Complicated behavior of dynamical systems. Mathematical methods and computer experiments. Kuznetsov N.V. 1, Leonov G.A. 1, and Seledzhi S.M. 1 St.Petersburg State University Universitetsky pr. 28 198504 St.Petersburg, Russia (e-mail: leonov@math.spbu.ru) Abstract. It is shown that the most suitable definition of chaos in dynamical systems is instability by Zhukovsky. There are considered computer experiments and analytical proofs are corresponding to this definition. Also it is shown that the Filter Hypothesis is untrue. Keywords: Zhukovsky stability, Harmonic linearization. 1 Introduction Complicated behavior in dynamical systems are attracting attentions of various researchers (Moon[1], Shuster[2], Kolesnikov[3], Magnitskii and Sidorov[4], Leonov[5]). In this work we consider some approaches for investigation such complicated behaviors of trajectories as strange attractors, complicated periodic solutions, flattering of trajectories. Here it is shown that from among the classical notions of instability for studying strange attractors in the continuous case, the most adequate ones are Zhukovsky instability. In order to investigate Zhukovsky stability, a new research tool a moving Poincaré section is introduced (Leonov[5]). The complicated behavior of periodic solutions and filter hypothesis are considered. Difficulties of numerical investigations of limit cycles of two dimensional autonomous systems are considered. 2 Zhukovsky stability It is well known (Leonov[5]) that the defining property of strange attractors is the sensitivity of their trajectories with respect to the initial data. The notion of Zhukovsky instability is adequate to the sensitivity of trajectories with respect to the initial data for continuous dynamical systems. In order to investigate Zhukovsky stability, a new research tool a moving Poincaré section is introduced. With the help of this tool, extensions of the widelyknown theorems of Andronov Witt, Demidovich, Borg, and Poincaré are carried out.
2 Kuznetsov, Leonov and Seledzhi Consider the dynamical systems, generated by the differential equations dx dt = f(x), t R1, x R n. (1) We introduce now the definition of Zhukovsky stability. For this purpose we need to consider the following set of homeomorphisms Hom = {τ( ) τ : [0, + ) [0, + ), τ(0) = 0}. The functions τ(t) from the set Hom play the role of the reparametrization of time for the trajectories of system (1). Definition. The trajectory x(t, x 0 ) of system (1) is said to be Zhukovsky stable if for any number ε > 0 there exists the number δ(ε) > 0 such that for any vector y 0, satisfying the inequality x 0 y 0 δ(ε), the function τ( ) Hom can be found such that the following inequality x(t, x 0 ) x(τ(t), y 0 ) ε, t 0 is valid. If, in addition, for the certain number δ 0 > 0 and any y 0 from the ball {y x 0 y δ 0 } the function τ( ) Hom can be found such that the relation holds lim t + x(t, x 0) x(τ(t), y 0 ) = 0, then we shall say that the trajectory x(t, x 0 ) is asymptotically stable by Zhukovsky. This means that the stability by Zhukovsky is a stability by Lyapunov for the suitable reparametrization of each of perturbed trajectories. Consider here relationships between the basic stability notions. There are well known examples of periodic trajectories of continuous systems that happen to be Lyapunov unstable but Poincaré stable. Example. Consider the Duffing equation ẍ + x + x 3 = 0 It is easily seen (Leonov[10]) that all solutions of are periodic and the period of the solutions is depend on initial date x 0. Here the trajectories of the equation remain wholly within small neighborhoods of each other for small changes in initial data; hence they are Poincaré stable. Since all trajectories considered are periodic, they are also Zhukovsky stable. However, in this case Lyapunov stability is lacking. Example. Consider the linearized equations of two decoupled pendula: ẋ 1 = y 1, ẏ 1 = ω1x 2 1, ẋ 2 = y 2, ẏ 2 = ω2 2x 2. (2) with ω 1 /ω 2 irrational. Change the flow of trajectories on the tori as follows. Cut the toroidal surface along a certain segment of the fixed trajectory from
Complicated behavior of dynamical systems 3 Fig.1. Heteroclinic trajectories with hole H. the point z 1 to the point z 2 (Fig. 1). Then the surface is stretched diffeomorphically along the torus so that a cut is mapped into the circle with the fixed points z 1 and z 2. Denote by H the interior of the circle. Outside the hole H, after the diffeomorphic stretching, the disposition of trajectories on the torus is the same. Outside the hole H, the trajectories are everywhere dense on torus. They are therefore, as before, asymptotically Poincaré stable. But it is possible here to choose two trajectories in a small neighborhood of z 1 that these trajectories are repelled from each other; hence, the trajectory leaving z 0 is Zhukovsky unstable. Thus, a trajectory can be asymptotically Poincaré stable and Zhukovsky unstable. Hence, from among the classical notions of instability for studying strange attractors in the continuous case, the most adequate ones are Zhukovsky instability. The reparametrization of trajectories permits us to introduce interesting and important tool of investigations, namely the moving Poincare section. The classical Poincare section is the transversal (n 1)-dimensional surface S in the phase space R n, which possesses a recurring property. We force now the Poincare section to move along the trajectory x(t, x 0 ) Fig. 2.
