Hidden attractors in dynamical systems: systems with no-equilibria equilibria, coexistence of attractors and multistability Fundamental problems (16t Hilbert problem, Aizerman and Kalman conjectures) and engineering models (PLL, aircraft control systems, drilling systems, Chua circuits)
Nikolay Kuznetsov , Gennady Leonov, http://www.math.spbu.ru/user/nk/
tutorial: http://www.math.spbu.ru/user/nk/PDF/Hidden-attractor-coexistence-multistability.pdf
N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 1/26 Content
I Classical computation of self-excited attractors
I Hidden attractors: hidden periodic oscillations and hidden chaotic attractors, e.g., in the systems with no-equilibria or with the only stable equilibrium ( multistability and coexistence of attractors )
I Hidden attractors in fundamental problems and applied models
I 16th Hilbert problem on nested limit cycles in the plane
I Aizerman conjecture and Kalman conjecture on absolute stability of nonlinear control systems
I Phase-locked loop (PLL) and Costas loop circuits
I Aircrafts control systems (windup and antiwindup)
I Drilling systems, electrical machines and induction motors
I Hidden chaotic attractor in Chua circuit
I Secure (chaotic) communications: synchronization of coupled chaotic oscillators Survey: Leonov G.A.,Kuznetsov N.V., Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bif. and Chaos 23(1) 2013, art. no. 1330002, doi: 10.1142/S0218127413300024 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 2/26 Engineering analysis of complex dynamical system x x˙ 1 f1(x1, x2, .., xn) 1 stable unstable x˙ 2 f2(x1, x2, .., xn) x˙ = = =F(x) ... ... x˙ f (x , x , .., x ) x n n 1 2 n 2 x 3 standard computational procedure: 1. Find equilibria xk : F(xk) = 0; 2. Computation of trajectories with local stability analysis initial data from neighborhoods of ˙ dF(x) unstable equilibria. y = dx x=xk y
3 Van der Pol Lorenz 2 45
40 . 1 . 35
x = y x = −s(x − y) 30 0 . 2 . 25 y = −x + e(1−x )y−1 y = rx − y − xz 20 . 15 −2 10 30 20 5 z = −bz + xy 10 −20 −15 0 −10 −5 −3 0 −10 5 10 −20 −4 −2 0 2 4 15 20 −30
only for localization of oscillations excited from unstable equilibria
N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 3/26 Computation of oscillations and attractors
Attractors classi cation: An attractor is called hidden attractor if its basin of attraction does not intersect with small neighborhoods of equilibria, otherwise it is called self-excited attractor. e.g. hidden attractors in the systems with no-equilibria or with the only stable equilibrium (special case of multistability& coexistence of attractors) standard computational procedure:[1. to nd equilibria; 15
2. trajectory, starting from a point of unstable manifold in a 10
5 neighborhood of unstable equilibrium, after transient process z 0 reaches an self-excited oscillation and localizes it] −5 5 −10 0 y allows to nd self-excited but not hidden oscillations. −15 −15 −10 −5 −5 x0 5 10 15 How to predict existence and choose initial data in attraction domain?
