Hidden Oscillations in Dynamical Systems

Recent Researches in System Science Hidden oscillations in dynamical systems G.A. LEONOV a, N.V. KUZNETSOV b,a, S.M. SELEDZHI a aSt.Petersburg State University, Universitetsky pr. 28, St.Petersburg, 198504, RUSSIA bUniversity of Jyväskylä, P.O. Box 35 (Agora), FIN-40014, FINLAND [email protected], [email protected], [email protected] Abstract :- The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. Study of hidden oscillations and attractors requires the development of new analytical and numerical methods which will be considered in this invited lecture. Key-Words:- Hidden oscillation, attractor localization, hidden attractor, harmonic balance, describing function method, Aizerman conjecture, Kalaman conjecture, Hilbert 16th problem 1 Introduction 5 In establishing and developing the theory of non- 4 linear oscillations in the first half of the twentieth century [1, 2, 3, 4] most attention was given to analyzing 3 and synthesizing oscillatory systems for which solving 2 the problem of the existence of the oscillation modes did not present any great difficulties. The structure 1 of many mechanical, electromechanical and electronic y 0 systems was such that there were oscillation modes in them, the existence of which was almost obvious −1 — oscillations are excited from unstable equilibria. From computational point of view this allows one to −2 use numerical method, in which after transient pro- −3 cess a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches −4 an attractor and identifies it. −5 −3 −2 −1 0 1 2 3 Consider corresponding classical examples. x Example 1 Van der Pol oscillator Figure 1: Numerical localization of limit cycle in Van Consider an oscillations arising in the electrical cir- der Pol oscillator cuit — the van der Pol oscillator [5] 2 Example 2 x¨ + µ(x − 1)˙x + x = 0 (1) Belousov-Zhabotinsky (BZ) reaction and carry out its simulation for the parameter µ = 2. In 1951 B.P. Belousov first discovered oscillations in the chemical reactions in liquid phase [6]. Consider ISBN: 978-1-61804-023-7 292 Recent Researches in System Science one of the Belousov-Zhabotinsky dynamic model ( ) ˙ = (1 ) + f q − x εx x − x + z q x (2) 50 z˙ = x − z 45 and carry out its simulation with standard parame- 40 35 ters f = 2/3, q = 8 × 10−4,4ε = × 10−2. 30 z 25 0.4 20 15 0.35 10 0.3 5 30 20 0 0.25 −20 10 −15 −10 0 −5 0 −10 y 0.2 5 x 10 −20 15 y 20 −30 0.15 Figure 3: Numerical localization of chaotic attractor 0.1 in Lorenz system 0.05 0 Here the function −0.05 f(x) = m1x( + m0 − m1)sat(= x) 1 (5) −0.1 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 = + ( )(1 + 1 ) x m1x 2 m0 − m1 |x | − |x − | Figure 2: Numerical localization of limit cycle in characterizes a nonlinear element, of the system, called Belousov-Zhabotinsky (BZ) reaction Chuas diode; α, β, γ, m0, m1 — are parameters of the system. In this system it was discovered the strange attractors [9] called then Chuas attractors. To date Now consider three-dimensional dynamic models. all known classical Chuas attractors are the attractors that are excited from unstable equilibria. This Example 3 Lorenz system makes it possible to compute different Chuas attractors [10] with relative easy. Consider Lorenz system [7] Here we simulate this system with parameters x˙ = σ(y − x) α = 9.35,, β = 14.79, γ = 0.016, m0 = −1.1384, = 0 7225. y˙ = x(ρ − z) − y (3) m1 . z˙ = xy − βz Here, in all the above examples, limit cycles and and carry out its simulation with standard parame- attractors are those excited from unstable equilibria. From computational point of view this allows one to ters σ = 10, β = 8/3, ρ = 28. use numerical method, in which after transient pro- Example 4 Chua system cess a trajectory, started from a point of unstable Consider the behavior of the classical Chua circuit manifold in the neighborhood of equilibrium, reaches [8]. Consider its dynamic model in dimensionless co- an attractor and identifies it. ordinates x˙ = α(y − x) − αf(x), 2 Hidden oscillations and attractors y˙ = x − y + z, (4) Further there came to light so called hidden oscilla- z˙ = −(βy + γz). tions - the oscillations, the existence itself of which ISBN: 978-1-61804-023-7 293 Recent Researches in System Science 4 numerical analysis. 