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Sensitivity and chaos control for the forced nonlinear oscillations
Chaos, Solitons and Fractals 26 (2005) 1437–1451 www.elsevier.com/locate/chaos
Sensitivity and chaos control for the forced nonlinear oscillations Irina Bashkirtseva, Lev Ryashko
*
Department of Mathematics, Ural State University, 620083 Ekaterinburg, Russian Federation Accepted 30 March 2005
Abstract This paper is devoted to study the problem of controlling chaos for forced nonlinear dynamic systems. We suggest a new control technique based on sensitivity analysis. With the help of approximation of nonequilibrium quasipotential, stochastic sensitivity function (SSF) is constructed. This function is used as basic tool of a quantitative description for a system response on the random external disturbances. The possibilities of SSF to predict chaotic dynamics for the periodic and stochastic forced Brusselator are shown. The problem of chaos control based on SSF is considered. A design of attractors with the desired features by feedback regulator is discussed. Analysis of controllability and effective technique for regulator synthesis is presented. An example of suppressing chaos for Brusselator is considered. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Control of stochastic and chaotic oscillations is a challenging and a fundamental problem of nonlinear engineering. After the pioneering work of Ott et al. [1] controlling chaos attracts many researchers. Various methods are used to suppress chaos to equilibria or periodic orbits [1–13]. Analysis of the forced nonlinear oscillations play an important role for understanding the corresponding dynamical phenomena for electronic generators, lasers, mechanical, chemical and biological systems. Even small external disturbances may change the behavior of dynamic systems essentially. The various noise-induced transitions through periodic to more complicated chaotic regimes are a central problem in modern nonlinear dynamics theory. Underlying reason of unexpected chaotic-looking response of regular oscillations under small perturbations is high sensitivity of nonlinear dynamic systems. Usually, one can justify the presence of chaos on the basis of system instability to the initial condition disturbances. In recent investigations a research interest has been focused on phenomena with so-called ‘‘sensitive dependence to noise without sensitive dependence to initial conditions’’ [14] and local or nonuniform stability [15,16]. Classic characteristics of oscillation stability are multiplicators. Unfortunately, this standard stability measure does not reflect the important details of dynamics on the interval of periodicity. Local sensitivity analysis of the forced oscillations is a key for understanding of chaotic response.
*
Corresponding author. E-mail address: [email protected] (L. Ryashko).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.029
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From this viewpoint, possibilities of controlling chaos are connected with controlling of sensitivity. The reason of chaotic response is high sensitivity of attractors. Decreasing sensitivity by control is a constructive approach for control of chaos. In this paper, we describe foundations of stochastic sensitivity analysis and propose a new feedback control method of designing a regulator that can modify the sensitivity of a given nonlinear system so as to suppress chaos. Analysis of nonlinear oscillations under the stochastic disturbances was started by Pontryagin et al. [17] and continued by many researchers [18–20]. The random trajectories of forced system leave the closed curve of deterministic limit cycle and due to cycle stability form some bundle around it. Stochastic cycles were considered both near and far from Hopf bifurcation point. A qualitative effect of external fluctuations on the Hopf scenario was found and investigated in [21,22]. A nonuniformity of stochastic bundles along cycles for nonlinear parameters far from bifurcation points attracts attention of researchers, see e.g., [23,16]. Small external noises acting on limit cycles may give rise to local phase-dependent response of the oscillations. Local instability of cycle is a reason of its significant sensitive dependence and can cause noise-induced transition to chaos [24]. A crucial role of sensitivity of non-normal dynamical systems for subcritical transitions from laminar to turbulent flow and generation of large-scale magnetic fields of galaxies is demonstrated in [25,26]. A variance of stochastic bundles perpendicular to the deterministic orbit is a natural measure of limit cycles sensitivity. Kolmogorov–Fokker–Planck (KFP) equation gives the most detailed probabilistic description. However, the direct use of this equation is very difficult even for the simplest situations. Under these circumstances asymptotics and approximations are used. Asymptotic analysis of distribution density for small noises based on quasipotential function is actively developed [27–31]. In Section 2, we give the first approximation of quasipotential in the vicinity of limit cycle. This approximation is an orbital quadratic form. Matrix of this quadratic form defines a covariance of the normal deviations of random trajectories for any point on a cycle. This matrix function plays a role of stochastic sensitivity function (SSF) of a cycle. SSF is a natural probabilistic measure [33] of stochastic 3D-cycles response to small random disturbances. The possibilities of SSF as an useful analytical tool for research of thin effects observed near chaos in a period-doubling bifurcations zone for stochastic Lorenz and Roessler models were demonstrated in [32,34]. In Section 3, for the case of cycle on a plane (2D-cycle) a scalar analytical representation of this matrix SSF is given. The possibilities of this scalar SSF to predict some peculiarities of 2D-cycles for the forced Brusselator are demonstrated in Section 4. The well-known phenomenon [35,36] of order to chaos transition for periodically forced Brusselator from a sensitivity point of view is discussed. On the basis of SSF analysis new critical (supersensitive) values of Brusselator parameters are shown [24]. For these values very small periodic and stochastic disturbances transfer Brusselator to chaotic regime. In Section 5, we consider the problem of control for stochastic and periodic forced cycles based on SSF. The possibilities for designing a limit cycle with wishful features by feedback regulator are discussed. Controllability analysis and effective algorithms for regulator synthesis for 2D-cycles are presented in Section 6. The examples of suppressing chaos for Brusselator are considered. A stabilization of the chaotic attractor to the regular orbit closed to the unforced limit cycle is demonstrated.
2. Stochastic sensitivity of limit cycles For many dynamical processes with regular oscillations, the basic mathematical model is the nonlinear deterministic system x_ ¼ f ðxÞ;
ð1Þ
with T-periodic solution x = n(t). Here x is a n-vector and f(x) is a n-vector function. Let c be a phase curve (limit cycle) of solution n(t) satisfying the following stability property. Definition 1. The cycle c is called exponentially stable if for small neighbourhood C of cycle c there exist constants K > 0, l > 0 such that for any solution x(t) of system (1) with x(0) = x0 2 C the following inequality holds kDðxðtÞÞk 6 Kelt kDðx0 Þk. Here D(x) = x c(x) is a deviation of a point x from a cycle c, c(x) is the point on cycle c that is nearest to x. A system of stochastic differential equations (in ItoÕs or StratonovichÕs sense) _ x_ ¼ f ðxÞ þ erðxÞw;
ð2Þ
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is a traditional mathematical model allowing to study quantitative description of results of external disturbances. Here x(t) is a n-dimensional Wiener process and r(x) is a n · n-matrix function of disturbances with intensity e. The random trajectories of the forced system (2) leave the closed curve of deterministic cycle c and due to cycle stability form some bundle around it. The detailed description of random distribution dynamics of this bundle is given by Kolmogorov–Fokker–Planck (KFP) equation. If the character of transient is inessential and the main interest is connected with the regime of steady-state stochastic auto-oscillations then it is possible to restrict an investigation by analysis of a stationary density function q(x, e). Analytical research of the stationary KFP equation for stochastic limit cycles considered here is a very difficult problem. Under these circumstances, asymptotics based on quasipotential v(x) = lime!0e2 log q(x, e) are actively used. After quadratic approximation of quasipotential v(x), the probabilistic distribution for the bundle of random trajectories localized near cycle has the Gaussian representation vðxÞ ðDðxÞ; Uþ ðcðxÞÞDðxÞÞ q Ke e2 K exp ; 2e2 with covariance matrix e2U(c). This covariance matrix characterizes a dispersion of the points of intersection of random trajectories with the hyperplane orthogonal to the cycle at the point c. A function U(c) is a stochastic sensitivity function (SSF) of limit cycle. This function allows to describe nonuniformity of a bundle width along cycle for all directions. It gives a simple way to indicate the most and the least sensitive parts of the cycle to external noises. It is convenient to search for a function U(c) in parametric form. The solution n(t) connecting the points of cycle c with points of an interval [0, T) gives the natural parametrization U(n(t)) = W(t). Matrix function W(t) is a solution of Lyapunov equation [33] W_ ¼ F ðtÞW þ WF T ðtÞ þ P ðtÞSðtÞP ðtÞ; ð3Þ with conditions W ð0Þ ¼ W ðT Þ;
ð4Þ
W ðtÞrðtÞ 0.