Survey: Leonov G.A.,Kuznetsov N.V., Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. Bif. and Chaos, 23(1), 2013, art. no. 1330002, doi: 10.1142/S0218127413300024
N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 4/26 Poincare visualization problem Poincare visualization problem, 1881: the problem ... to construct the curves de ned by dierential equations 16th Hilbert problem (second part) 1900: number and mutual disposition of limit cycles for 2 2 x˙ = Pn(x, y) = a1x + b1xy + c1y + α1x + β1y + ... 2 2 y˙ = Qn(x, y) = a2x + b2xy + c2y + α2x + β2y + ... V. Arnold (2005):To estimate the number of LCs of square vector elds on plane, A.N. Kolmogorov had distributed several hundreds of such elds (with randomly chosen coe cients of quadratic expressions) among a few hundreds of students of Mech.&Math. Faculty of Moscow Univ. as a mathematical practice. Each student had to nd the number of LCs of his/her eld. The result of this experiment was absolutely unexpected: not a single eld had a LC!... Visualization problem: nested limit cycles are hidden oscillations
N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 5/26 Hidden oscillations in R2: nested limit cycles in QS x˙ = x2 + xy + y, y˙ = ax2 + bxy + cy2 + αx + βy c∈(1/3,1), α=−ε−1, bc<1, b>a + c, 2c
Kuznetsov N.V., Kuznetsova O.A., Leonov G.A., Visualization of four normal size limit cycles in two-dimensional polynomial quadratic system, Dierential equations and Dynamical systems, 21(1-2), 2013, pp. 29-34 (doi:10.1007/s12591-012-0118-6)
N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 6/26 Hidden oscillations in R3: Aizerman and Kalman conjectures ∗ if z˙ =Az+bkc z, is asympt. stable ∀ k ∈(k1, k2): z(t, z0)→0 ∀z0, then ∗ is x˙ =Ax+bϕ(σ), σ =c x, ϕ(0)=0, k1 <ϕ(σ)/σ k ϕ(σ) 0 1949 : k1 < ϕ(σ)/σ < k2 1957 : k1 < ϕ (σ) < k2 In general, conjectures are not true (Aizerman's in R2 , Kalman's in R4) Periodic solution can coexist with the only stable equilibria for nonlinearity from the sector of linear stability hidden oscillation Aizerman'sconj.: N.Krasovsky 1952: R2, x(t)→∞; V.Pliss 1958: R3, periodic x(t) Why such conjectures: by describing function method (DFM) there are no periodic solutions in the case of Aizerman's and Kalman's conditions N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 7/26 Kalman conjecture (Kalman problem) 1957 ∗ 0 x˙ =Ax+bϕ(σ), σ =c x, ϕ(0)=0, 0<ϕ (σ) I Bernat J. & Llibre J. 1996: analytical-numerical `counterexamples' 4 0 construction in R , ϕ(σ) `close' to sat(σ) (0≤ϕ (σ)) I Leonov G., Kuznetsov N.: some of Fitts counterexamples are true; smooth counterex. in 4: 0 ; R ϕ(σ) = tanh(σ): 0 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 8/26 Synthesis of scenario of transition to deterministic and chaotic hidden oscillations: an example ˙ (1) x˙ 1 =Px1 +ϕ1(x1)(2) x˙ ε =Pxε +εϕε(xε) (3) xεj =Pxεj +εjϕεj (xεj) small ε allows one to determine analytically a stable nontrivial periodic solution xε(t) for system (2) oscillating attractor A0. 