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Hidden Attractors in Fundamental Problems and Applied Models Hidden attractors in fundamental problems and applied models Nikolay V. Kuznetsov Department of Applied Cybernetics, Saint-Petersburg State University (Russia) University of Jyvaskyla (Finland) nkuznetsov239©gmail.com http://www.math.spbu.ru/user/nk/ D Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 1/37 Gennady A. Leonov received his Candidate degree and D.Sc. degree from Saint-Petersburg State University, USSR, in 1971 and 1983, respectively. In 1986 he was awarded the USSR State Prize for development of the theory of phase synchronization for radiotechnics and communications. Since 1988 he has been Dean of the Faculty of Mathematics and Mechanics at Saint-Petersburg State University and since 2007 Head of the Department of Applied Cybernetics. Since 2006 he is member (corresponding) of the Russian Academy of Science (the section of Machine-building and Control processes). In 2016 and 2017 he was proclaimed the Highly Cited Researchers in the Russian Federation by Clarivate Analytics (Tomson Reutors, Web of Science) In 2017 he became foreign member of the Finnish Academy of Science and Letters. http:/ /www.math.spbu.ru/user /leonov / naMRTVI reHHaP, VIR AneKceeBvt '-la JleOHOBa. https://spbu. ru/ news-events/ novosti/ pamyati-gennadiya-alekseevicha-leonova HeKponor. reHHap,vtiA AneKceeBVl'-1 fleOHOB. http://www. ipme . ru/ ipme/ru/ news/ Leonov _Nekrolog. pdf In Memoriam: Gennady Alekseevich Leonov. N.V. Kuznetsov, S. Abramovich, A.L. Fradkov, G. Chen, lnt. Journal of Bifurcation and Chaos, 2018, http://www.math. spbu.ru/user/nk/ PDF/2018-IJBC-in-Memoriam-GA-Leonov .pdf D Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 2/37 Hidden and self-excited attractors History: stability and oscillation in applied mathematical models Chaotic oscillations in electronic Chua circuit and Chua conjecture Hidden attractor in engineering models Hidden attractors coexistence in Sommerfeld effect Hidden attractors in drilling models Hidden attractors in aircraft control and Keldysh model Hidden attractors in physical models Hidden attractors in fundamental problems Necessary and sufficient conditions of stability Aizerman and Kalman conjecture 16th Hilbert problem Numerical search of hidden attractors Hilbert-Kolmogorov visualization problem Hidden attractors and "Virtual Reality": PLL in SPICE and SIMULINK History, Research groups, and Publications D Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 3/37 James Watt's flyingball regulator (1788). Working regime: w(t) wo, x(t) xo ~ x (t) == x(t) -xo, ~w(t) ==w(t)-wo Technology of metal processing allowed to reduce friction and heat generation: Engine breakdowns began to occur Vyshnegradsky I.A. (1877): linearized model: ~w '" + a~w " + (3 ~w ' + 1~w == 0 Without friction there is no control! af3>1 * loca I ly stable eq u i Ii bria (trivia I attractor) Limit cycles (periodic attractors) as working regimes: ../ Rayleigh J. W. S. (1877) The Theory of Sound (MacMillan, London) . ../ Leaute H. (1885) Sur les oscillations a longues periodes dans les machines actionnees par des moteurs hydrauliques et sur les moyens de prevenir ces oscillations, C.R. Acad. Sci. Paris, Vol. 100, 154-156 Working (locally stable in some sense) regimes: 1.stable equilibria (trivial attractor), limit cycles (periodic attractors), "oscillating" attractors 2.stability in large (basin of attraction), coexistence & multistability, global stability ../ Poincare H., Les methodes nouvelles de la mecanique celeste. V.1-3, Gauthiers-Villars, Paris, 1892,93,99 ../ Lyapunov A.M. (1892) The General Problem of the Stability of Motion, Kharkov, ../ Andronov A.A., (Vitt E.A.,) Khaikin S.E., Theory of Oscillators, ONTI NKTP SSSR, 1937 Watt regulator: multistability (e.g. limit cycles existence)/ global stability? Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 4/37 H. Barkhausen (192x) selbsterregten Schwingungen; (Mandelstam, Papalexi) A. Andronov {1929) self( excited) oscillations: the generation and mainte­ nance of a periodic motion by a source of power that lacks any corresponding periodicity. [transition process to the limit cycle in Van der Pol model] Van der Pol 2 Lorenz x=y 1 x = -s(x - y) ·0 y = -x + e(1-x2)y_: y=rx-y-xz 20 -2 z = -bz + xy _',.:,..,____ _7,o ----:>-..,..__-' o x 10 20-M -2 0 2 4 An oscillation can be visualized numerically if initial data from its vicinity lead to a long­ term behavior that approaches the oscillation. From a computational point of view, such an oscillation (or a set osc.) is called an attractor, and its attracting set is called the basin of attraction. How to choose in itia I data in the basin of attraction? . x, - X1 f1 (x1, x2, .. , Xn) e stable . e unstable . X2 f2(x1, x2 , .. , Xn) - x == ==F (x ) Xt+1 == F (x t) - • . Xn fn(x1,x2, .. ,xn) 3 I ~X2 If attractors's basin of attraction doesn 't overlap with vicinities of equilibria? Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 5/37 Leonov-Kuznetsov, 2009: An attractor is called a self-excited attractor if its basin of attraction intersects with small neighborhood of an equilibrium, otherwise it is called a hidden attractor. E.g. hidden attractors are attractors in the systems without equilibria, or with the only stable equilibrium (a case of multistability). x == a(y - x - f(x )) , iJ == x - y + z, + i == -((3y + 1z), L.Chua, 1990 (conjecture): If the zero equilibrium is stable* no attractors. Two symmetric hidden chaotic attractors ( A~i~den - green domains) in the classica I Chua system. trajectories (red) from unstable manifolds M!~s t of two saddle points 8 1 ,2 are either attracted to the locally stable zero equilibrium Fo , or tend to infinity; trajectories (black) from stable manifolds Mo~± tend to Fo or 81 ,2 ./ Leonov G.A.,Kuznetsov N.V., Vagaitsev V.I, Localization of hidden Chua's attractors, Physics Letters A, 375(23), 2011, 2230-2233 x 15 -5 y Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 6/37 Criteria of periodic solution existence (DFM) x = Px+ q1/;(r *x ) : W(p)=r *(P - pl)-Iq, lmW(iw)=O, k=-(ReW(iw))-I · - P ( * ) · P - P k * ,Po_± . R..... ,Po 0 -r)f, k * x - ox + qcp r x . o- + qr , AI,2 - iw, t:Aj>2 < , cp- '+' - r x x = Sy, A = s-I PoS, b = (bI ,b2,b 3)* = s-Iq , c*= (1,0,c3)* = r *S Harmonic linearization, linear transformation, small parameter method YI =COS w YI E YI= - wy2+bIEC-P(YI + c3y 3) Y2 (t) = sin(wt)yI (0) + O(E ) iJ2 = WY I + b2ccp(yI + c3y3) y3 (t) = eA 3 ty3(0) + On-2 (c) Y3 = A3Y3 + b3ccp(yI + c3y 3) t E (0, T] Y3(A3 + A 3)y3 < - 2dlY31 2 y (O) E 0 = {YI E [aI, a2], Y2 = 0, IY31< D E}, Fy(O) = y (T), T= 2;: + O(c) 27r/ w Theorem. If there exists a0 > 0 : <P(a0 ) = J cp( cos(wt)a0 ) cos(wt)dt = 0 0 and b1 d~~a) la= ao < 0 then exists periodic solution x(t) = Sy(t) with initial data x(O) =S (ao +O(c:), 0, On- 2(c))*. After the discovery of hidden Chua attractor, Leon Chua wrote (2012): the su bcritica I Hopf bifurcation leads to the birth of a hidden attractor = Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 7 /37 Hidden oscillation localization: analytical-numerical procedure (1) x==F(x) In the procedure, computational methods and the engineering notion of a transient process are combined to study oscillations: 1) Choose or add a parameter A in such a way that for A == a analytically or computationally one can find a nontrivial attractor Ao in (2) (often this attractor has a simple form, e.g.: periodic, self-excited) 2) Localize an attractor A in (1): numerically follow transformation of~ with in­ creasing j == O: Ao == a, ... , j == m: Am == b (A== Am in (3)) 1. If a II points of Ao are in the basin of attraction of Ai ( osci 1lati ng attractor of ( 3) with j == 1), then a trajectory xi (t) with an initial data xi (0) == x 0 (o) E Ao is attracted and identified Ai of (3) with j == 1 . 2. Numerically investigate changes to the shape of Ai. If the change in A. (from Aj to Aj +i in (3)) does not cause a loss of the stability bifurcation of Aj : i.e. if, in the computation, the trajectory xj+i(t) is not attracted to equilibria or CX) (for suff. large time interval [O, T ]) , then 2 x j+i (t) identify an attractor Aj+i 1 and one can start trajectory x j+ (t) with x j+2 (0) == x j +i(T) (last point on previous step). ~ Leonov G.A., Kuznetsov N.V., Mokaev T.N., Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like syst em describing convective fluid motion, European Physical Journal Special Topics, 224, 2015, 1421-1458 Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg 2018 8/37 t = 0. 1 (b) t = 0.3 ( c) t = 0.5 (a) .... ... -.. .. .. ..... _ .. ,.... ., ... .. ....... ~ . ., .. ... .... .' . ·.· . .. .... ... ·' ... .... ... ,­ ·' ... .. •.. .. .. .. •' '.. · .. .. .. .. .. , .• .. , . ' .. '" .. ~ ,. .. .. .. ". .. .. '• •' ... --·' ' ' .. N ,. .. ' N ... .. ' . ' ' : ·: ., •' ., ... ,• .. ... .. , . -5 . - .. ... .. .. .. .. .. ~.. ' .. ·, . ,_ ..... '. ....... .. .. - 10 ... ... .. -15 .. .. ..~ t . .. -15 ... 5 (d) t = 0.8 ( e) 1: = o.9 t = 1. 0 ... ... .. .. ,..... .. .. ,.. (f) -..... '· . .. ., .. .. ,. .. •' •' '· . , . .. -.. .. '.• .... '.. .... ' ., .. .• '" .. ~ . .. .. .... .. .... N J. '· -:· '.. : ·:. ... ... ,• ,. .. .... .. .... .. .. ' ,, ',. ... .. .. .. ... ' ... " .. " .... ..·;·" ... x x x D Nikolay Kuznetsov Hidden attractors in dynamical systems IPME RAS, St.Petersburg
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