ð5Þ
Here F ðtÞ ¼
of ðnðtÞÞ; ox
SðtÞ ¼ rðnðtÞÞrT ðnðtÞÞ;
rðtÞ ¼ f ðnðtÞÞ;
P ðtÞ ¼ P rðtÞ ;
P r ¼ I rrT =rT r;
where Pr is a projection matrix onto the subspace orthogonal to the vector r 5 0.
3. Sensitivity analysis of 2D-cycles For the case n = 2 the projection matrix is given by P(t) = p(t)pT(t), where p(t) is a normalized vector orthogonal to f(n(t)). As a result, the matrix W(t) can be written as W(t) = l(t)P(t). Here l(t) > 0 is T-periodic scalar stochastic sensitivity function (SSF). Multiplying the matrix differential equation (3) by pT from the left and by p from the right and using the projective matrices properties we get equation l_ ¼ aðtÞl þ bðtÞ; with T-periodic coefficients aðtÞ ¼ pT ðtÞðF T ðtÞ þ F ðtÞÞpðtÞ;
bðtÞ ¼ pT ðtÞSðtÞpðtÞ.
The explicit formula for solution l(t) is given by the following: lðtÞ ¼ gðtÞðc þ hðtÞÞ; where gðtÞ ¼ exp
Z 0
t
aðsÞ ds ;
hðtÞ ¼
Z 0
t
bðsÞ ds; gðsÞ
c¼
gðT ÞhðT Þ . 1 gðT Þ
Note that Lyapunov exponent k is connected with function a(t) by the following: Z T 1 k¼ aðsÞ ds. 2T 0
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The value M = max l(t), t 2 [0, T] plays an important role in the analysis of stochastic dynamics about a limit cycle. We shall consider M as a sensitivity factor of a cycle c response to random disturbances.
4. Forced Brusselator: nonuniform sensitivity and chaos Consider forced system x_ 1 ¼ a ðb þ 1Þx1 þ x21 x2 þ eh; x_ 2 ¼ bx1 x21 x2 ;
ð6Þ
received by the addition of small disturbances eh(t) to classical Brusselator. For b > b ¼ 1 þ a2 the unforced system (e = 0) has a stable limit cycle c ( b is bifurcation value). It is well-known that periodically forced Brusselator [35,36] with h(t) = cos xt demonstrates for a = 0.2, b = 1.2 a sequence of period doubling bifurcations with transition to chaos (see Fig. 1). _ Compare Brusselator response on periodic disturbances with stochastic ones. Let hðtÞ ¼ wðtÞ, where w(t) is an independent Wiener process. The random trajectories of stochastically forced Brusselator leave the closed curve of deterministic cycle and form some bundle around it. In Fig. 2, the random trajectories found by a direct numerical simulation for three values of noise intensity e = 104, e = 103, e = 102 are presented. The value of intensity e = 102 is close to the stochastic bifurcation point when disturbances destroy limit cycle [22].
Fig. 1. Periodic force Brusselator: bifurcations and chaos a = 0.4, b = 1.2, e = 0.05. (a) x = 0.8015, (b) x = 0.803 and (c) x = 0.81.
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Fig. 2. Stochastic forced Brusselator: destruction of cycle a = 0.4, b = 1.2. (a) e = 0.0001, (b) e = 0.001 and (c) e = 0.01.
In Fig. 3, the plot of stochastic sensitivity function l(t) along with values (asterisks) of empirical sensitivity function ~ðtÞ for the limit cycle are shown. Here l ~ðt; eÞ ¼ e12 Dðt; eÞ, where D(t, e) is an empirical dispersion of intersection points l ~ were calculated for 100 rotations around the cycle with for random trajectories with a line orthogonal to c. Values l
Fig. 3. Sensitivity function of Brusselator with a = 0.4, b = 1.2.
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~ of empirical sennoise e = 103. From Fig. 3, one can see that the theoretical curve l(t) is arranged near the values l sitivity function and describes the main features (sharp peaks, monotonicity intervals) of this function clearly. Essential overfall of l(t) values on the periodicity interval reflects a nonuniformity of random trajectories dispersion (See Fig. 2(b)). The widest part of a bundle is placed on the bottom part of a cycle. The other part of the bundle is much more close to ideal deterministic trajectory c. A large value of the sensitivity factor M = 568 means that Brusselator amplifies both periodic and stochastic input disturbances essentially and leads to chaotic response (see Figs. 1(c) and 2(c)). From an extended investigation of Brusselator sensitivity, we have found the following interesting domain of parameters. Consider the Brusselator behavior for a fixed a = 0.2 and various values of parameter b > b ¼ 1.04 from an interval [1.06, 1.07]. _ Let disturbances in (6) be stochastic: hðtÞ ¼ w. For the Lyapunov exponent k and sensitivity factor M, dependence on values b is shown in Fig. 4. As we can see in Fig. 4(a), a parameter k monotonically decreases with growth b. This means increase of a stability degree of a cycle to disturbances of initial data. One should think it should be accompanied by the appropriate decrease in the sensitivity of a cycle to random disturbances. However, here the converse is observed. The value M behaves absolutely otherwise (see Fig. 4(b)). Here we have a typical example of ‘‘sensitive dependence to noise without sensitive dependence to initial conditions’’ [14]. On the considered interval the function M(b) is not monotonic. Its graph has a sharp high peak. As a result, the function M(b) has an essential overfall of values. By critical value of parameter b, one has b* = arg maxb M(b) = 1.064082, M(b*) = 4.4 · 1010. We compare the bundles of random trajectories and plots of sensitivity functions of stochastic Brusselator for values b close to b*. For three values of parameter, b = 1.06, b* = 1.064082, b = 1.065 in Figs. 5 and 6 our results are presented. In Fig. 5 the random trajectories found by direct numerical simulation for e = 105 are demonstrated. As we see the Brusselator with b = b* (Fig. 5(b)) is supersensitive. For small background stochastic disturbances the burst of response amplitude is shown. As is visible in Fig. 6, the sensitivity function reflects these noticed peculiarities clearly. Note that bottom part of the cycle is most sensitive (see Fig. 5(a)). Let disturbances in (6) be periodic: h(t) = cos (xt) with x = 0.5. For these periodic disturbances, value b* = 1.064082 is also critical. Really, increase of periodic force intensity e results in the period-doubling bifurcations of system (6) (see Fig. 7): 1-cycle (e = 0.0005) ! 2-cycle (e = 0.0007) ! 4-cycle (e = 0.000763) and so on. For e = 0.00085 the bundle of trajectories (see Fig. 7(d)) looks chaotic. For a critical parameter value b* = 1.064082 the forced Brusselator is a generator of chaos. It is significant to stress the following geometrical details. A top part of attractors both for period-doubling and chaotic zone is more stable and practically coincides with the ideal trajectory of the unforced cycle. Arising chaotic structure is connected with bottom part of the cycle. Note that huge peak of SSF (see Fig. 6(b)) corresponds to this bottom part of the cycle.