1. if all points of A0 are in the attraction domain of A1 (oscillating attractor of (3) with ), then solution can be determined numerically by starting a j = 1 xε1 (t) trajectory of (3) with j =1 from initial point xε(0). If in computational process is not fallen to equilibria and is not (on su. large ), then xε1 (t) → ∞ [0,T ] xε1 (t) computes attractor A1. Then perform similar procedure for (3) with j =2: by starting trajectory of (3) with from init. point (last point on xε2 (t) j = 2 xε1 (T ) previous step) A2 can be computed. And so on. Localization of attractor A in (1):numerically follow transformation of Aj with increasing j=0, ..,m (εm =1, Am=A) [special continuation method] 2. in the change from system (2) to (3) with j =1, it's observed loss of stability bifurcation and vanishing of attractor A0(or Aj−1 on j-th step). N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 9/26 Synthesis of scenario of transition to hidden oscillations: counterexample to Aizerman conjecture ε = 0.3 30 0.4 0.3 x˙ =−x −10ϕ(σ), x˙ =x −10.1ϕ(σ) 20 1 2 2 1 0.2 10 0.1 x˙ 3 =x4, x˙ 4 =−x3 −x4 +ϕ(σ) ) 2 x 0 σ 0 ( 3 −0.1 −10 σ =x −10.1x −0.1x −0.2 1 3 4 −20 −0.3 −30 −0.4 j σ, |σ| ≤ εj; −30 −20 −10 0 10 20 30 −0.4 −0.2 0 0.2 0.4 x σ ϕ (σ)= 1 3 ε = 0.7 sign(σ)ε , |σ| > εj 1 j 30 , 20 ε0 = ε << 1 εj = 0.1, ..., 1 0.5 10 ) 2 x 0 0 σ ( 7 Thm: For su. small periodic solution: ε ∃ −10 −0.5 −20 x1(0)=x3(0)=x4(0)=0, x2(0)=−1.7513 −30 −1 −30 −20 −10 0 10 20 30 −1 −0.5 0 0.5 1 x ε σ 1 = 1 20 1.5 30 15 1 20 10 0.5 10 5 ) 2 σ σ x 0 0 0 ( 0 1 −5 −10 −0.5 −10 −20 −1 −15 −30 −20 N.V.Kuznetsov,−30 G.A.Leonov−20 −10 0 Hidden10 20 attractors:30 systems0 20 without40 60 equilibria,80 100 attractors coexistence−1 −0.5 0 & multistability0.5 1 1.5 10/26 x t σ 1 Smooth counterexample to Kalman conjecture 20 1.5 30 x˙ 1 = −x2 − 10 ϕ(σ) 15 1 20 10 x˙ = x − 10.1 ϕ(σ) 0.5 2 1 10 5 2 ( σ x 0 0 0 σ ( 1 x˙ = x θ 3 4 −5 −10 −0.5 −10 −20 x˙ = −x − x + ϕ(σ) −1 4 3 4 −15 −30 −20 −1.5 −30 −20 −10 0 10 20 30 0 20 40 60 80 100 −2 −1 0 1 2 σ = x − 10.1 x − 0.1 x x t σ 1 3 4 1 20 1.5 30 i 15 1 ϕ(σ) = θ (σ)= 20 10 0.5 10 5 sat(σ)+i(tanh(σ)−sat(σ))/10 2 ( σ x 0 0 0 σ ( 5 θ −10 −5 i = 1, .., 10 −0.5 −10 σ −σ −20 −1 e −e −15 tanh(σ) = σ −σ −30 e +e −20 −1.5 −30 −20 −10 0 10 20 30 0 20 40 60 80 100 −2 −1 0 1 2 x t σ 1 20 1.5 Smooth counterexample 30 15 1 20 10 to Kalman conj-re (i=10): 0.5 10 5 ( d 2 σ x 0 0 0 σ ( ), ( 0< tanh(σ) ≤ 1 10 −5 θ dσ −10 −0.5 −10 periodic solution exists, −20 −1 −15 −30 −20 −1.5 linear system is stable. −30 −20 −10 0 10 20 30 0 20 40 60 80 100 −2 −1 0 1 2 x t σ 1 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 11/26 Hidden oscillation:Phase-locked loop (PLL), Costas loop PLL and Costas loop circuits: synchronization of two periodic signals Turbo regime - in microprocessor i486DX2-50 (1992, I.Young) PLL doubles frequency - stable operation was not guaranteed x˙ =−ax−(1−ab)(sin θ−γ), hidden oscillation: θ˙ = kx + kb(sin θ − γ). stable & unstable or semistable regimes αs+β lead-lag lter tr.f.(s)= ; x x s+λ 1.4 1.4 - describes lter state; 1.2 1.2 x(t) 1 1 k-vco gain; ωvco , ωref -freq. 0.8 0.8 free 0.6 0.6 θ(t) - phase dierence 0.4 0.4 ωvco −ωref 1.2 1.2 free 0 0 σ σ σ σ σ σ σ σ σ σ σ σ σ 1 σ σ σ 1 σ 1 σ σ σ γ = 1 2 1 2 1 2 1 2 1 2 2 1 2 2 2 1 2 kα+k 1 (β−aλ) -0.2 -0.2 λ 130 135 140 145 150 155 130 135 140 145 150 155 in simulation all trajectories tend to equilibria (global stability - in nite pull-in range), but only bounded pull-in range in real operation N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 12/26 Hidden oscillation in aircraft control systems Lauvdal, Murray, Fossen (CDC, 1997): Since stability in simulations does not imply stability