Hidden oscillations in dnamical sems G.A. LEONOV a, N.V. KUZNETSOV b,a, S.M. SELEDZHI a a St.Petersburg State Universit, Universitetsk pr. 28, St.Petersburg, 19854, RUSSIA b Universit of Jväsklä, P.O. Bo 35 (Agora), FIN-414, FINLAND leonov@math.spbu.ru, nkuznetsov239@gmail.com, ssm@ss1563.spb.edu Abract:- The classical attractors of Lorenz, Rossler, Chua, Chen, and other widel-known attractors are those ecited from able equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajector, arted from a point of able manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another tpe: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. Stud of hidden oscillations and attractors requires the development of new analtical and numerical methods which will be considered in this invited lecture. Ke-Words:- Hidden oscillation, attractor localization, hidden attractor, harmonic balance, describing function method, Aizerman conjecture, Kalaman conjecture, Hilbert 16th problem 1 Introduction In eablishing and developing the theor of nonlinear oscillations in the fir half of the twentieth centur [1, 2, 3, 4] mo attention was given to analzing and snthesizing oscillator sems for which solving the problem of the eience of the oscillation modes did not present an great difficulties. The ructure of man mechanical, electromechanical and electronic sems was such that there were oscillation modes in them, the eience of which was almo obvious oscillations are ecited from able equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajector, arted from a point of able manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. Consider corresponding classical eamples. Eample 1 Van der Pol oscillator Consider an oscillations arising in the electrical circuit the van der Pol oscillator [5] ẍ + µ( 2 1)ẋ + = (1) and carr out its simulation for the parameter µ = 2. 5 4 3 2 1 1 2 3 4 5 3 2 1 1 2 3 Figure 1: Numerical localization of limit ccle in Van der Pol oscillator Eample 2 Belousov-Zhabotinsk (BZ) reaction In 1951 B.P. Belousov fir discovered oscillations in the chemical reactions in liquid phase [6]. Consider ISBN: 978-1-6184-23-7 292
one of the Belousov-Zhabotinsk dnamic model εẋ = (1 ) + ż = z f(q ) q + z (2) and carr out its simulation with andard parameters f = 2/3, q = 8 1 4, ε = 4 1 2. z 5 45 4 35 3 25.4 2.35 15 1.3 5 3.25.2.15.1.5 2 15 1 5 5 1 15 2 3 Figure 3: Numerical localization of chaotic attractor in Lorenz sem 2 1 1 2.5.1.1.1.2.3.4.5.6.7.8.9 Here the function f() = m 1 + (m m 1 )sat() = = m 1 + 1 2 (m m 1 )( + 1 1 ) (5) Figure 2: Numerical localization of limit ccle in Belousov-Zhabotinsk (BZ) reaction Now consider three-dimensional dnamic models. Eample 3 Lorenz sem Consider Lorenz sem [7] ẋ = σ( ) ẏ = (ρ z) ż = βz (3) and carr out its simulation with andard parameters σ = 1, β = 8/3, ρ = 28. Eample 4 Chua sem Consider the behavior of the classical Chua circuit [8]. Consider its dnamic model in dimensionless coordinates ẋ = α( ) αf(), ẏ = + z, ż = (β + γz). (4) characterizes a nonlinear element, of the sem, called Chuas diode; α,β,γ,m,m 1 are parameters of the sem. In this sem it was discovered the range attractors [9] called then Chuas attractors. To date all known classical Chuas attractors are the attractors that are ecited from able equilibria. This makes it possible to compute different Chuas attractors [1] with relative eas. Here we simulate this sem with parameters α = 9.35,, β = 14.79, γ =.16, m = 1.1384, m 1 =.7225. Here, in all the above eamples, limit ccles and attractors are those ecited from able equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajector, arted from a point of able manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. 2 Hidden oscillations and attractors Further there came to light so called hidden oscillations - the oscillations, the eience itself of which ISBN: 978-1-6184-23-7 293