Fig. 4. Lyapunov exponent (a) and sensitivity factor (b) for Brusselator with a = 0.2.
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Fig. 5. Stochastic forced Brusselator with a = 0.2, e = 105. (a) b = 1.06, (b) b* = 1.064082 and (c) b=1.065.
Thus, the function of sensitivity is the useful analytical tool for the prediction of singular responses of a nonlinear system both to stochastic and periodic disturbances.
5. Sensitivity control Consider a stochastic control system _ x_ ¼ f ðx; uÞ þ erðx; uÞw;
ð7Þ
where x is a n-dimensional state variable, u is a r-dimensional vector of control functions, f(x, u), r(x, u) are vector functions, w(t) is a n-dimensional Wiener process and e is a scalar parameter of disturbances intensity. It is supposed that for e = 0 and u = 0 the system (7) has T-periodic solution x = n(t) with a phase trajectory c (cycle). A stability of c is not assumed. The stabilizing regulator will be selected from the class U of admissible feedbacks u = u(x) satisfying the conditions (a) u(x) is sufficiently smooth and ujc = 0; (b) for the deterministic system x_ ¼ f ðx; uÞ; the solution x = n(t) is exponentially stable in the neighbourhood C of cycle c.
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Fig. 6. Sensitivity functions for a = 0.2, (a) b = 1.06, (b) b* = 1.064082 and (c) b=1.065.
Our aim is to control the stochastic sensitivity function W. For control system (7) it follows from (3) that W_ ¼ F ðt; uÞW þ WF T ðt; uÞ þ P ðtÞSðtÞP ðtÞ;
ð8Þ
where of of ou ðnðtÞ; 0Þ þ ðnðtÞ; 0Þ ðnðtÞÞ; ox ou ox SðtÞ ¼ rðnðtÞ; 0ÞrT ðnðtÞ; 0Þ;
F ðt; uÞ ¼
As we see, a variation of control u allows to change in Eq. (8) the coefficient F(t, u) only. Note that the outcome of the control depends only on values of the derivative ou . It gives us the possibility to simplify the structure of the used ox regulator. 5.1. Choice of regulator structure Consider TaylorÕs expansion of control function u(x) at a point c uðxÞ ¼ uðcÞ þ
ou ðcÞðx cÞ þ Oðkx ck3 Þ. ox
For c = c(x) taking into account condition (a), we get uðxÞ ¼
ou ðcðxÞÞDðxÞ þ OðkDðxÞk3 Þ. ox
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Fig. 7. Periodic forced Brusselator: bifurcations and chaos a = 0.2, b* = 1.064082, x = 0.5 (a) e = 0.0005, (b) e = 0.0007, (c) e = 0.000763 and (d) e = 0.00085.
As we see, a first approximation u1(x) for an arbitrary smooth control function u 2 U for small deviations D(x) = x c(x) is the feedback u1 ðxÞ ¼ UðcðxÞÞDðxÞ.
ð9Þ
Here U(c(x)) is the feedback matrix coefficient. Appropriate t-parametric representation for this matrix looks like K(t) = U(n(t)). As mentioned above, capabilities of control by sensitivity function W(t) are completely determined by linear approximation of a function u(x) and independent of higher order terms. It allows to restrict our consideration without loss of generality by more simple regulators (9) in the following form: u ¼ KðtðxÞÞDðxÞ.
ð10Þ
where t(x) = n1(c(x)). Thus, the feedback matrix K(t) completely determines capabilities of the regulator (10) to synthesize SSF W(t).
6. Control of SSF for 2D-cycles Consider in detail the case n = 2. Sensitivity function l for u 2 U is a solution of the boundary-value problem l_ ¼ aðtÞl þ bðtÞ;
lð0Þ ¼ lðT Þ.
ð11Þ
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Here aðtÞ ¼ pT ðtÞðF T ðt; uÞ þ F ðt; uÞÞpðtÞ; bðtÞ ¼ pT ðtÞSðtÞpðtÞ.
ð12Þ
p(t) is normalized vector orthogonal to cycle c at a point n(t). Connect controlled coefficient a(t) in Eq. (11) with feedback matrix K(t) directly. Really, it follows from (12) that aðtÞ ¼ a0 ðtÞ þ a1 ðtÞ; a0 ðtÞ ¼ 2qT ðtÞpðtÞ;
ð13Þ
T
a1 ðtÞ ¼ 2b ðtÞkðtÞ; where qðtÞ ¼ AðtÞpðtÞ; bðtÞ ¼ BðtÞpðtÞ; T of ðnðtÞ; 0Þ ; AðtÞ ¼ ox T of ðnðtÞ; 0Þ ; BðtÞ ¼ ou ou kðtÞ ¼ ðnðtÞÞpðtÞ ¼ KðtÞpðtÞ. ox
ð14Þ
Note that the vector k is a derivative of a control function u from (10) in the direction of normal vector p. From (10) and relations D = PD, P = ppT, k = Kp it follows that u ¼ kðtðxÞÞdðxÞ;
ð15Þ
T
where d(x) = p (t(x))D(x) is a scalar function characterizing a distance of point x from cycle c. Thus, the control properties of the regulator (15) is defined by choice of vector function k(t) only. Varying a feedback gain in (15), one can design various SSF for the guided limit cycle. 6.1. Control goal and choice of regulator parameters ðtÞ 2 M be some asThe aim of control is the synthesis of desired SSF for cycle c of the stochastic system (7). Let l signed SSF. Here M ¼ fl 2 C 1½0;T jlðtÞ > 0; lð0Þ ¼ lðT Þg. Denote by lu a SSF of cycle c for stochastic system (7) with control u 2 U. . 2 M there exists l 2 U such that lu ¼ l Definition 2. A cycle c is called completely stochastic controllable if for all l One can prove the following criterion. Proposition. A cycle c is completely stochastic controllable if and only if bðtÞ 6¼ 0 8t 2 ½0; T .
ð16Þ
ðtÞ is connected with control parameter k(t) by the following equation Function l bT ðtÞkðtÞ ¼
a1 ðtÞ ; 2
ð17Þ
where _ ðtÞ a0 ðtÞ a1 ðtÞ ¼ ðl lðtÞ bðtÞÞ= lðtÞ. Eq. (17) under condition (16) has infinite set of the solutions (control is not unique). Consider additional optimal criterion kkðtÞk2 ! min . An optimal regulator problem (17) and (18) has the unique solution.
ð18Þ
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Here optimal feedback gain for the regulator (15) can be found as follows: _ ðtÞ a0 ðtÞ lðtÞ bðtÞÞbðtÞ kðtÞ ¼ a1 ðtÞbðtÞ ¼ ðl . T 2b ðtÞbðtÞ 2 lðtÞbT ðtÞbðtÞ
ð19Þ
ðtÞ Note that Lyapunov exponent k for the deterministic system with regulator (15), (19) is connected with function l by an explicit formula Z T 1 bðtÞ dt. ð20Þ k¼ ðtÞ 2T 0 l ðtÞ > 0 we have k < 0. It means stability of cycle c and u 2 U. Formula (20) allows to control the Thus, for any l ðtÞ > 0 easily. Lyapunov exponent by choice of sensitivity function l 6.2. Control of stochastic and chaotic oscillation for Brusselator Consider forced Brusselator (6) with control x_ ¼ a ðb þ 1Þx1 þ x21 x2 þ u1 þ eh; x_ 2 ¼ bx1 x21 x2 þ u2 ;
ð21Þ
where e is an external force intensity, u1 and u2 are control functions. For suppressing force-induced stochastic and chaotic oscillations of Brusselator, we shall use a sensitivity control theory presented above.