of the physical control system (an example is the crash of the YF22 [Raptor Boeing crash in 1992]) stronger theoretical understanding is required The angle-of-attack control system of the unstable aircraft model in the presence of saturation in magnitude of the control input: stable zero coexists with undesired stable oscillation hidden oscillation B.R. Andrievskii, N.V. Kuznetsov, G.A. Leonov et. al., Hidden oscillations in aircraft ight control system with input saturation, IFAC Proceedings Volumes (IFAC-PapersOnline), 5(1), 2013, pp. 75-79 (doi:10.3182/20130703-3-FR-4039.00026) N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 13/26 Hidden oscillation: Drilling system with induction motor Drill string failure ≈ 1 out of 7 drilling rigs, costs $100 000 (www.oilgasprod.com) ˙ π 16 Stable limit cycle Stable equilibrium Li˙1 + Ri1 =nBSθu cos θu + , Drive part 14 2 θ ≈ 3.7 π 12 ω ˙ ˙ − θ u≈6.65 Li2 + Ri2 =nBSθu cos θu , 10 = 0 ω ≈6.65 6 ωu=8 l π l 8 ˙ ˙ 5 θ Upper disc ω u ω =8 Li3 + Ri3 =nBSθu cos θu − 6 , 6 l ¨ ˙ ˙ 4 Juθu + k(θu − θl) + b(θu − θl) 2 θl Lower disc 0 3 (7−4k)π −2 P θ = 0.5 +nBS k=1 ik cos θu + 6 =0, 8 ω Friction torque 6 u=1 4 ωl =1 ω 10 15 ¨ ˙ ˙ ˙ u 2 5 Jlθl − k(θu − θl) − b(θu − θl) − Tfl(ω − θl) = 0. 0 −5 0 θ θu, θl angular displacements relative to ω (speed of rotation of both discs in idle mode); i1, i2, i3 currents in the rotor coils; Tfl friction between lower disk and the bedrock; ωu = (ω − θ˙u), ωl = (ω − θ˙l) velocities with respect to xed coordinate system. Operating mode: angular displacement θ = θu − θl → const, ωu and ωl → const. Hidden oscillation (stable limit cycle coexists with stable equilibrium) of angular displacement θ may lead to breakdown Leonov G.A., Kuznetsov N.V., Kiseleva M.A., Solovyeva E.P., Zaretskiy A.M., Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor, Nonlinear dynamics, 2014 (doi:10.1007/s11071-014-1292-6) N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 14/26 Self-excited and hidden attractors in Chua's circuits x˙ = α y − x − f(x), y˙ = x − y + z, z˙ = −(βy + γz), 1 f(x)=m x+sat(x)=m x+ (m −m )(|x+1|+|x−1|) 1 1 2 0 1 L.Chua (1983) Chua circuit is used in chaotic communications 1983now: computations of Chua self-excited attractors by standard procedure: trajectory from neighborhood of unstable zero equilibrium traces the chaotic attractor. [Bilotta&Pantano, A gallery of Chua attractors, WorldSci. 2008] Existence of a chaotic attractor if zero equilibrium is stable? L.Chua, 1990: zero equilibrium is stable ⇒ no chaotic attractor N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 15/26 Hidden attractor in classical Chua's system In 2009-2010 the classi cation self-excited and hidden attractor was suggested and hidden chaotic attractor for the rst time was found in Chua circuit: [Leonov G.A.,Kuznetsov N.V., Vagaitsev V.I, Localization of hidden Chua's attractors, Physics Letters A, 375(23), 2011, 2230-2233] x˙ =α(y−x−m x−ψ(x)) 15 1 Ahidden y˙ =x−y+z, z˙ =−(βy+γz) 10 5 S2 z 0 unst ψ(x)=(m0 −m1)sat(x) M −5 1 −10 α=8.4562, β =12.0732 −15 st 5 M1 Mst γ =0.0052 4 2 m0 =−0.1768, m1 =−1.1468 3 2 equilibria: stable zero F0 & 2 saddles S1,2 1 Munst trajectories: 'from' S1,2 tend (black) 2 y 0 S1 to zero F0 or tend (red) to in nity; F −1 0 Hidden chaotic attractor (in green) 15 −2 10 5 x with positive Lyapunov exponent −3 0 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 16/26 Scenario of transition to chaos: from self-excited periodic oscillations to hidden chaotic attractor ε=0.1 ε=0.2 ε=0.3 15 15 15 10 10 10 5 5 5 z 0 z 0 z 0 −5 −5 −5 −10 −10 −10 −15 −15 −15 5 5 5 0 0 0 10 10 10 y 5 y 5 y 5 0 0 0 −5 −5 −5 −5 −10 x −5 −10 x −5 −10 x ε=0.5 ε=0.7 ε=0.9 15 15 15 10 10 10 5 5 5 z 0 z 0 z 0 −5 −5 −5 −10 −10 −10 −15 −15 −15 5 5 5 0 0 0 10 10 10 y 5 y 5 y 5 0 0 0 −5 −5 −5 N.V.Kuznetsov,−5 −10 G.A.Leonovx Hidden attractors: systems−5 −10 without equilibria,x attractors coexistence−5 −10 & multistabilityx 17/26 Hidden attractors in secure communication Yang T., A Survey of Chaotic Secure Communication Systems, 2004 Li C. et al. Chaos synchronization of the Chua system, 2006 Transmitter (Chua circuit 1) Receiver (Chua circuit 2) x˙ = α(y − x − ψ(x)) x˜˙ = α(˜y − x˜ − ψ(˜x)) y˙ = x − y + z y˜˙ =x ˜ − y˜ +z ˜ + K(y − y˜) z˙ = −(βy + γz) z˜˙ = −(βy˜ + γz˜) ψ(x) = m1x + (m0 − m1)sat(x) ψ(˜x) = m1x˜ + (m0 − m1)sat(˜x). Synchronization on self-excited attractor No synchronization on hidden attracton K = 1.8 K = 1.8 x(0) = 2.0848, y(0) = 0.0868, z(0) = −2.819 x(0) = -3.7727, y(0) = -1.3511, z(0) = 4.6657 ~ ~ ~ 5 ~ ~ ~ 10 x(0) = 0.01, y(0) = 0, z(0) = 0 ~ x(0) = 0.01, y(0) = 0, z(0) = 0 ~ x(t) x(t) − 0 − 0 Circuit 1 −5 Circuit 1 −10 x(t) x(t) x(t) 4 Circuit 2 0 50 100 150 200 250 300 10 Circuit 2 0 50 100 150 200 250 300 t t 2 0.5 5 2 ~ ~ ~ ~ y(t) y(t) − 0 − 0 0 0 y(t) y(t) −0.5 y(t) −2 0 50 100 150 200 250 300 0 50 100 150 200 250 300 −2 t −5 t z(t) and z(t) z(t) and z(t) z(t) 5 10 −4 −10 0.4 ~ 10 ~ z(t) 0 z(t) 0 0.2 5 − 5 10 − 0 −5 0 −10 0 z(t) 0 z(t) ~ −0.2 ~ 0 50 100 150 200 250 300 ~ −5 ~ 0 50 100 150 200 250 300 y(t) and y(t) −0.4 −5 x(t) and x(t) t y(t) and y(t) −10 −10 x(t) and x(t) t N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 18/26 Further examples of hidden attractors J.C. Sprott, A conservative dynamical system with a strange attractor, Physics Letters A, 2014 V.-T. Pham, F. Rahma, M. Frasca, L. Fortuna, Dynamics and synchronization of a novel hyperchaotic system without equilibrium, Int. J. of Bif.&Chaos, 24(6), 2014 Q. Li, H. Zeng, X.-S. Yang, On hidden twin attractors and bifurcation in the Chua's circuit, Nonlinear Dynamics, 2014 Wei Z., A new nding of the existence of hyperchaotic attractors with no equilibria, Mathematics and Computers in Simulation, 100, 2014 C. Li, J.C. Sprott, Coexisting hidden attractors in a 4-d simpli ed Lorenz system, IJBC, 2014 C. Li, J.C. Sprott, Chaotic ows with a single nonquadratic term, Physics Letters A, 378, 2014 Z. Wei, Hidden hyperchaotic attractors in a modi ed Lorenz-Sten o system with only one stable equilibrium, Int. J. of Bif.&Chaos, 2014 Q. Li, H. Zeng, X.-S. Yang, On hidden twin attractors and bifurcation in the Chua's circuit, Nonlinear Dynamics, 2014 H. Zhao, Y. Lin, Y. Dai, Hidden attractors and dynamics of a general autonomous van der Pol-Du ng oscillator, Int. J. of Bif.&Chaos, 2014 S.-K. Lao, Y. Shekofteh, S. Jafari, J.C. Sprott, Cost function based on Gaussian mixture model for parameter estimation of a chaotic circuit with a hidden attractor, Int. J. of Bif.&Chaos, 24(1), 2014 U. Chaudhuri, A. Prasad , Complicated basins and the phenomenon of amplitude death in coupled hidden attractors, Physics Letters A, 378, 2014 M. Molaie, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Simple chaotic ows with one stable equilibrium, Int. J. of Bif.