4 numerical analsis. 3 Eample 5 Four limit ccles in quadratic sem z 2 1 1 2 3 4.5.5 2.5 2 1.5 1.5.5 1 1.5 2 2.5 Consider the following quadratic sem d dt = 2 + +, d dt = a 2 2 + b 2 + c 2 2 + α 2 + β 2. (6) Application of special analtical methods [12, 17] allow us to visualize in this sem four limit ccle. In Fig. 5 for set of the coefficients b 2 = 2.7, c 2 =.4, a 2 = 1, α 2 = 437.5, β 2 =.3 three large limit ccles around zero point and 1 large limit ccle to the left of raight line = 1 can be observed [18]. Figure 4: Numerical localization of chaotic attractor in Chua circuit is not obvious (which are small and, therefore, are difficult for numerical analsis or are not connected with equilibrium, i.e. the creation of numerical procedure of integration of trajectories for the passage from equilibrium to periodic solution is impossible). For the fir time the problem of finding hidden oscillations had been ated b D. Hilbert in 19 (Hilbert s 16 th problem) for two-dimensional polnomial sems. For a more than centur hior, in the framework of the solution of this problem the numerous theoretical and numerical results were obtained. However the problem is ill far from being resolved even for the simple class of quadratic sems. In 4-5s of the 2 th centur A.N. Kolmogorov became the initiator of a few hundreds of computational eperiments, in the result of which the limit ccles in two-dimensional quadratic sems would been found. The result was absolutel unepected: in not a single eperiment a limit ccle was found, though it is known that quadratic sems with limit ccles form open domains in the space of coefficients and, therefore, for a random choice of polnomial coefficients, the probabilit of hitting in these sets is positive. It should be noted also that small and need ccles [11, 12, 13, 14, 15, 16] are difficult to Figure 5: Visualization of 4 limit ccles in quadratic sem ISBN: 978-1-6184-23-7 294
Further the problem of analsis of hidden oscillations arose in applied problems of automatic control. In the process of inveigation, connected with Aizerman s (1949) and Kalman s (1957) conjectures, it was ated that the differential equations of sems of automatic control, which satisf generalized Routh-Hurwitz abilit criterion, can also have hidden periodic regimes [19]. Eample 6 Countereample to Kalman conjecture Consider a sem [2] ẋ 1 = 2 1ϕ( 1 1.1 3.1 4 ), ẋ 2 = 1 1.1ϕ( 1 1.1 3.1 4 ), ẋ 3 = 4, ẋ 4 = 3 4 + ϕ( 1 1.1 3.1 4 ) with with smooth rictl increasing nonlinearit (7) ϕ(σ) = tanh(σ) = eσ e σ e σ + e σ, < d tanh(σ) 1. dσ (8) Here for ϕ(σ) = kσ linear sem (7) is able for k (, 9.9) and b the above-mentioned theorem for piecewise-continuous nonlinearit ϕ(σ) = ϕ (σ) with sufficientl small ε there eis periodic solution. For sem (7) there eis a periodic solution (Fig. 6). Similar situation arises in attractors localization. The classical attractors of Lorenz, Rossler, Chua, Chen, and other widel-known attractors are those ecited from able equilibria. However there are attractors of another tpe [25]: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. Numerical localization, computation, and analtical inveigation of such attractors are much more difficult problems. Recentl such hidden attractors were discovered [25] in classical Chua s circuit. Consider Chua sem (4) with the parameters α = 8.4562, β = 12.732, γ =.52, m =.1768, m 1 = 1.1468. (9) Note that for the considered values of parameters there are three equilibria in the sem: a locall able zero equilibrium and two saddle equilibria. Here application of special analtical-numerical algorithm [26] allow us to find hidden attractor A hidden (see Fig. 7). A hidden S 1 z S 2 F Figure 6: The projection of trajector with the initial data 1 () = 3 () = 4 () =, 2 () = 2 of sem (8) on the plane ( 1, 2 ) Further, the issues analsis of hidden oscillations arose in the ud of dnamical phase locked loops [21, 22, 23]. In 1961 semi-able limit ccle were founded in two-dimensional model of PLL [24] (which also can not be detected b numerical simulations). 3 Conclusion Stud of hidden oscillations and attractors requires the development of new analtical and numerical methods. At this invited lecture there are discussed the new analtic-numerical approaches to inveigation of hidden oscillations in dnamical sems, based on the development of numerical methods, computers, and applied bifurcation theor, which sugges ISBN: 978-1-6184-23-7 295