Fig. 8. Ideal cycle (dotted) and attractors of the controlled periodic forced Brusselator (solid) with a = 0.4, b = 1.2, e = 0.05, x = 0.81: ¼ 1, (b) l ¼ 0.1 and (c) l ¼ 0.01. (a) l
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For (21) a matrix B from (14) has the following elements b11 = 1, b12 = 0, b21 = 0, b22 = 1 and b(t) p(t). Hence, feedback gain in (15) is kðtÞ ¼ a1 ðtÞpðtÞ 2
ð22Þ
and regulator (15) can be written in a simple form a1 ðtðx1 ; x2 ÞÞ ðx1 n1 ðtðx1 ; x2 ÞÞÞ; 2 a1 ðtðx1 ; x2 ÞÞ ðx2 n2 ðtðx1 ; x2 ÞÞÞ. u2 ¼ 2
u1 ¼
ð23Þ
ðtÞ by an explicit formula Here scalar feedback coefficient a1 ðtÞ is connected with desired sensitivity function l a1 ðtÞ ¼
_ ðtÞ a0 ðtÞ ðl lðtÞ bðtÞÞ ; ðtÞ l
a0 ðtÞ ¼
2f 11 f22 ðf12 þ f21 Þf1 f2 þ 2f 22 f12 ; f12 þ f22
where
f 11 ¼ 2n1 n2 ðb þ 1Þ;
f 12 ¼ n21 ;
bðtÞ ¼
f22 ; f12 þ f22
f 21 ¼ b 2n1 n2 ;
f 1 ¼ a ðb þ 1Þn1 þ n21 n2 ;
f 2 ¼ bn1 n21 n2 ;
f 22 ¼ n21 ;
¼ 1, (b) l ¼ 0.1 and Fig. 9. Random trajectories of the controlled stochastic forced Brusselator with a = 0.4, b=1.2, e = 0.01: (a) l ¼ 0.01. (c) l
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Fig. 10. Ideal cycle (dotted) and attractors of the controlled periodic forced Brusselator (solid) with a = 0.2, b* = 1.064082, x = 0.5, l ¼ 1: (a) e = 0.001; (b) e = 0.01.
Fig. 11. Random trajectories of the controlled stochastic forced Brusselator with a = 0.2, b* = 1.064082, l ¼ 1: (a) e = 0.001; (b) e = 0.01.
n1(t), n2(t) are coordinates of a T-periodic vector function n(t) = (n1(t), n2(t))T defining a limit cycle c of unforced Brusselator. Let us consider two sets of parameters presented above for chaotic Brusselator without control. 6.2.1. Control of Brusselator with a = 0.4, b = 1.2 Potentialities of a regulator (23) depends on the choice of sensitivity function l(t). Here we consider three constant values l(t) = 1, l(t) = 0.1, l(t) = 0.01. Such choice of SSF is dictated by a desire to have oscillations with small and uniform sensitivity. In Figs. 8 and 9 results of control for periodic disturbances h(t) = cos xt with e = 0.05, x = 0.81 and stochastic dis_ turbances hðtÞ ¼ wðtÞ with e = 0.01 are shown. In Fig. 8, a dotted line represents unforced limit cycle (designed goal) and solid lines represent the control forced Brusselator attractors. As we can see, our regulators for any l stabilize the chaotic attractor (see Fig. 1(c)) to the periodic orbits. Moreover, these periodic orbits tend to an unforced limit cycle with a decreased l. In Fig. 9, the bundles of random trajectories of the control stochastic Brusselator found by a direct numerical simulation for the same three values of l are presented. As we can see, our regulators for any l stabilize stochastic attractor (see Fig. 2(b)) to the periodic bundles with small and uniform dispersion of the forced trajectories. Furthermore, the dispersions of these bundles vanish with a decreased l. Thus, constructed regulators suppress chaos for Brusselator forced by periodic and stochastic disturbances successfully.
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6.2.2. Control of supersensitive Brusselator with a = 0.2, b* = 1.064082 For this set of parameters, the forced Brusselator is a generator of chaos (see Section 4, Figs. 5(b) and 7(d)). An extraordinary sensitivity of Brusselator is connected with huge values of SSF m = max l(t) ’ 1010. ðtÞ 1, construct regulator (23) and consider Now we demonstrate the controlling chaos for this model. Let us take l dynamics of forced and controlled Brusselator. Because of the sensitivity drop produced by our regulator, we suppress chaos. Really, forced trajectories of the control system (23) with small disturbances (e = 105 for stochastic and e = 0.85 · 103 for periodic) considered in Section 4 coincide with the ideal cycle. Moreover, a controlling chaos may be observed in Figs. 10 and 11 for periodic _ (h(t) = cos xt, x = 0.5) and stochastic ðhðtÞ ¼ wðtÞÞ disturbances of larger intensity e = 103, e = 102. For e = 103 both periodically and stochastically forced trajectories of controlled Brusselator lie very close to unforced limit cycle (see Figs. 10(a) and 11(a)). A scatter of points of corresponding forced attractors monotonically increase with growth e (see Figs. 10(b) and 11(b)). But our technique allows to decrease these dispersions with the help of regulators providing smaller sensitivity. As we see, the sensitivity analysis worked out in this paper allows to construct a feedback regulator based on sensitivity function control and really provides the solution of a controlling chaos problem.
Acknowledgment This work was partially supported by a grant RFBR04-01-96098ural.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
Ott E, Grebodi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Chen G, Dong H. On feedback control of chaotic nonlinear dynamic system. Int J Bifurcat Chaos 1992;2(2):407–11. Pyragas K. Continuous control of chaos by self-controlling feedback. Phys Lett A 1992;170:421–8. Shinbrot T. Progress in the control of chaos. Adv Phys 1995;44(2):73–111. Hwang C, Chow H, Wang Y. A new feedback control of a modified ChuaÕs circuit system. Physica D 1996;92:95–100. Chen G. On some controllability conditions for chaotic dynamics control. Chaos, Solitons & Fractals 1997;8(9):1461– 70. Fradkov AL, Pogromsky AY. Introduction to control of oscillations and chaos. World Scientific Series of Nonlinear Science 1998. Boccaletti S, Grebodi C, Lay YC, Mancini H, Maza D. The control of chaos: theory and applications. Phys Rep 2000;329:103–97. Luchinsky DG, Beri S, Mannella R, McClintock PVE. Optimal fluctuations and the control of chaos. Int J Bifurcat Chaos 2002;12:583–604. Chen G, Yu X, editors. Chaos control: theory and applications. New York: Springer-Verlag; 2003. Luo XS, Chen G, Wang BH, Fang JQ. Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical system. Chaos, Solitons & Fractals 2003;18:775–83. Yassen MT. Chaos control of Chen chaotic dynamical system. Chaos, Solitons & Fractals 2003;15:271–83. Liu ST, Chen G. Nonlinear feedback-controlled generalized synchronization of spatial chaos. Chaos, Solitons & Fractals 2004;22:35–46. Deissler RJ, Farmer JD. Deterministic noise amplifiers. Physica D 1992;55:155–65. Nicolis JS, Meyer-Kress G, Haubs G. Non-uniform chaotic dynamics with implications to information processing. Z Naturforsch 1983;38(a):1157–69. Ali F, Menzinger M. On the local stability of limit cycles. Chaos 1999;9:348–56. Pontryagin LS, Andronov AA, Witt AA. On the statistical treatment of dynamical systems. Zh Eksp Teor Fiz 1933;3:165–80 [in Russian]. Stratonovich RL. Topics in the theory of random noise. New York: Gordon and Breach; 1963. Ibrahim RA. Parametric random vibration. New York: John Wiley and Sons; 1985. Soong TT, Grigoriu M. Random vibration of mechanical and structural systems. Englewood Cliffs, NJ: RTL Prentice-Hall; 1993. Moss F, McClintock PVE. Noise in nonlinear dynamical systems. Cambridge, MA: Cambridge University Press; 1989. Arnold L, Bleckert G, Schenk-Hoppe K. The stochastic Brusselator: parametric noise destroys Hopf bifurcation. In: Stochastic dynamics 1997, Bremen. NewYork: Springer; 1999. p. 71–92. Kurrer C, Schulten K. Effect of noise and perturbations on limit cycle systems. Physica D 1991;50:311–20. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of the stochastically and periodically forced Brusselator. Physica A 2000;278:126–39. Fedotov S, Bashkirtseva I, Ryashko L. Stochastic analysis of a non-normal dynamical system mimicking a laminar-to-turbulent subcritical transition. Phys Rev E 2002;66:066310.