&Chaos, 23(11), 2013 Jafari S., Sprott J., Simple chaotic ows with a line equilibrium, Chaos, Sol & Fra, 57, 2013 C. Li, J.C. Sprott, Multistability in a butter y ow, Int. J. of Bif.&Chaos, 23(12), 2013 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 19/26 Our publications on hidden oscillations: 2014 X Leonov G.A., Kuznetsov N.V., Kiseleva M.A., Solovyeva E.P., Zaretskiy A.M., Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor, Nonlinear dynamics, 77(1-2), 2014, 277-288 (doi: 10.1007/s11071-014-1292-6) X Kuznetsov N.V., Leonov G.A. Hidden attractors in dynamical systems: systems with no equilibria, multistability and coexisting attractors IFAC Proceedings Volumes (IFAC-PapersOnline), 2014 (accepted) X Kiseleva M.A., Kondratyeva N.V., Kuznetsov N.V., Leonov G.A., Solovyeva E.P., Hidden periodic oscillations in drilling system driven by induction motor IFAC Proceedings Volumes (IFAC-PapersOnline), 2014 (accepted) X Kuznetsov N.V., Leonov G.A., Yuldashev M.V., Yuldashev R.V., Nonlinear analysis of classical phase-locked loops in signal's phase space IFAC Proceedings Volumes (IFAC-PapersOnline), 2014 (accepted) N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 20/26 Our publications on hidden oscillations: 2013 X Leonov G.A., Kuznetsov N.V. Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits, Int. J. of Bifurcation and Chaos, 23(1), 2013, 169 (doi:10.1142/S0218127413300024) X Leonov G.A., Kuznetsov N.V. Prediction of hidden oscillations existence in nonlinear dynamical systems: analytics and simulation, Advances in Intelligent Systems and Computing, Volume 210 AISC, Springer, 2013, 5-13 (doi:10.1007/978-3-319-00542-3_3) X Kuznetsov N., Kuznetsova O., Leonov G., Vagaitsev V., Analytical-numerical localization of hidden attractor in electrical Chua's circuit, Lecture Notes in Electrical Engineering, Volume 174 LNEE, 2013, Springer, 149-158 (doi:10.1007/978-3-642-31353-0_11) X Leonov G.A., Kiseleva M.A., Kuznetsov N.V., Neittaanmaki P., Hidden Oscillations in Drilling systems: Torsional Vibrations, Journal of Applied Nonlinear Dynamics, 2(1), 2013, 83-94 (doi:10.5890/JAND.2012.09.006) X Kuznetsov N.V., Kuznetsova O.A., Leonov G.A., Visualization of four normal size limit cycles in two-dimensional polynomial quadratic system, Dierential equations and Dynamical systems, 21(1-2), 2013, pp. 29-34 (doi:10.1007/s12591-012-0118-6) X Kiseleva M.A., Kuznetsov N.V., Leonov G.A., Neittaanmaki P., Hidden oscillations in drilling system actuated by induction motor, IFAC Proceedings Volumes (IFAC-PapersOnline), 5(1), 2013, 86-89 (doi:10.3182/20130703-3-FR-4039.00028) X Andrievsky B.R., Kuznetsov N.V., Leonov G.A., Pogromsky A., Hidden Oscillations in Aircraft Flight Control System with Input Saturation, IFAC Proceedings Volumes (IFAC-PapersOnline), 5(1), 2013, 75-79 (doi:10.3182/20130703-3-FR-4039.00026) X Andrievsky B.R., Kuznetsov N.V., Leonov G.A., Seledzhi S.M., Hidden oscillations in stabilization system of exible launcher with saturating actuators, IFAC Proceedings Volumes (IFAC-PapersOnline), 19(1), 2013, 37-41 (doi: 10.3182/20130902-5-DE-2040.00040) N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 21/26 Our publications on hidden oscillations: 2012 