S 2 F A hidden [6] B. Belousov, Collection of short papers on radiation medicine for 1958. Moscow: Med. Publ., 1959, ch. A periodic reaction and its mechanism. [7] E. Lorenz, Determiniic nonperiodic flow, J. Atmos. Sci., vol. 2, no. 2, pp. 13 141, 1963. [8] L. Chua, A zoo of range attractors from the canonical chua s circuits, Proceedings of the IEEE 35th Midwe Smposium on Circuits and Sems (Cat. No.92CH399-9), vol. 2, pp. 916 926, 1992. S 1 [9] T. Matsumoto, A chaotic attractor from chuas circuit, IEEE Transaction on Circuits and Sems, vol. 31, pp. 155 158, 199. Figure 7: Equilibrium, able manifolds of saddles, and localization of hidden attractor. revisiting and revising earl ideas on the application of the small parameter method and the harmonic linearization [13, 26, 2, 25]. Acknowledgements This work was supported b the Academ of Finland, Russian Minir of Education and Science (FTP programm), and Saint-Petersburg State Universit. References [1] S. Timoshenko, Vibration Problems in Engineering. N.Y: Van Norand, 1928. [2] A. Krlov, The Vibration of Ships [in Russian]. Moscow: Gl Red Sudoroit Lit, 1936. [3] A. Andronov, A. Vitt, and S. Khaikin, Theor of Oscillators. Oford: Pergamon, 1966. [4] J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Sems. N.Y: L.: Interscience, 195. [5] B. van der Pol, On relaation-oscillations, Philosophical Magazine and Journal of Science, vol. 7, no. 2, pp. 978 992, 1926. [1] E. Bilotta and P. Pantano, A galler of chua attractors, World scientific series on nonlinear science, Series A, vol. 61, 28. [11] N. V. Kuznetsov and G. A. Leonov, Limit ccles and range behavior of trajectories in two dimension quadratic sems, Journal of Vibroengineering, no. 1(4), pp. 46 467, 28. [12] G. A. Leonov and O. A. Kuznetsova, Evaluation of the fir five lapunov eponents for the lienard sem, Doklad Phsics, no. 54(3), pp. 131 133, 29. [13] G. Leonov, Effective methods for periodic oscillations search in dnamical sems, App. math. & mech., no. 74(1), pp. 37 73, 21. [14] G. Leonov and O. Kuznetsova, Lapunov quantities and limit ccles of two-dimensional dnamical sems. analtical methods and smbolic computation, Regular and Chaotic Dnamics, vol. 15, no. 2-3, pp. 354 377, 21. [15] G. Leonov and N. Kuznetsov, Limit ccles of quadratic sems with a perturbed weak focus of order 3 and a saddle equilibrium at infinit, Doklad Mathematics, vol. 82, no. 2, pp. 693 696, 21. [16] G. Leonov, N. Kuznetsov, and E. Kudrashova, A direct method for calculating lapunov quantities of two-dimensional dnamical sems, Proceedings of the STEKLOV INSTITUTE OF ISBN: 978-1-6184-23-7 296
MATHEMATICS, vol. 272, Suppl. 1, pp. S119 S127, 211. on the harmonic linearization method, Doklad Mathematics, vol. 82, no. 1, pp. 663 666, 21. [17] G. Leonov, Four normal size limit ccle in twodimensional quadratic sem, IJBC, no. 21(2), pp. 425 429, 211. [18] N. Kuznetsov, O. Kuznetsova, and G. Leonov, Inveigation of limit ccles in two-dimensional quadratic sems, in Proceedings of 2nd International smposium Rare Attractors and Rare Phenomena in Nonlinear Dnamics (RA 11), 211, pp. 12 123. [19] V. Pliss, Some Problems in the Theor of the Stabilit of Motion. Leningrad: Izd LGU, 1958. [2] G. Leonov, V. Bragin, and N. Kuznetsov, Algorithm for conructing countereamples to the kalman problem, Doklad Mathematics, vol. 82, no. 1, pp. 54 542, 21. [21] N. Kuznetsov, G. Leonov, and S. Seledzhi, Phase locked loops design and analsis, ICINCO 28-5th International Conference on Informatics in Control, Automation and Robotics, Proceedings, vol. SPSMC, pp. 114 118. [22], Nonlinear analsis of the coas loop and phase-locked loop with squarer, Proceedings of the IASTED International Conference on Signal and Image Processing, SIP 29, pp. 1 7, 29. [23] G. Leonov, S. Seledzhi, N. Kuznetsov, and P. Neittaanmaki, Asmptotic analsis of phase control sem for clocks in multiprocessor arras, in ICINCO 21 - Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, vol. 3. IN- STICC Press, 21, pp. 99 12. [24] N. Gubar, Inveigation of a piecewise linear dnamical sem with three parameters, J. Appl. Math. Mech., no. 25, pp. 1519 1535, 25. [25] G. Leonov, N. Kuznetsov, and V. Vagaitsev, Localization of hidden chua s attractors, Phsics Letters A, vol. 375, pp. 223 2233, 211 (doi:1.116/j.phsleta.211.4.37). [26] G. Leonov, V. Vagaitsev, and N. Kuznetsov, Algorithm for localizing chua attractors based ISBN: 978-1-6184-23-7 297