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[26] Fedotov S, Bashkirtseva I, Ryashko L. Stochastic analysis of subcritical amplification of magnetic energy in a turbulent dynamo. Physica A 2004;342:491–506. [27] Freidlin MI, Wentzell AD. Random perturbations of dynamical systems. New York: Springer; 1984. [28] Dembo M, Zeitouni O. Large deviations techniques and applications. Boston: Jones and Bartlett Publishers; 1995. [29] Naeh T, Klosek MM, Matkowsky BJ, Schuss Z. A direct approach to the exit problem. SIAM J Appl Math 1990;50:595–627. [30] Roy RV. Asymptotic analysis of first passage problem. Int J Non-Linear Mech 1997;32:173–86. [31] Smelyanskiy VN, Dykman MI, Maier RS. Topological features of large fluctuations to the interior of a limit cycle. Phys Rev E 1997;55:2369–91. [32] Bashkirtseva IA, Ryashko LB. Sensitivity analysis of stochastically forced Lorenz model cycles under period-doubling bifurcations. Dyn Syst Appl 2002;11:293–309. [33] Bashkirtseva IA, Ryashko LB. Stochastic sensitivity of 3D-cycles. Math Comput Simul 2004;66:55–67. [34] Ryagin Yu, Ryashko L. The analysis of the stochastically forced periodic attractors for ChuaÕs circuit. Int J Bifurcat Chaos 2004;14(11):3981–7. [35] Tomita K, Kai T. Stroboscopic phase portrait and strange attractors. Phys Lett 1978;66(a):91–3. [36] Tomita K, Kai T. Chaotic response of a limit cycle. J Stat Phys 1979;21:65–86.
Sensitivity and chaos control for the forced nonlinear oscillations Irina Bashkirtseva, Lev Ryashko
*
Department of Mathematics, Ural State University, 620083 Ekaterinburg, Russian Federation Accepted 30 March 2005
Abstract This paper is devoted to study the problem of controlling chaos for forced nonlinear dynamic systems. We suggest a new control technique based on sensitivity analysis. With the help of approximation of nonequilibrium quasipotential, stochastic sensitivity function (SSF) is constructed. This function is used as basic tool of a quantitative description for a system response on the random external disturbances. The possibilities of SSF to predict chaotic dynamics for the periodic and stochastic forced Brusselator are shown. The problem of chaos control based on SSF is considered. A design of attractors with the desired features by feedback regulator is discussed. Analysis of controllability and effective technique for regulator synthesis is presented. An example of suppressing chaos for Brusselator is considered. 2005 Elsevier Ltd. All rights reserved.
1. Introduction Control of stochastic and chaotic oscillations is a challenging and a fundamental problem of nonlinear engineering. After the pioneering work of Ott et al. [1] controlling chaos attracts many researchers. Various methods are used to suppress chaos to equilibria or periodic orbits [1–13]. Analysis of the forced nonlinear oscillations play an important role for understanding the corresponding dynamical phenomena for electronic generators, lasers, mechanical, chemical and biological systems. Even small external disturbances may change the behavior of dynamic systems essentially. The various noise-induced transitions through periodic to more complicated chaotic regimes are a central problem in modern nonlinear dynamics theory. Underlying reason of unexpected chaotic-looking response of regular oscillations under small perturbations is high sensitivity of nonlinear dynamic systems. Usually, one can justify the presence of chaos on the basis of system instability to the initial condition disturbances. In recent investigations a research interest has been focused on phenomena with so-called ‘‘sensitive dependence to noise without sensitive dependence to initial conditions’’ [14] and local or nonuniform stability [15,16]. Classic characteristics of oscillation stability are multiplicators. Unfortunately, this standard stability measure does not reflect the important details of dynamics on the interval of periodicity. Local sensitivity analysis of the forced oscillations is a key for understanding of chaotic response.
*
Corresponding author. E-mail address: [email protected] (L. Ryashko).
0960-0779/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.029
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From this viewpoint, possibilities of controlling chaos are connected with controlling of sensitivity. The reason of chaotic response is high sensitivity of attractors. Decreasing sensitivity by control is a constructive approach for control of chaos. In this paper, we describe foundations of stochastic sensitivity analysis and propose a new feedback control method of designing a regulator that can modify the sensitivity of a given nonlinear system so as to suppress chaos. Analysis of nonlinear oscillations under the stochastic disturbances was started by Pontryagin et al. [17] and continued by many researchers [18–20]. The random trajectories of forced system leave the closed curve of deterministic limit cycle and due to cycle stability form some bundle around it. Stochastic cycles were considered both near and far from Hopf bifurcation point. A qualitative effect of external fluctuations on the Hopf scenario was found and investigated in [21,22]. A nonuniformity of stochastic bundles along cycles for nonlinear parameters far from bifurcation points attracts attention of researchers, see e.g., [23,16]. Small external noises acting on limit cycles may give rise to local phase-dependent response of the oscillations. Local instability of cycle is a reason of its significant sensitive dependence and can cause noise-induced transition to chaos [24]. A crucial role of sensitivity of non-normal dynamical systems for subcritical transitions from laminar to turbulent flow and generation of large-scale magnetic fields of galaxies is demonstrated in [25,26]. A variance of stochastic bundles perpendicular to the deterministic orbit is a natural measure of limit cycles sensitivity. Kolmogorov–Fokker–Planck (KFP) equation gives the most detailed probabilistic description. However, the direct use of this equation is very difficult even for the simplest situations. Under these circumstances asymptotics and approximations are used. Asymptotic analysis of distribution density for small noises based on quasipotential function is actively developed [27–31]. In Section 2, we give the first approximation of quasipotential in the vicinity of limit cycle. This approximation is an orbital quadratic form. Matrix of this quadratic form defines a covariance of the normal deviations of random trajectories for any point on a cycle. This matrix function plays a role of stochastic sensitivity function (SSF) of a cycle. SSF is a natural probabilistic measure [33] of stochastic 3D-cycles response to small random disturbances. The possibilities of SSF as an useful analytical tool for research of thin effects observed near chaos in a period-doubling bifurcations zone for stochastic Lorenz and Roessler models were demonstrated in [32,34]. In Section 3, for the case of cycle on a plane (2D-cycle) a scalar analytical representation of this matrix SSF is given. The possibilities of this scalar SSF to predict some peculiarities of 2D-cycles for the forced Brusselator are demonstrated in Section 4. The well-known phenomenon [35,36] of order to chaos transition for periodically forced Brusselator from a sensitivity point of view is discussed. On the basis of SSF analysis new critical (supersensitive) values of Brusselator parameters are shown [24]. For these values very small periodic and stochastic disturbances transfer Brusselator to chaotic regime. In Section 5, we consider the problem of control for stochastic and periodic forced cycles based on SSF. The possibilities for designing a limit cycle with wishful features by feedback regulator are discussed. Controllability analysis and effective algorithms for regulator synthesis for 2D-cycles are presented in Section 6. The examples of suppressing chaos for Brusselator are considered. A stabilization of the chaotic attractor to the regular orbit closed to the unforced limit cycle is demonstrated.