X Leonov G.A., Kuznetsov N.V., Vagaitsev V.I., Hidden attractors in smooth Chua's systems, Physica D, 241(18) 2012, 1482-1486 (doi:10.1016/j.physd.2012.05.016) X G.A. Leonov, B.R. Andrievskii, N.V. Kuznetsov, A.Yu. Pogromskii, Aircraft control with anti-windup compensation, Dierential equations, 48(13), 2012, pp. 1700-1720 (doi:10.1134/S001226611213) X Andrievsky B.R., Kuznetsov N.V., Leonov G.A., Pogromsky A.Yu., Convergence Based Anti-windup Design Method and Its Application to Flight Control, 2012 4th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (ICUMT), 2012, IEEE, pp. 212-218 (doi: 10.1109/ICUMT.2012.6459667) Leonov G.A., Kuznetsov N.V., Pogromskii A.Yu., Stability domain analysis of an antiwindup control system for an unstable object, Doklady Mathematics, 86(1), 2012, pp. 587-590 (doi: 10.1134/S1064562412040035) X Leonov G.A., Kuznetsov N.V., Yuldashev M.V., Yuldashev R.V., Analytical method for computation of phase-detector characteristic, IEEE Transactions on Circuits and Systems Part II, 59(10), 2012 (doi:10.1109/TCSII.2012.2213362) X Kiseleva M.A., Kuznetsov N.V., Leonov G.A., Neittaanmaki P., Drilling Systems Failures and Hidden Oscillations, NSC 2012 - 4th IEEE Int. Conf. on Nonlinear Science and Complexity, 2012, 109-112 (doi:10.1109/NSC.2012.6304736) Leonov G.A., Kuznetsov N.V., IWCFTA2012 Keynote Speech I - Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits, Chaos-Fractals Theories and Applications (IWCFTA), 2012 Fifth International Workshop on, 2012, pp. XV-XVII (doi: 10.1109/IWCFTA.2012.8) N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 22/26 Our publications on hidden oscillations: 2011 X V.O. Bragin, V.I. Vagaitsev, N.V. Kuznetsov, G.A. Leonov Algorithms for nding hidden oscillations in nonlinear systems. The Aizerman and Kalman Conjectures and Chua's Circuits, J. of Computer and Systems Sciences Int.,50(4),2011, 511-544 (doi:10.1134/S106423071104006X)(survey) X Leonov G.A., Kuznetsov N.V., Vagaitsev V.I, Localization of hidden Chua's attractors , Physics Letters A, 375(23), 2011, 2230-2233 (doi:10.1016/j.physleta.2011.04.037) X G.A. Leonov, N.V. Kuznetsov, O.A. Kuznetsova, S.M. Seledzhi, V.I.Vagaitsev, Hidden oscillations in dynamical systems, Transaction on Systems and Control, 6(2), 2011, 5467 (survey) X G.A. Leonov, N.V. Kuznetsov, Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems, Doklady Mathematics, 84(1), 2011, 475-481 (doi:10.1134/S1064562411040120) X G.A. Leonov, N.V. Kuznetsov, and E.V. Kudryashova, A Direct Method for Calculating Lyapunov Quantities of Two-Dimensional Dynamical Systems, Proceedings of the Steklov Institute of Mathematics, 272(Suppl. 1), 2011, 119-127 (doi:10.1134/S008154381102009X) X Leonov G.A., Four normal size limit cycle in two-dimensional quadratic system, Int. J. of Bifurcation and Chaos, 21(2), 2011, 425-429 X Leonov G.A., Kuznetsov N.V., Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems, IFAC Proceedings Volumes (IFAC-PapersOnline), 18(1), 2011, 2494-2505 (doi:10.3182/20110828-6-IT-1002.03315) (survey paper) X Kuznetsov N.V., Leonov G.A., Seledzhi S.M., Hidden oscillations in nonlinear control systems, IFAC Proceedings Volumes (IFAC-PapersOnline), 18(1), 2011, 2506-2510 (doi:10.3182/20110828-6-IT-1002.03316) X Kuznetsov N.V., Kuznetsovà O.A., Leonov G.A., Vagaitsev V.I., Hidden attractor in