3 Example 5 Four limit cycles in quadratic system 2 Consider the following quadratic system 1 dx 2 = x y+ x + y, z 0 dt (6) dy 2 2 = a2x + b2xy + c2y + α2x + β2y. −1 dt −2 Application of special analytical methods [12, 17] allow us to visualize in this system four limit cycle. In −3 Fig. 5 for set of the coefficients b2 c= 2.7, 2 =, 0.4 a2 0= −1 , α2 = −437.5, β2 = 0.003 three “large” −4 0.5 limit cycles around zero point and 1 “large” limit cycle to the left of straight line x = −1 can be observed y 0 [18]. −0.5 −0.5 0 0.5 1 1.5 2 2.5 −2.5 −2 −1.5 −1 x Figure 4: Numerical localization of chaotic attractor in Chua circuit is not obvious (which are “small” and, therefore, are difficult for numerical analysis or are not “connected” with equilibrium, i.e. the creation of numerical pro- cedure of integration of trajectories for the passage from equilibrium to periodic solution is impossible). For the first time the problem of finding hidden oscillations had been stated by D. Hilbert in 1900 (Hilbert’s 16th problem) for two-dimensional polynomial systems. For a more than century history, in the framework of the solution of this problem the numerous theoretical and numerical results were ob- tained. However the problem is still far from being resolved even for the simple class of quadratic systems. In 40-50s of the 20th century A.N. Kolmogorov became the initiator of a few hundreds of computational experiments, in the result of which the limit cycles in two-dimensional quadratic systems would been found. The result was absolutely unexpected: in not a single experiment a limit cycle was found, though it is known that quadratic systems with limit cycles form open domains in the space of coefficients and, therefore, for a random choice of polynomial coefficients, the probability of hitting in these sets is positive. It should be noted also that small and Figure 5: Visualization of 4 limit cycles in quadratic nested cycles [11, 12, 13, 14, 15, 16] are difficult to system ISBN: 978-1-61804-023-7 294 Recent Researches in System Science Further the problem of analysis of hidden oscil- Similar situation arises in attractors localization. lations arose in applied problems of automatic con- The classical attractors of Lorenz, Rossler, Chua, trol. In the process of investigation, connected with Chen, and other widely-known attractors are those Aizerman’s (1949) and Kalman’s (1957) conjectures, excited from unstable equilibria. However there are it was stated that the differential equations of sys- attractors of another type [25]: hidden attractors, a tems of automatic control, which satisfy generalized basin of attraction of which does not contain neigh- Routh-Hurwitz stability criterion, can also have hid- borhoods of equilibria. Numerical localization, com- den periodic regimes [19]. putation, and analytical investigation of such attractors are much more difficult problems. Example 6 Counterexample to Kalman conjecture Recently such hidden attractors were discovered [25] in classical Chua’s circuit. Consider a system [20] Consider Chua system (4) with the parameters ˙ = 0 1 ( 0 1 1 0 1 ) x1 −x2 − ϕ x1 − . x3 − . x4 , α = 8.4562, β = 12.0732, γ = 0.0052, (9) x˙ 2 = x1 −0 1 .1 ϕ(x1 −0 1 .1 x3 − 0.1 x4), (7) m0 = −0.1768, m1 = −1.1468. x˙ 3 = x4, Note that for the considered values of parameters ˙ = + ( 0 1 1 0 1 ) x4 −x3 − x4 ϕ x1 − . x3 − . x4 there are three equilibria in the system: a locally with with smooth strictly increasing nonlinearity stable zero equilibrium and two saddle equilibria. Here application of special analytical-numerical σ −σ algorithm [26] allow us to find hidden attractor — ( ) = tanh( ) = e − e 0 d tanh( ) 1 ϕ σ σ − , < σ ≤ . eσ + e σ dσ Ahidden (see Fig. 7). (8) Here for ϕ(σ) = kσ linear system (7) is stable for k ∈ (0, 9.9) and by the above-mentioned theorem for piecewise-continuous nonlinearity ( ) = 0( ) with ϕ σ ϕ σ unst A M hidden sufficiently small ε there exists periodic solution. For 2 system (7) there exists a periodic solution (Fig. 6). S 1 st M1 st M2 z S2 F0 y unst M1 x Figure 6: The projection of trajectory with the ini- tial data x1(0) = x3(0) = x4x(0) = 0, 2(0) = −20 of system (8) on the plane (x1, x2) 3 Conclusion Study of hidden oscillations and attractors requires the development of new analytical and numerical meth- Further, the issues analysis of hidden oscillations ods.

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