2. Stochastic sensitivity of limit cycles For many dynamical processes with regular oscillations, the basic mathematical model is the nonlinear deterministic system x_ ¼ f ðxÞ;
ð1Þ
with T-periodic solution x = n(t). Here x is a n-vector and f(x) is a n-vector function. Let c be a phase curve (limit cycle) of solution n(t) satisfying the following stability property. Definition 1. The cycle c is called exponentially stable if for small neighbourhood C of cycle c there exist constants K > 0, l > 0 such that for any solution x(t) of system (1) with x(0) = x0 2 C the following inequality holds kDðxðtÞÞk 6 Kelt kDðx0 Þk. Here D(x) = x c(x) is a deviation of a point x from a cycle c, c(x) is the point on cycle c that is nearest to x. A system of stochastic differential equations (in ItoÕs or StratonovichÕs sense) _ x_ ¼ f ðxÞ þ erðxÞw;
ð2Þ
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is a traditional mathematical model allowing to study quantitative description of results of external disturbances. Here x(t) is a n-dimensional Wiener process and r(x) is a n · n-matrix function of disturbances with intensity e. The random trajectories of the forced system (2) leave the closed curve of deterministic cycle c and due to cycle stability form some bundle around it. The detailed description of random distribution dynamics of this bundle is given by Kolmogorov–Fokker–Planck (KFP) equation. If the character of transient is inessential and the main interest is connected with the regime of steady-state stochastic auto-oscillations then it is possible to restrict an investigation by analysis of a stationary density function q(x, e). Analytical research of the stationary KFP equation for stochastic limit cycles considered here is a very difficult problem. Under these circumstances, asymptotics based on quasipotential v(x) = lime!0e2 log q(x, e) are actively used. After quadratic approximation of quasipotential v(x), the probabilistic distribution for the bundle of random trajectories localized near cycle has the Gaussian representation vðxÞ ðDðxÞ; Uþ ðcðxÞÞDðxÞÞ q Ke e2 K exp ; 2e2 with covariance matrix e2U(c). This covariance matrix characterizes a dispersion of the points of intersection of random trajectories with the hyperplane orthogonal to the cycle at the point c. A function U(c) is a stochastic sensitivity function (SSF) of limit cycle. This function allows to describe nonuniformity of a bundle width along cycle for all directions. It gives a simple way to indicate the most and the least sensitive parts of the cycle to external noises. It is convenient to search for a function U(c) in parametric form. The solution n(t) connecting the points of cycle c with points of an interval [0, T) gives the natural parametrization U(n(t)) = W(t). Matrix function W(t) is a solution of Lyapunov equation [33] W_ ¼ F ðtÞW þ WF T ðtÞ þ P ðtÞSðtÞP ðtÞ; ð3Þ with conditions W ð0Þ ¼ W ðT Þ;
ð4Þ
W ðtÞrðtÞ 0.
ð5Þ
Here F ðtÞ ¼
of ðnðtÞÞ; ox
SðtÞ ¼ rðnðtÞÞrT ðnðtÞÞ;
rðtÞ ¼ f ðnðtÞÞ;
P ðtÞ ¼ P rðtÞ ;
P r ¼ I rrT =rT r;
where Pr is a projection matrix onto the subspace orthogonal to the vector r 5 0.
3. Sensitivity analysis of 2D-cycles For the case n = 2 the projection matrix is given by P(t) = p(t)pT(t), where p(t) is a normalized vector orthogonal to f(n(t)). As a result, the matrix W(t) can be written as W(t) = l(t)P(t). Here l(t) > 0 is T-periodic scalar stochastic sensitivity function (SSF). Multiplying the matrix differential equation (3) by pT from the left and by p from the right and using the projective matrices properties we get equation l_ ¼ aðtÞl þ bðtÞ; with T-periodic coefficients aðtÞ ¼ pT ðtÞðF T ðtÞ þ F ðtÞÞpðtÞ;
bðtÞ ¼ pT ðtÞSðtÞpðtÞ.
The explicit formula for solution l(t) is given by the following: lðtÞ ¼ gðtÞðc þ hðtÞÞ; where gðtÞ ¼ exp
Z 0
t
aðsÞ ds ;
hðtÞ ¼
Z 0
t
bðsÞ ds; gðsÞ
c¼
gðT ÞhðT Þ . 1 gðT Þ
Note that Lyapunov exponent k is connected with function a(t) by the following: Z T 1 k¼ aðsÞ ds. 2T 0
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The value M = max l(t), t 2 [0, T] plays an important role in the analysis of stochastic dynamics about a limit cycle. We shall consider M as a sensitivity factor of a cycle c response to random disturbances.
4. Forced Brusselator: nonuniform sensitivity and chaos Consider forced system x_ 1 ¼ a ðb þ 1Þx1 þ x21 x2 þ eh; x_ 2 ¼ bx1 x21 x2 ;
ð6Þ
received by the addition of small disturbances eh(t) to classical Brusselator. For b > b ¼ 1 þ a2 the unforced system (e = 0) has a stable limit cycle c ( b is bifurcation value). It is well-known that periodically forced Brusselator [35,36] with h(t) = cos xt demonstrates for a = 0.2, b = 1.2 a sequence of period doubling bifurcations with transition to chaos (see Fig. 1). _ Compare Brusselator response on periodic disturbances with stochastic ones. Let hðtÞ ¼ wðtÞ, where w(t) is an independent Wiener process. The random trajectories of stochastically forced Brusselator leave the closed curve of deterministic cycle and form some bundle around it. In Fig. 2, the random trajectories found by a direct numerical simulation for three values of noise intensity e = 104, e = 103, e = 102 are presented. The value of intensity e = 102 is close to the stochastic bifurcation point when disturbances destroy limit cycle [22].
Fig. 1. Periodic force Brusselator: bifurcations and chaos a = 0.4, b = 1.2, e = 0.05. (a) x = 0.8015, (b) x = 0.803 and (c) x = 0.81.
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Fig. 2. Stochastic forced Brusselator: destruction of cycle a = 0.4, b = 1.2. (a) e = 0.0001, (b) e = 0.001 and (c) e = 0.01.
In Fig. 3, the plot of stochastic sensitivity function l(t) along with values (asterisks) of empirical sensitivity function ~ðtÞ for the limit cycle are shown. Here l ~ðt; eÞ ¼ e12 Dðt; eÞ, where D(t, e) is an empirical dispersion of intersection points l ~ were calculated for 100 rotations around the cycle with for random trajectories with a line orthogonal to c. Values l
Fig. 3. Sensitivity function of Brusselator with a = 0.4, b = 1.2.
