 (3.19) X l ( t , U ) = -x,(t,a) + 7x2(t,a) 2 u3
Figure 2. The action of exp(aU,) on x,(t,a), which is a mapping through u,, space. The initial trajectory a t a = 0 corresponds to a harmonic oscillator with unit amplitude and frequency, and x I o= 1, x: = 0. The range in t is for one period of oscillation. Figures 2-5 have all been normalized to the same size cube so that the scales should not be compared. The relative behavior is the relevant observation.
u0(a)
= --xl(t,a) + -x,(t,a) 2 2 It is not hard to show from the analytical solution to eq 3.10 what action each generator is going to have on the initial oscillator. The effect of UI acting upon uo is to change the real part of the eigenvalue which satisfies the secular equation for eq 3.10. Recall that the solutions to eq 3.10 have the form x,(t) Aeuf (3.20) X,(t,a)
-
-
Bew' with A and B being constants. Equation 3.20 is a valid form of the solution only for w # 0, and the form of w is given by eq 3.7b. Therefore, the action of exp(aU,) on the solution with the initial conditions stated above results in the following form for the frequency uo(a) iu3 w=f2 2 with uo(a) = uo(0) a = a. Thus xl(t,a)has the form x,(t,a) wAeQt12eiWI2 and similarly for x,(~,a).This shows that the amplitude varies exponentially with both a and t , and this behavior can be seen by looking at the surface shown in Figure 2. Thus, U I changes the amplitude but not the frequency of the oscillator. The generators U, and U3affect both the frequency and the amplitude of the initial oscillator. The action of exp(aU2) on x,(t,O)is shown in Figure 3. This operator linearly increases or decreases u I , and after a finite transformation eq 3.18 becomes
xZ(t)
+
The oscillator frequency decreases in values as u,(a) increases, until u l ( a ) = u3. At that point the solutions to eq 3.21 become linear in t: x , ( t , a ) = x l 0+ u3(xIo + x,')t with x,(O,a) = xio
x,(t,a) = x20 - u3(xIo + x20)t x,(O,a) = x20 The x , ( r , a ) trajectory for u l ( n )= u3 is shown as a darkened line in Figure 3. The eigenvalue, w, of the secular determinant changes from pure imaginary to real for a values which give u l ( a ) > u3. At that point the solution, x,(t,a), shown in Figure 3 changes from an oscillatory to an exponential form. This is easily seen on the
X l ( W
Figure 3. The action of exp(bU2) on x,(t,b),which is a mapping through L), space. Refer to caption on Figure 2.
surface for the a values larger than the value represented by the darkened line. Equations 3.18 under the action of exp(aU3) become XI(t,U)
=
u*(a)
2
+ u3 x2
(3.22)
with 0, linearly increasing or decreasing. Figure 4 shows the action of exp(aU3) on x,(t,O), which has an effect similar to the one U, has on the frequency of the initial oscillator. The solutions to eq 3.22 become linear in t for x,(t,c)and constant in t for x z ( t , c ) when uz(a) = u3. The trajectory, X , ( Z , U ) ~=~ x= l~ o= 1, is shown as a darkened line in Figure 4. Beyond this value in a the solution becomes exponential for the same reasons discussed above. Both finite transformations due to U, and U3change the amplitude of the initial oscillator, but not in the same way as UI does. These former transformations have action on the preexponential amplitude factor in the solution, so that the amplitude is a-dependent but not t-dependent. The action of exp(aU,) on xl(t,O)is shown in Figure 5 . The transformation changes u3 linearly in a, so that the only effect
Parameter Space Mapping of Differential Equations
The Journal of Physical Chemistry, Vol. 90, No. 10, 1986 2279
X
Figure 4. The action of exp(cU3)on x , ( t , c ) , which is a mapping through u2 space. Refer to caption on Figure 2.
-
l
W
Figure 5. The action of exp(dU4)on x , ( t , d ) , which is a mapping through v3 space.