Chua's circuits, ICINCO 2011 - Proc. of the 8th Int. Conf. on Informatics in Control, Automation and Robotics, Vol. 1, 2011, 279-282 (doi:10.5220/0003530702790283) N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 23/26 Our publications on hidden oscillations: 2010-... X G.A. Leonov, V.O. Bragin, N.V. Kuznetsov, Algorithm for constructing counterexamples to the Kalman problem. Doklady Mathematics, 82(1), 2010, 540-542 (doi:10.1134/S1064562410040101) X G. A. Leonov, Four limit cycles in quadratic two-dimensional systems with a perturbed rst order weak focus, Doklady Mathematics, 81(2), 2010, 248-250 (doi:10.1134/S1064562410020237). X G.A. Leonov, N.V. Kuznetsov, Limit cycles of quadratic systems with a perturbed weak focus of order 3 and a saddle equilibrium at in nity, Doklady Mathematics, 82(2), 2010, 693-696 (doi:10.1134/S1064562410050042) X G.A. Leonov, V.I. Vagaitsev, N.V. Kuznetsov, Algorithm for localizing Chua attractors based on the harmonic linearization method, Doklady Mathematics, 82(1), 2010, 663-666 (doi:10.1134/S1064562410040411) X Leonov G.À., Kuznetsova O.A., Lyapunov quantities and limit cycles of two-dimensional dynamical systems. Analytical methods and symbolic computation, R&C dynamics, 15(2-3), 2010, 354-377 X Leonov G.A., E cient methods for search of periodical oscillations in dynamic systems. Appl. mathematics and mechanics, 1, 2010, 37-73 X Leonov G.A., Kuznetsov N.V., Bragin V.O., On Problems of Aizerman and Kalman, Vestnik St. Petersburg University. Mathematics, 43(3), 2010, 148-162 (doi:10.3103/S1063454110030052) X N.V. Kuznetsov, G.A. Leonov and V.I. Vagaitsev, Analytical-numerical method for attractor localization of generalized Chua's system, IFAC Proceedings Volumes (IFAC-PapersOnline), 4(1), 2010, 29-33 (doi:10.3182/20100826-3-TR-4016.00009) X Leonov G.A., Kuznetsov N.V., Kudryashova E.V., Cycles of two-dimensional systems: computer calculations, proofs, and experiments, Vestnik St. Petersburg University. Mathematics, 41(3), 2008, 216-250 (doi:10.3103/S1063454108030047) X Kuznetsov N.V., Leonov G.A., Lyapunov quantities, limit cycles and strange behavior of trajectories in two-dimensional quadratic systems, J. of Vibroengineering, 10(4), 2008, 460-467 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 24/26 Lyapunov exponent: sign inversions, Perron eects, chaos, linearization ( n x˙ = F (x), x ∈ R ,F (x0) = 0 y˙ = Ay + o(y) z˙ =Az dF (x) x(t) ≡ x0,A = y(t) ≡ 0, (y =x−x0) z(t)≡0 dx x=x0 Xstationary: z(t) = 0 is exp. stable ⇒ y(t) = 0 is asympt. stable x˙ = F (x), x˙(t) = F (x(t)) 6≡ 0 dF (x) y˙ =A(t)y + o(y) z˙ =A(t)z x(t) 6≡ x ,A(t) = 0 y(t) ≡ 0, (y =x−x(t)) z(t)≡0 dx x=x(t) ? nonstationary: z(t) = 0 is exp. stable ⇒? y(t) = 0 is asympt. stable ! Perron eects:z(t)=0 is exp. stable(unst), y(t)=0 is exp. unstable(st) Positive largest Lyapunov exponent doesn't, in general, indicate chaos Survey: G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron eects, Int. Journal of Bifurcation and Chaos, 17(4), 2007, 1079-1107 (doi:10.1142/S0218127407017732) N.V. Kuznetsov, G.A. Leonov, On stability by the rst approximation for discrete systems, 2005 Int. Conf. on Physics and Control, PhysCon 2005, Proc. Vol. 2005, 2005, 596-599 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 25/26 N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 26/26