4 Kuznetsov, Leonov and Seledzhi We assume further that the vector-function f(x) is twice continuously differentiable and the trajectory x(t, x 0 ), whose the Zhukovsky stability (or instability) will be considered, is wholly situated in the certain bounded domain Ω R n for t 0. Suppose also that f(x) 0, x Ω. Here Ω is a closure of the domain Ω. Under these assumptions there exist the positive numbers δ and ε such that the following relation f(y) f(x) δ, y S(x, ε), x Ω is satisfied. Here S(x, ε) = {y (y x) f(x) = 0, x y < ε}. Definition. The set S(x(t, x 0 ), ε) is called a moving Poincare section. The classical Poincare section allow us to clear up the behavior of trajectories using the information at discrete time of their crossings with the Poincare section. The reparametrization of trajectories makes it possible to organize the motion of trajectories so that at time t all trajectories are situated on the same moving Poincare section S(x(t, x 0 ), ε): x(ϕ(t), y 0 ) S(x(t, x 0 ), ε). (3) Here ϕ(t) is a reparametrization of the trajectory x(t, y 0 ), y 0 S(x 0, ε). This consideration has, of course, a local property and it is only possible for the values t such that x(ϕ(t), y 0 ) x(t, x 0 ) < ε. Conside system (1) in the form where dz dt = A(x(t, x 0))z + g(t, z), z f(x(t, x 0 )) = 0, (4) A(x) = f [ ( ) ] f(x)f(x) f f (x) x f(x) 2 x (x) + x (x). Here the following relations g(t, z) f(x(t, x 0 )) = 0, g(t, z) = O( z 2 ) are valid. Thus, for system (4) we have the system of the first approximation dv dt = A(x(t, x 0))v, f(x(t, x 0 )) v = 0. (5) It differs from the usual system of the first approximation dw dt = f x (x(t, x 0))w (6) in that we introduce here the projector v = (I f(x(t, x 0)f(x(t, x 0 )) ) f(x(t, x 0 ) 2 w. (7) Theorem. Let Λ is the higher characteristic exponent of system (5), Γ is a coefficient of irregularity. Then if for system (5) the inequality Λ + Γ < 0,
Complicated behavior of dynamical systems 5 is satisfied, then the trajectory x(t, x 0 ) is asymptotically Zhukovsky stable. It possible to show that system (6) has the one null characteristic exponent λ 1. Then denote by λ 2 λ n the rest of characteristic exponents of system (6). From relations (7) it follows that the characteristic exponents of system (5) are not greater than the corresponding characteristic exponents of system (6). From this we have the inequality γ Γ. (8) Here γ is the coefficient of irregularity of system (6). Besides we have λ 2 Λ. (9) Theorem 4 and inequalities (8) and (9) give the following Theorem 5. If for system (6) the following inequality λ 2 + γ < 0 (10) is satisfied, then the trajectory x(t, x 0 ) is asymptotically Zhukovsky stable. This result is the generalization of the well-known Andronov Witt theorem. Theorem (Andronov and Witt[6]) 6. If the trajectory x(t, x 0 ) is periodic, differs from equilibria and for system (6) the inequality λ 2 < 0 is satisfied, then the trajectory x(t, x 0 ) is asymptotically orbitally stable (asymptotically Poincare stable). Theorem 6 is a corollary of Theorem 5 since system (6) with the periodic matrix f x (x(t, x 0)) is regular (Demidovich[7]). Recall that for the periodic trajectories the asymptotic stability by Zhukovsky and by Poincare are equivalent. The theorem of Demidovich is also a corollary of Theorem 5. Theorem 7 (Demidovich[8]). If system (6) is regular (i.e. γ = 0) and the inequality λ 2 < 0 is satisfied, then the trajectory x(t, x 0 ) is asymptotically orbitally stable. 3 The harmonic linearization method describing function method Consider a system dx dt = Px + qϕ(r x), (11) where P is a constant n n-matrix, q and r are constant n-vectors, ϕ(σ) is a piecewise-continuous function, and is the operation of transposition. When applied the harmonic linearization method (Khalil[9]) to this system, standard assumptions are the existence of a pair of purely imaginary eigen