on eq 3.18 is that the coefficient becomes parameter-dependent, Le., u3 +(a). The oscillator frequency, for uo uI + u2 = 0, is equal to u,. Thus the solution to eq 3.10 under the action of U, will remain a harmonic oscillator, with variable frequency, Le., U, tunes through the frequency space with no effect on the amplitude. Any linear combination of the four generators given above can also be used to interconvert solutions to eq 3.10. For example, the combination of U1+ U, will act upon the parameter cI1in eq 2.3a. The resulting finite transformations will give the solutions, x,(t,a), as a function of change in the c l l coefficient. Also of possible interest with regard to eq 2.3a are the kinetic equations which result when cI1= -cI2 and c22= -cZ1. The rate constant dependencies of the concentrations can be easily monitored by using appropriate linear combinations of the four independent generators given in eq 3.15. Consecutive application of different generators can also be used to generate different kinds of mappings. For example, a given solution xl(t,O), can be transformed by action of Ul for a certain value of a and then by action of U2 for a different b value, Le.,
gives the time-dependent solution for the coefficients in the generator. The formalism allows complete control over the choice of $(c) and x(0,c). The derivation in section IIIb proceeded with x(0,c) = 0 and each $,(c) = 1. Notice that for t = 0 all of the coefficients ~,,(t,c)J,=~ in each of the four generators in eq 3.15 are zero. The particular choice of x(0,c) = 0 therefore results in time-dependent invariance transformations which do not alter the initial values, x,(O,a), while changing the system parameters u,, i.e.,
R,(t,a,b) = exp(bU2) exp(aU1) x,(t,O,O) = exp(bU2) ni(t,a,O)
is to look at the solution to eq 2.12 for x(0,c) # 0. The symbol Vwill be used in what follows for generators which are obtained by considering the initial conditions to be variable parameters. If we choose +(c) = 0, so that eq 2.12 becomes
+
The choice of of the transformations is arbitrary here. Alternatively we could use
R,(t,a,b)
~ , ( t , 6 ,= ~ )exp(iiU,) exp(W2) xi(t,0,o) = exp(iiU,) ~ , ( t , 6 , 0 )
If the generators commute, [U,,U2]= 0, then ii = a and 6 = b. This is the case for the time-dependent generators of eq 3.15. By considering such linear combinations or consecutive applications of the Lie operators, it is possible to move at will throughout the entire parameter space. There may be practical reasons for considering either one or the other type of application of the Lie operators. For example, for some desired mapping of the trajectories there may be a region in the hyperdimensional space of all system variables where the solutions are highly unstable. By the appropriate application of Lie operators it may be possible to avoid this particular region of the space during the mapping. An analogy can be made with thermodynamic state variables, where the value of a reversible state depends on where you are and not how you got there. The same idea applies to the mapping procedure discussed here, because the particular path chosen for mapping the solution from the initial region of system parameter space to some other region is, in principle, irrelevant. c. Initial Conditions. The response of a dynamic system to variations in initial conditions can also be monitored by using the Lie group formalism demonstrated above. Recall eq 2.12 which
Z,(O,a) = x,(O,a)
This is useful when studying the effect of system parameter variations for a given fixed set of initial conditions, since the parameter space mapping resulting from the time-dependent transformations in section IIIb will not transform the vectors of initial values x(0,a). One way to obtain generators which do not vanish a t t = 0, i.e. Z,(O,a) = exp(aV x,(O,O) (3.23)
x(t,c) = exp(Kt) x(0,c)
(3.24)
choices for x(0,c) can be made such that the resulting vector, x(t,c),yields generators which can either independently act on each of xl(O,O)and x2(0,0), or can act simultaneously on them both. As an example of the latter case, consider the choice of x(0,c) = (l,O,O,l), and $(c) = 0. This choice of x(O,c),with K given by eq 3.1, results in the generator
which gives a simple dilation for the finite transformation of the initial value of each system variable:
ni(O,b) = ebxi(O,O) where b is the parameter of a one-parameter Lie group associated with the initial condition transformation. This particular choice for x(0,c) yields the operator exp(bV which has equivalent action on each of the initial conditions. It is more interesting and useful, however, to obtain a generator which treats the initial conditions for each system variable independently, e.g., varying one initial condition while the others remain constant, etc. It would then be necessary to construct the appropriate vectors, x(O,c), to retain the desired terms in the
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The Journal of Physical Chemistry, Vol. 90, No. 10, 1986
x = cx into any other phase portrait of these same equations with different c .
generator, V. The general form of Vis given by
and only differs from the expression for U given in eq 2.4 in that the components of x(t,c) in the above expression are made up of contributions from the first term on the right-hand side of eq 2.12. Choosing the x(0,c) in an analogous way as the L(v) where chosen in section IIIb results in four new independent generators. Two of these choices, for x(0,c) = (1,0,0,0) and x(0,c) = (O,O,O,l), result in generators which act independently on either x,(O,O) or x2(0,0), respectively. The resultant finite transformations then give the evolution of the system of equations from this change in initial conditions. The choice of x(0,c) = (0,1,0,0) and x(0,c) = (0,0,1,0) results in generators which couple the initial conditions. Assuming this coupling will not be of interest in most cases, we will give here the generators, V I and V2,which act independently on the initial conditions for either xl(t,a) or x 2 ( t , a ) . An expression for x(t,c) can be obtained from eq 3.24 for x(0,c) = (1,0,0,0) using K = TDT-I, where the matrices T , D , and T-I were given in eq 3.12. Performing the appropriate matrix multiplications results in the following expressions for the four components of x(t,c):
xIl(t,c) = w2 + uI2 + x12(t,c)= -w(uz
+
(3.26) u 3 ) sinh ( u t ) - u1(u2+ u3)(cosh ( w t ) - I ) (
~
-2~ 3 ’~) cash (ut)
~ ~ ~ ( t=, w(02 c ) - u3) sinh ( u t ) - u I ( u 2 ~ 2 2 ( t , ~=)w2
+ ui2+ (
~
+ u3)(cosh ( u t ) - I )
--2~
~3
cash ~ )
(ut)
Substituting these components of x(t,c) into the expression given in eq 3.25 results in the generator VI. Notice that x(t,c) given in eq 3.26 is equal to (1,0,0,0) at t = 0, so that only the initial condition, xl(O,O) is affected by VI. In a similar way, V2 can be constructed by using x(0,c) = (0,0,0,1), which gives the following result for the four components of x(t,c):
xll(t,c) = w 2 - u j 2 - ( 0 2 , - ~
3
cash ~ ) (wt)
Hubbard et al.