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~ of empirical sennoise e = 103. From Fig. 3, one can see that the theoretical curve l(t) is arranged near the values l sitivity function and describes the main features (sharp peaks, monotonicity intervals) of this function clearly. Essential overfall of l(t) values on the periodicity interval reflects a nonuniformity of random trajectories dispersion (See Fig. 2(b)). The widest part of a bundle is placed on the bottom part of a cycle. The other part of the bundle is much more close to ideal deterministic trajectory c. A large value of the sensitivity factor M = 568 means that Brusselator amplifies both periodic and stochastic input disturbances essentially and leads to chaotic response (see Figs. 1(c) and 2(c)). From an extended investigation of Brusselator sensitivity, we have found the following interesting domain of parameters. Consider the Brusselator behavior for a fixed a = 0.2 and various values of parameter b > b ¼ 1.04 from an interval [1.06, 1.07]. _ Let disturbances in (6) be stochastic: hðtÞ ¼ w. For the Lyapunov exponent k and sensitivity factor M, dependence on values b is shown in Fig. 4. As we can see in Fig. 4(a), a parameter k monotonically decreases with growth b. This means increase of a stability degree of a cycle to disturbances of initial data. One should think it should be accompanied by the appropriate decrease in the sensitivity of a cycle to random disturbances. However, here the converse is observed. The value M behaves absolutely otherwise (see Fig. 4(b)). Here we have a typical example of ‘‘sensitive dependence to noise without sensitive dependence to initial conditions’’ [14]. On the considered interval the function M(b) is not monotonic. Its graph has a sharp high peak. As a result, the function M(b) has an essential overfall of values. By critical value of parameter b, one has b* = arg maxb M(b) = 1.064082, M(b*) = 4.4 · 1010. We compare the bundles of random trajectories and plots of sensitivity functions of stochastic Brusselator for values b close to b*. For three values of parameter, b = 1.06, b* = 1.064082, b = 1.065 in Figs. 5 and 6 our results are presented. In Fig. 5 the random trajectories found by direct numerical simulation for e = 105 are demonstrated. As we see the Brusselator with b = b* (Fig. 5(b)) is supersensitive. For small background stochastic disturbances the burst of response amplitude is shown. As is visible in Fig. 6, the sensitivity function reflects these noticed peculiarities clearly. Note that bottom part of the cycle is most sensitive (see Fig. 5(a)). Let disturbances in (6) be periodic: h(t) = cos (xt) with x = 0.5. For these periodic disturbances, value b* = 1.064082 is also critical. Really, increase of periodic force intensity e results in the period-doubling bifurcations of system (6) (see Fig. 7): 1-cycle (e = 0.0005) ! 2-cycle (e = 0.0007) ! 4-cycle (e = 0.000763) and so on. For e = 0.00085 the bundle of trajectories (see Fig. 7(d)) looks chaotic. For a critical parameter value b* = 1.064082 the forced Brusselator is a generator of chaos. It is significant to stress the following geometrical details. A top part of attractors both for period-doubling and chaotic zone is more stable and practically coincides with the ideal trajectory of the unforced cycle. Arising chaotic structure is connected with bottom part of the cycle. Note that huge peak of SSF (see Fig. 6(b)) corresponds to this bottom part of the cycle.
Fig. 4. Lyapunov exponent (a) and sensitivity factor (b) for Brusselator with a = 0.2.
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Fig. 5. Stochastic forced Brusselator with a = 0.2, e = 105. (a) b = 1.06, (b) b* = 1.064082 and (c) b=1.065.
Thus, the function of sensitivity is the useful analytical tool for the prediction of singular responses of a nonlinear system both to stochastic and periodic disturbances.
5. Sensitivity control Consider a stochastic control system _ x_ ¼ f ðx; uÞ þ erðx; uÞw;
ð7Þ
where x is a n-dimensional state variable, u is a r-dimensional vector of control functions, f(x, u), r(x, u) are vector functions, w(t) is a n-dimensional Wiener process and e is a scalar parameter of disturbances intensity. It is supposed that for e = 0 and u = 0 the system (7) has T-periodic solution x = n(t) with a phase trajectory c (cycle). A stability of c is not assumed. The stabilizing regulator will be selected from the class U of admissible feedbacks u = u(x) satisfying the conditions (a) u(x) is sufficiently smooth and ujc = 0; (b) for the deterministic system x_ ¼ f ðx; uÞ; the solution x = n(t) is exponentially stable in the neighbourhood C of cycle c.
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Fig. 6. Sensitivity functions for a = 0.2, (a) b = 1.06, (b) b* = 1.064082 and (c) b=1.065.
Our aim is to control the stochastic sensitivity function W. For control system (7) it follows from (3) that W_ ¼ F ðt; uÞW þ WF T ðt; uÞ þ P ðtÞSðtÞP ðtÞ;
ð8Þ
where of of ou ðnðtÞ; 0Þ þ ðnðtÞ; 0Þ ðnðtÞÞ; ox ou ox SðtÞ ¼ rðnðtÞ; 0ÞrT ðnðtÞ; 0Þ;
F ðt; uÞ ¼
As we see, a variation of control u allows to change in Eq. (8) the coefficient F(t, u) only. Note that the outcome of the control depends only on values of the derivative ou . It gives us the possibility to simplify the structure of the used ox regulator. 5.1. Choice of regulator structure Consider TaylorÕs expansion of control function u(x) at a point c uðxÞ ¼ uðcÞ þ
ou ðcÞðx cÞ þ Oðkx ck3 Þ. ox
For c = c(x) taking into account condition (a), we get uðxÞ ¼
ou ðcðxÞÞDðxÞ þ OðkDðxÞk3 Þ. ox
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Fig. 7. Periodic forced Brusselator: bifurcations and chaos a = 0.2, b* = 1.064082, x = 0.5 (a) e = 0.0005, (b) e = 0.0007, (c) e = 0.000763 and (d) e = 0.00085.
As we see, a first approximation u1(x) for an arbitrary smooth control function u 2 U for small deviations D(x) = x c(x) is the feedback u1 ðxÞ ¼ UðcðxÞÞDðxÞ.
ð9Þ
Here U(c(x)) is the feedback matrix coefficient. Appropriate t-parametric representation for this matrix looks like K(t) = U(n(t)). As mentioned above, capabilities of control by sensitivity function W(t) are completely determined by linear approximation of a function u(x) and independent of higher order terms. It allows to restrict our consideration without loss of generality by more simple regulators (9) in the following form: u ¼ KðtðxÞÞDðxÞ.
ð10Þ
where t(x) = n1(c(x)). Thus, the feedback matrix K(t) completely determines capabilities of the regulator (10) to synthesize SSF W(t).
6. Control of SSF for 2D-cycles Consider in detail the case n = 2. Sensitivity function l for u 2 U is a solution of the boundary-value problem l_ ¼ aðtÞl þ bðtÞ;
lð0Þ ¼ lðT Þ.
ð11Þ
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Here aðtÞ ¼ pT ðtÞðF T ðt; uÞ þ F ðt; uÞÞpðtÞ; bðtÞ ¼ pT ðtÞSðtÞpðtÞ.
ð12Þ
p(t) is normalized vector orthogonal to cycle c at a point n(t). Connect controlled coefficient a(t) in Eq. (11) with feedback matrix K(t) directly. Really, it follows from (12) that aðtÞ ¼ a0 ðtÞ þ a1 ðtÞ; a0 ðtÞ ¼ 2qT ðtÞpðtÞ;
ð13Þ
T
a1 ðtÞ ¼ 2b ðtÞkðtÞ; where qðtÞ ¼ AðtÞpðtÞ; bðtÞ ¼ BðtÞpðtÞ; T of ðnðtÞ; 0Þ ; AðtÞ ¼ ox T of ðnðtÞ; 0Þ ; BðtÞ ¼ ou ou kðtÞ ¼ ðnðtÞÞpðtÞ ¼ KðtÞpðtÞ. ox
ð14Þ
Note that the vector k is a derivative of a control function u from (10) in the direction of normal vector p. From (10) and relations D = PD, P = ppT, k = Kp it follows that u ¼ kðtðxÞÞdðxÞ;
ð15Þ
T
where d(x) = p (t(x))D(x) is a scalar function characterizing a distance of point x from cycle c. Thus, the control properties of the regulator (15) is defined by choice of vector function k(t) only. Varying a feedback gain in (15), one can design various SSF for the guided limit cycle. 6.1. Control goal and choice of regulator parameters ðtÞ 2 M be some asThe aim of control is the synthesis of desired SSF for cycle c of the stochastic system (7). Let l signed SSF. Here M ¼ fl 2 C 1½0;T jlðtÞ > 0; lð0Þ ¼ lðT Þg. Denote by lu a SSF of cycle c for stochastic system (7) with control u 2 U. . 2 M there exists l 2 U such that lu ¼ l Definition 2. A cycle c is called completely stochastic controllable if for all l One can prove the following criterion. Proposition. A cycle c is completely stochastic controllable if and only if bðtÞ 6¼ 0 8t 2 ½0; T .
