16568
J. Phys. Chem. 1996, 100, 16568-16570
Stabilizing Unstable Fixed Points Using Derivative Control P. Parmananda and M. Eiswirth* Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany ReceiVed: June 4, 1996X
We report the stabilization of unstable fixed points (saddles) in model systems exhibiting bistability. Stabilization is achieved using the derivative control strategy proposed in a different context by Biewlaski et al. (Phys. ReV. 1993, A47, 3276). This strategy does not change the location of fixed points in the system, but does alter their stability (eigenvalues), enabling us to stabilize the previously unstable fixed points (saddles). Maintaining the dynamics on the saddle fixed point could be desirable in certain experimental systems.
1. Introduction Altering the system dynamics by judiciously perturbing an accessible control parameter has been a subject of extensive research lately. Most of the reported results involve the control of observed chaotic behavior2-6 by converting it into periodic responses using flexible control strategies.7-9 Recently more work has been done on stabilizing the system dynamics on unstable fixed points using feedback techniques.1,10-12 This involves the conversion of the observed periodic or chaotic behavior to a fixed point response, namely, the initially unstable focus or node inside the limit cycle. Controlling the dynamics on these nonoscillatory steady states could prove to be useful in experimental situations where the emergence of oscillatory behavior results in the degradation of system performance. We report the stabilization of saddle points in various numerical models exhibiting bistability using the derivative control strategy. Control on saddle fixed points has been reported previously by Ross and co-workers13,14 using a system dependent control strategy. In the following section a brief discussion of the derivative control strategy is presented. Then the control strategy is applied to several model systems. We start with the generic form of a bistable system near a cusp. Though one variable is sufficient for bistability, generally more than one is required in chemically realistic systems. An example with two species, which is treated in section 3.2, exhibits only bistability. Finally we show that the strategy also works in systems with two dynamical degrees of freedom, which exhibit oscillations in a regime adjacent to the bistable one. 2. Derivative Control Strategy Derivative control strategy as proposed by Bielawski et al.1 can be applied to autonomous model systems of the form
x˘ ) F(x,P)
(1)
where x Rn and p Rm. The fixed point solutions (x0) of the system at a fixed parameter vector (p0) are found by solving
F(x0,p0) ) 0
(2)
The number of fixed points and their respective stabilities can be calculated using linear stability analysis. The goal is to stabilize the unstable fixed points by adding a continuous perturbation to one of the system equations. The perturbation (control) signal is proportional to the derivative of one of the components of the state space vector x and can be superimposed X
Abstract published in AdVance ACS Abstracts, September 15, 1996.
S0022-3654(96)01637-1 CCC: $12.00
either directly to one of the system equations (as in section 3.2) or via one of the control parameters (as in section 3.3). A control of this type does not change the location of the fixed points, but it can alter their stability. To implement the derivative control strategy in bistable systems, one first needs to apply an initial perturbation (parameter spike) to knock the systems out of one of the existing stable nodes. 3. Application to Models The control strategy has been implemented in several models. We first describe a strictly one-dimensional bistable system, which can be mapped onto the normal form of a (codimension 2) cusp bifurcation. Next control is achieved in a two-variable model with one fast-relaxing variable, which is effectively onedimensional; we chose a simple model of a LangmuirHinshelwood mechanism. Finally a model with two dynamic degrees of freedom, which exhibits oscillations at slightly different parameters, is controlled in the bistable regime. The model (electrochemical corrosion) was chosen because earlier derivative control was applied to it in order to achieve stabilization of an unstable focus inside a limit cycle.12 3.1. Bistability near a Cusp. It can be shown analytically that implementation of the derivative control strategy would stabilize the saddle of a bistable system in the vicinity of a cusp bifurcation. An analytically solvable form of a cusp is chosen of the form