6 Kuznetsov, Leonov and Seledzhi values ±iω 0 (ω 0 > 0) of the matrix P and a negativeness of the rest of eigen values. By nonsingular linear transformation, under the above assumptions system (1) can be reduced to the form ẋ 1 = ω 0 x 2 + b 1 ϕ(x 1 + c x 3 ) ẋ 2 = ω 0 x 1 + b 2 ϕ(x 1 + c x 3 ) ẋ 3 = Ax 3 + bϕ(x 1 + c x 3 ). (12) Here A is a constant (n 2) (n 2)-matrix, all eigen values of which have negative real parts, b and c are (n 2)-dimensional vectors, b 1 and b 2 are certain numbers. Combined application of the harmonic linearization method, the classical method of small parameter, and numerical methods permit us to compute periodic oscillations of certain multistage procedure, where at the first step the harmonic linearization method is applied. In the basic, noncritical, case we suppose that the relation ϕ(σ) = εψ(σ), where ε is a small parameter, is satisfied. In the sequel, without loss of generality, we can assume that for A there exists the number α > 0 such that x 3 (A + A )x 3 2α x 3 2, x 3 R n 2. We introduce the function K(a) = 2π/ω 0 0 ψ(cos(ω 0 t)a)cos(ω 0 t)dt. Theorem. If the conditions K(a) = 0, da < 0, are satisfied, then for sufficiently small ε > 0 system (11) has T-periodic solution such that b 1 dk(a) r x(t) = a cos(ω 0 t) + O(ε), T = 2π ω 0 + O(ε). This periodic solution is stable in the sense that there exists its certain ε- neighbourhood such that all solutions with the initial data from this ε-neighbourhood remain in it in increasing time t. The described here standard, basic, method of harmonic linearization turns out too rough to locate periodic oscillations in nonlinear systems, satisfying the generalized Routh-Hurwitz conditions. The extension of this Theorem in the spirit of classical research of critical cases in the theory of motion stability makes it possible to obtain effective estimates for periodic oscillations in systems, satisfying the generalized Routh-Hurwitz conditions. The described periodic solution can be considered as certain support (basic) periodic oscillations and system (11) with the considered above nonlinearities as generating start system in the algorithms of seeking the periodic solutions of another system, namely dx dt = P 0x + qf(r x), (13)
Complicated behavior of dynamical systems 7 In this case, we can organize a finite sequence of the functions ϕ j (σ) j = 1,...,m, such that the graphs of each pair ϕ j and ϕ j+1 are close to each other and ϕ 1 (σ) = ϕ(σ), ϕ m (σ) = f(σ). Then, for the system dx dt = Px + qϕ j(r x) (14) with ϕ 1 (σ) = ϕ(σ) and small ε we take the periodic solution g 1 (t), described in either Theorem 1 or Theorem 2. Two cases occur: either all points of this periodic solution are situated in a domain of attraction of the stable periodic solution g 2 (t) of system (14) with j = 2, or in passing from system (14) with j = 1 to system (14) with j = 2 we have a bifurcation of stability loss and a vanishing of periodic solution. In the first case, we can numerically find g 2 (t) when the trajectory of system (14) with j = 2 begins at the initial point x(0) = g 1 (0). Starting from the point g 1 (0), after transient process the computational procedure outputs into the periodic solution g 2 (t) and calculates it. For this purpose the interval [0, τ], on which the computation occurs, must be sufficiently large. After computation of g 2 (t) it is possible to go to the following system (14) with j = 3 and to organize a similar procedure for computation of the periodic solution g 3 (t) when a trajectory, which in increasing t tends to the periodic trajectory g 3 (t), starts from the initial point x(0) = g 2 (τ). Proceeding then this procedure for sequential computing g j (t) and making use of trajectories of system (14) with the initial data x(0) = g j 1 (τ), we arrive by numerical computation of periodic solution of system (13) or observe, at a certain step, a bifurcation of stability loss and a vanishing of periodic solution. We give two examples of applying this procedure. Fig. 3. Periodic solution of system for ε = 1 Example (Leonov[10], Leonov[11]). Consider system (12) with the function ϕ(σ) = εψ(σ), ψ(σ) = k 1 σ + k 3 σ 3. Then we have K(a) = (k 1 a + 3 4 k 3a 3 ) π ω 0. It follows that a can be determined from the equation K(a) = 0
8 Kuznetsov, Leonov and Seledzhi in such way a = a 1 = 4k1 3k 3, and a stability condition takes the form b 1 k 1 < 0. Let be k 1 = 3, k 3 = 4, ω 0 = 1, b 1 = 1, b 2 = 1, A = 1, c = 1, b = 1. Then a 1 = 1. Using the classical harmonic linearization method (Khalil[9]), we obtain that for any ε > 0, system (12) has a periodic solution and in this case σ(t) = r x(t) cost. By Theorem 1 for small ε > 0, system (12) has a periodic solution of the form x 1 (t) = cost + O(ε), x 2 (t) = sint + O(ε), x 3 (t) = O(ε). Further, using the above computational procedure, we obtain a periodic solution of system (12) for ϕ j (σ) = ε j ϕ(σ), ε 1 = 0, 1, ε 2 = 0, 3, ε 3 = 0, 6, ε 4 = 0, 7, ε 5 = 0, 9, ε 6 = 1. In Fig. 3 it is shown a projection of periodic solution thus computed on the plane {x 1, x 2 } when ε = 1. For these periodic solution the graph σ(t) = x 1 (t) + x 3 (t) is represented. In this case for ε = 1, the output σ(t) is substantially not harmonic and the filter hypothesis is untrue. Therefore, here it is impossible, in principle, to justify a standard harmonic linearization method, founding on the filter hypothesis. References 1.F. Moon. Chaotic Vibrations. John Wiley. New York, 1987. 2.H. G. Schuster. Deterministic Chaos. Physik Verlag, Weinheim, 1984. 3.A. Kolesnikov. Synergetics Methods of Complex Systems Control: System Synthesis Theory. Moscow: URSS, 2006. 4.N. A. Magnitskii, S. V. Sidorov. New Methods for Chaotic Dynamics, World Scientific, 2006 5.G. A. Leonov. Strange attractors and classical stability theory. Saint-Petersburg State University Press, 2008 6.A. A. Andronov, A. A. Witt. On Lyapunov Stability. Journal of Experimental and Theoretical Physics. Vol. 3, N 3, 1933. 7.B.P. Demidovich. Lectures on Mathematical Theory of Stability. Nauka, Moscow, 1967. 8.B. P. Demidovich. Orbital Stability of the Solutions Autonomous Systems. I, II// Differential Equations, 1968, vol. 4, p. 575 588, N 8, p. 1359 1373 (Translated from Differenzialnya Uravneniya). 9.H. K. Khalil. Nonlinear Systems. Prentice Hall, 2002. 10.G. A. Leonov. Method of Harmonic Linearization. Automation and Remote Control, 2008. 11.G. A. Leonov. Aizerman Problem. Automation and Remote Control, 2008. [in print]