ð16Þ
ðtÞ is connected with control parameter k(t) by the following equation Function l bT ðtÞkðtÞ ¼
a1 ðtÞ ; 2
ð17Þ
where _ ðtÞ a0 ðtÞ a1 ðtÞ ¼ ðl lðtÞ bðtÞÞ= lðtÞ. Eq. (17) under condition (16) has infinite set of the solutions (control is not unique). Consider additional optimal criterion kkðtÞk2 ! min . An optimal regulator problem (17) and (18) has the unique solution.
ð18Þ
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Here optimal feedback gain for the regulator (15) can be found as follows: _ ðtÞ a0 ðtÞ lðtÞ bðtÞÞbðtÞ kðtÞ ¼ a1 ðtÞbðtÞ ¼ ðl . T 2b ðtÞbðtÞ 2 lðtÞbT ðtÞbðtÞ
ð19Þ
ðtÞ Note that Lyapunov exponent k for the deterministic system with regulator (15), (19) is connected with function l by an explicit formula Z T 1 bðtÞ dt. ð20Þ k¼ ðtÞ 2T 0 l ðtÞ > 0 we have k < 0. It means stability of cycle c and u 2 U. Formula (20) allows to control the Thus, for any l ðtÞ > 0 easily. Lyapunov exponent by choice of sensitivity function l 6.2. Control of stochastic and chaotic oscillation for Brusselator Consider forced Brusselator (6) with control x_ ¼ a ðb þ 1Þx1 þ x21 x2 þ u1 þ eh; x_ 2 ¼ bx1 x21 x2 þ u2 ;
ð21Þ
where e is an external force intensity, u1 and u2 are control functions. For suppressing force-induced stochastic and chaotic oscillations of Brusselator, we shall use a sensitivity control theory presented above.
Fig. 8. Ideal cycle (dotted) and attractors of the controlled periodic forced Brusselator (solid) with a = 0.4, b = 1.2, e = 0.05, x = 0.81: ¼ 1, (b) l ¼ 0.1 and (c) l ¼ 0.01. (a) l
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For (21) a matrix B from (14) has the following elements b11 = 1, b12 = 0, b21 = 0, b22 = 1 and b(t) p(t). Hence, feedback gain in (15) is kðtÞ ¼ a1 ðtÞpðtÞ 2
ð22Þ
and regulator (15) can be written in a simple form a1 ðtðx1 ; x2 ÞÞ ðx1 n1 ðtðx1 ; x2 ÞÞÞ; 2 a1 ðtðx1 ; x2 ÞÞ ðx2 n2 ðtðx1 ; x2 ÞÞÞ. u2 ¼ 2
u1 ¼
ð23Þ
ðtÞ by an explicit formula Here scalar feedback coefficient a1 ðtÞ is connected with desired sensitivity function l a1 ðtÞ ¼
_ ðtÞ a0 ðtÞ ðl lðtÞ bðtÞÞ ; ðtÞ l
a0 ðtÞ ¼
2f 11 f22 ðf12 þ f21 Þf1 f2 þ 2f 22 f12 ; f12 þ f22
where
f 11 ¼ 2n1 n2 ðb þ 1Þ;
f 12 ¼ n21 ;
bðtÞ ¼
f22 ; f12 þ f22
f 21 ¼ b 2n1 n2 ;
f 1 ¼ a ðb þ 1Þn1 þ n21 n2 ;
f 2 ¼ bn1 n21 n2 ;
f 22 ¼ n21 ;
¼ 1, (b) l ¼ 0.1 and Fig. 9. Random trajectories of the controlled stochastic forced Brusselator with a = 0.4, b=1.2, e = 0.01: (a) l ¼ 0.01. (c) l
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Fig. 10. Ideal cycle (dotted) and attractors of the controlled periodic forced Brusselator (solid) with a = 0.2, b* = 1.064082, x = 0.5, l ¼ 1: (a) e = 0.001; (b) e = 0.01.
Fig. 11. Random trajectories of the controlled stochastic forced Brusselator with a = 0.2, b* = 1.064082, l ¼ 1: (a) e = 0.001; (b) e = 0.01.
n1(t), n2(t) are coordinates of a T-periodic vector function n(t) = (n1(t), n2(t))T defining a limit cycle c of unforced Brusselator. Let us consider two sets of parameters presented above for chaotic Brusselator without control. 6.2.1. Control of Brusselator with a = 0.4, b = 1.2 Potentialities of a regulator (23) depends on the choice of sensitivity function l(t). Here we consider three constant values l(t) = 1, l(t) = 0.1, l(t) = 0.01. Such choice of SSF is dictated by a desire to have oscillations with small and uniform sensitivity. In Figs. 8 and 9 results of control for periodic disturbances h(t) = cos xt with e = 0.05, x = 0.81 and stochastic dis_ turbances hðtÞ ¼ wðtÞ with e = 0.01 are shown. In Fig. 8, a dotted line represents unforced limit cycle (designed goal) and solid lines represent the control forced Brusselator attractors. As we can see, our regulators for any l stabilize the chaotic attractor (see Fig. 1(c)) to the periodic orbits. Moreover, these periodic orbits tend to an unforced limit cycle with a decreased l. In Fig. 9, the bundles of random trajectories of the control stochastic Brusselator found by a direct numerical simulation for the same three values of l are presented. As we can see, our regulators for any l stabilize stochastic attractor (see Fig. 2(b)) to the periodic bundles with small and uniform dispersion of the forced trajectories. Furthermore, the dispersions of these bundles vanish with a decreased l. Thus, constructed regulators suppress chaos for Brusselator forced by periodic and stochastic disturbances successfully.
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6.2.2. Control of supersensitive Brusselator with a = 0.2, b* = 1.064082 For this set of parameters, the forced Brusselator is a generator of chaos (see Section 4, Figs. 5(b) and 7(d)). An extraordinary sensitivity of Brusselator is connected with huge values of SSF m = max l(t) ’ 1010. ðtÞ 1, construct regulator (23) and consider Now we demonstrate the controlling chaos for this model. Let us take l dynamics of forced and controlled Brusselator. Because of the sensitivity drop produced by our regulator, we suppress chaos. Really, forced trajectories of the control system (23) with small disturbances (e = 105 for stochastic and e = 0.85 · 103 for periodic) considered in Section 4 coincide with the ideal cycle. Moreover, a controlling chaos may be observed in Figs. 10 and 11 for periodic _ (h(t) = cos xt, x = 0.5) and stochastic ðhðtÞ ¼ wðtÞÞ disturbances of larger intensity e = 103, e = 102. For e = 103 both periodically and stochastically forced trajectories of controlled Brusselator lie very close to unforced limit cycle (see Figs. 10(a) and 11(a)). A scatter of points of corresponding forced attractors monotonically increase with growth e (see Figs. 10(b) and 11(b)). But our technique allows to decrease these dispersions with the help of regulators providing smaller sensitivity. As we see, the sensitivity analysis worked out in this paper allows to construct a feedback regulator based on sensitivity function control and really provides the solution of a controlling chaos problem.
Acknowledgment This work was partially supported by a grant RFBR04-01-96098ural.
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