y˘ ) (µy - y3)
µ>0
(3)
The three fixed points evaluated by equating the rhs ) 0 of the above equation are (0, ( xµ). Initially the eigenvalues corresponding to (0, + xµ, - xµ) calculated by evaluating the Jacobian µ - 3y2 are (µ, -2µ, -2µ) respectively. Hence 0 corresponds to the unstable saddle we wish to stabilize. Using the derivative control strategy the modified equation with the control implemented is
y˘ ) (µy - y3) - γ(µy - y3) ) (1 - γ)(µy - y3)
(4)
It is evident that for values of γ > 1 the saddle fixed point (0) is stabilized and the two other fixed points (+ xµ, - xµ) have positive eigenvalues associated with them and thus are rendered unstable. In the generic case, a bistable system includes a nonzero constant term
y˘ ) + µy - y3 © 1996 American Chemical Society
(5)
Stabilizing Unstable Fixed Points
J. Phys. Chem., Vol. 100, No. 41, 1996 16569
The control strategy for such systems has been checked numerically. The saddle of the original system was reached after an initial perturbation, which brings the system to some state between the two stable fixed points. Switching on the control caused the trajectory to home in on the saddle point. 3.2. Langmuir-Hinshelwood Mechanism. A numerical model simulating a binary LH mechanism on catalytic surfaces exhibits effectively one-dimensional behavior. The dynamics of the model system are determined by the following set of two coupled ODEs, where x and y denote the coverages of the reactants.15
x˘ ) k1(1 - x - y) - k2x - k3xy
(6)
y˘ ) k4(1 - x - y)2 - k5y2 - k3xy
(7)
The variables describe the coverages of the two reactants, x and y. Each equation contains the respective terms for adsorption and desorption and the binary surface reaction. It is assumed that y needs two adsorption sites. In the bistable regime there are two stable states with predominant x and y coverages, respectively, separated by a saddle point where appreciable amounts of x and y coexist on the surface. The control signal is proportional to the derivative of the state space variable and hence would be zero if the system were resting on its fixed point solution. Thus, it is necessary to give the system a parameter spike such that it kicks the system out of its steady state. The size and the sign of the spike are important and must be such that the system dynamics after the spike are somewhere between the two stable states, i.e. in the basin of attraction of the now stabilized saddle fixed point. As the system tries to return to either of the two stable states, control of the form
x˘ ) k1(1 - x - y) - k2x - k3xy - γx˘
(8)
y˘ ) k4(1 - x - y)2 - k5y2 - k3xy
(9)
is implemented. For a suitable value of γ, the control strategy stabilizes the previously unstable saddle fixed point. This implies that the real parts of the eigenvalues of the saddle fixed point are all negative. It should be pointed out that stabilization is not possible for all possible combinations. For example feeding γy˘ in eq 8 would not stabilize the system dynamics on the saddle fixed point. At parameters [k1,k2,k3,k4,k5] ) [8,1,60,15,0.2] the system exhibits bistable behavior. The two stable fixed points are (x, y) ) (0.82, 9.1 × 10-3) and (x, y) ) (0.25, 0.247), respectively. However, when control is implemented, the system dynamics approach the saddle fixed point (x, y) ) (0.335, 0.177), as shown in Figure 1. The points A, B, and C correspond to the three coexisting fixed point solutions of the system. The arrowed trajectories are the evolution of the system from different initial conditions to converge on the saddle point B (x, y) ) (0.335, 0.177). 3.3. Model for Electrochemical Corrosion. The numerical model we choose to implement the control strategy on is the Talbot-Oriani16,17 model, describing electrochemical corrosion. The two dimensionless differential equations describing the model dynamics are16,17
dY ) p(1 - θOH) - qY dτ
(10)
dθOH ) Y(1 - θOH) - exp(-βθOH)θOH dτ
(11)
Figure 1. State space portrait for the effectively one-dimensional LH model (eqs 8, 9) after implementation of the derivative control strategy. Trajectories from different initial conditions under the influence of the control converge to the saddle fixed point (x, y) ) (0.335, 0.177). Superimposed are the two nullclines confirming the position of the three fixed points.