(3.27)
+ uI(fi2 + u3)(cosh ( u t ) - 1) xzI(t,c)= -w(u2 - u3) sinh ( u t ) + u,(c2- c3)(cosh ( w t ) - I ) ~ 2 2 ( t , ~=) w 2 + I- ( ~ -2~ ~3 cash ~ ) (ut) xI2(t,c)= w(u2 + u 3 ) sinh ( w t )
fi12
Again, at t = 0, the above components of x(t,c) reduce to (0,0,0,1) and the finite transformations which result from V2 all vanish except for the transformation for x2(0,0). The finite transformations which result form the generators VI and V2, i.e. XI = exp(bV,)x, .t2= exp(cV2)x2 give the evolution of the system as a function of arbitrary changes in the initial conditions. There is another equivalent approach to initial condition transformations that may be established. As it stands, the initial conditions are unique among the system parameters in that they do not explicitly enter into the differential equations. However, by a simple transformation such as y , ( t ) = x , ( t ) - x,(O), we may produce a new inhomogeneous set of differential equations for y , which have null initial conditions. If we now impose the condition on the determining equations that x,(O) = 0 then the new “trivial” initial condition will remain fixed and the true initial condition, x,(O), will enter as ordinary parameters in the new differential equations. Linear combinations ~ , a , U , of the generators U, derived in sections IIIa-c give rise to group operators exp U ( & Y , ~ , ) that enable one to map any solution of x = cx into any solution of x = c’x. Instead of carrying out the mapping with operators of the form exp a(C,a,y,) one can use operators of the form exp a l U i exp azU2 ... In either case one has available a set of operators which act transitively in the space of c and in the space of solutions to x = cx. Thus one is also able to map any phase portrait of
IV. Conclusion A global parameter space mapping procedure has been demonstrated which is appropriate for any autonomous system of differential equations. We have applied the method to a coupled linear system as an illustrative example, although the basic interpretation will remain when going to a nonlinear system. Both time-independent and time-dependent invariance transformations have been obtained analytically for the case of two coupled equations. The time-independent transformations give simple dilations or rotations of the initial trajectories, x ( t , O ) . The time-independent transformations have no action on the characteristic frequency of the dynamical system. Because all points along the trajectory, x ( t , O ) , are transformed equivalently, the vector of initial conditions, x(O,O), is also transformed. There are thus no time-independent transformations which do not alter the initial conditions. The time-dependent transformations have much more interesting behavior. In section ITIb we saw that these transformations can alter the amplitude and/or the frequency of the initial trajectories, depending on what parameter was being varied. Thus the time-dependent transformations can map directly from a trajectory which is oscillatory to one which is either exponentially growing or decaying. This is a direct consequence of the action the generator has on the frequency of the dynamical system. Section I l k showed how the initial conditions of the system variables can also be treated as parameters, resulting in the ability to map through the space of initial conditions. Finally, any mapping can be performed which results from a linear combination or successive application of any or all of the generators mentioned above. The complete set of time-independent, time-dependent, and initial condition generators allow mappings of the trajectories, xi(t,u),through the entire space of initial conditions and parameters. A next significant extension of this work is to develop the machinery for application of the Lie group formalism to nonlinear systems. It will be extremely interesting to study the mappings which transform solutions from regular to chaotic regimes. There should be no ambiguity in the mapping at a bifurcation in phase space, because the mapping generated by the Lie group formalism occurs in the combined space of phase and timesi4 A central problem in treating nonlinear systems will be the evaluation of the coefficients &(x,c,t). For example, it will not be appropriate to truncate a power series expansion in x at the linear term, as was the case in the example presented in this paper. However, once the technical details are worked out, we expect the parameter space mapping of nonlinear systems to be quite informative. Another application of interest will be to constrained problems, which are notoriously difficult to treat. For example, dynamical systems exist to which a particular set of initial conditions must be found which will satisfy a given set of constraints, or final conditions. Instead of performing a search through all of initial condition space to find points which satisfy the constraint, one could imagine building the constraint into the generator itself. This would result in a mapping which would only project onto the appropriate portion of initial condition space, eliminating the need for a blind global search. There are aspects of the formalism itself which require further investigation. For example, it is of interest to investigate when retaining the coefficient r(x,c,t) in the determining equations will be of benefit physically in the finite transformations for the system variables and parameters. In conclusion, a variety of interesting developments and applications can be envisioned for the L.ie mapping technique. Acknowledgment. We thank Dr. Shenghua Shi and Dr. R. Sukeyuki Kumei for helpful and stimulating conversations. This research was supported by the National Science Foundation Grant CHE014165 to the University of the Pacific and a grant from the Air Force Office of Scientific Research to Princeton [Jniversity. (14) C. E. Wulfmanand R. D. Levine, Chem. P h w Lert.. 87, 105 (1982).