where Y is the concentration of metal ions in the electrolyte, θOH is the fractional coverage of the metal surface covered by MOH, p and q are parameters related to chemical rate constants, and β represents the non-Langmuir nature of MOH film formation. The system has been studied in some detail16,17 and exhibits bistability for appropriate parameter values adjacent to the oscillatory regime. To reiterate, in order to achieve control, the system has to be given a parameter spike (p was incremented) so as to kick it out of either of its steady fixed point solutions. As the system returns to either of the two stable states, control of the form
(
)
dθOH dY (1 - θOH) - qY ) p+γ dτ dτ
(12)
dθOH ) Y(1 - θOH) - exp(-βθOH)θOH dτ
(13)
is implemented to stabilize the previously unstable saddle fixed point. The control chosen here is analogous to the one in ref 6. For parameters [p,q,β] ) [0.000 06, 0.001, 8], the uncontrolled system exhibits bistability. The two stable fixed points corresponding to low and high coverage of the electrode are (θOH, Y) ) (0.1245, 0.0525) and (θOH, Y) ) (0.89, 0.00655), respectively. However, when control is implemented, the system dynamics approach the saddle fixed point (θOH, Y) ) (0.251, 0.0448), as shown in Figure 2a. The points A, B, and C correspond to the three coexisting fixed point solutions of the system. The curved part of the upper trajectory is the evolution of the system when it is given a parameter spike to kick it out from the fixed point corresponding to the low coverage A ((θOH, Y) ) (0.1245, 0.0525)), and the straight part is the path taken by the system to converge to the saddle point B ((θOH, Y) ) (0.251, 0.0448)) upon implementation of derivative control. Similarly, the other trajectory corresponds to evolution of the system from the fixed point corresponding to high coverage C (θOH, Y) ) (0.89, 0.006 55), while it is being spiked (curve) and then as control is being implemented (straight). Figure 2b is a similar state space portrait but at a different value in parameter space ([p,q,β) ) [0.00004, 0.001, 8]). The
16570 J. Phys. Chem., Vol. 100, No. 41, 1996
Parmananda and Eiswirth method to control onto a saddle point in the bistable regime. The stabilization of saddle fixed points could prove advantageous in experimental situations where stabilizing the dynamics on the saddle fixed points leads to improved system performance. For example, a binary surface reaction between two reactants x and y may exhibit two stable states, in which the surface is almost exclusively covered by x or y, respectively. Both would have small reactivity (for lack of the respective reaction partner), while high coverages of x and y coexist for the saddle point, which therefore exhibits a higher reaction rate. It should be emphasized that in systems involving more than one variable not every type of derivative control will lead to the stabilization of the saddle; that is, the feedback has to be applied to the appropriate variable. The only known previous work dealing with stabilization of unstable steady states (saddles) in bistable systems, that we are aware of, is simulations and experiments performed by J. Ross and co-workers.13,14 They, however, use a different, system dependent feedback technique to achieve control on the unstable branch of the bistable curve. With their method, they were able to map out the entire unstable branch using different values of the gain γ. With the present strategy, tracking is also possible using an intrinsic parameter (rather than gain γ) as the control parameter. Since the described control strategy does not depend on the details of the underlying system, not even on the number of variables, it should be applicable in experiments. Acknowledgment. We would like to thank G. Ertl for many helpful discussions. One of us (P.P.) acknowledges financial support from the Alexander von Humboldt Foundation and from CONACyT (Mexico) under Project Contract Ref 4873E. References and Notes
Figure 2. Control onto the saddle point of the two-dimensional model of electrochemical corrosion (eqs 12, 13). The dashed nullclines show the position of the fixed points, Full lines (arrowed) show the trajectories after application of a parameter spike and subsequent application of the derivative control. (a) Parameter values p ) 0.000 06, q ) 0.001, and β ) 8.0. The control parameter p was spiked (decremented) by 7% for 11 units in dimensionless time for the upper trajectory and was incremented by 5% for 20 time units for the lower trajectory. The values of γ used to achieve subsequent control for the upper and the lower trajectories were 0.34 and 0.24, respectively. (b) Parameter values p ) 0.000 04, q ) 0.001, and β ) 8.0. The control parameter p was spiked (decremented) by 11% for 10 time units for the upper trajectory and was incremented by 8% for 5 time units for the lower trajectory. The values of γ used to achieve subsequent control for the upper and the lower trajectories were 0.23 and 0.20, respectively.
dynamics under the combined effect of spike and derivative control approach the saddle fixed point. 4. Discussion Until now, derivative control strategy had been used only to suppress the observed oscillatory behavior (periodic and/or chaotic) by stabilizing fixed points,4,12 i.e. an unstable focus or a node localized inside the oscillatory trajectories. Here it was shown that derivative control in combination with an initial perturbation also provides a generally applicable
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