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

ContentI Classical computation of self-excited attractors

I Hidden attractors: hidden periodic oscillations and hidden chaoticattractors, e.g., in the systems with no-equilibria or with the onlystable equilibrium (multistability and coexistence of attractors )

I Hidden attractors in fundamental problems and applied modelsI 16th Hilbert problem on nested limit cycles in the planeI Aizerman conjecture and Kalman conjecture

on absolute stability of nonlinear control systemsI Phase-locked loop (PLL) and Costas loop circuitsI Aircrafts control systems (windup and antiwindup)I Drilling systems, electrical machines and induction motorsI Hidden chaotic attractor in Chua circuitI Secure (chaotic) communications:

synchronization of coupled chaotic oscillatorsSurvey: Leonov G.A.,Kuznetsov N.V., Hidden attractors in dynamical systems. From hiddenoscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor inChua circuits, Int. J. Bif. andChaos 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=

x1x2. . .xn

=f1(x1, x2, .., xn)f2(x1, x2, .., xn)

. . .fn(x1, x2, .., xn)

=F(x)stable

unstable

x 3

x 1

x 2

standard computational procedure:1. Find equilibria xk : F(xk) = 0;local stability analysisy = dF(x)

dx

∣∣x=xk

y

2. Computation of trajectories withinitial data from neighborhoods ofunstable equilibria.

−4 −2 0 2 4−3

−2

−1

0

1

2

3

x = yy = −x + e(1−x2)y..

−20−15

−10−5

05

1015

20 −30−20

−100

1020

30

5

10

15

20

25

30

35

40

45

x = −s(x − y)y = rx − y − xzz = −bz + xy

Van der Pol Lorenz...

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 onlystable equilibrium (special case ofmultistability& coexistence of attractors)

standard computational procedure: [1. to �nd equilibria;

2. trajectory, starting from a point of unstable manifold in a

neighborhood of unstable equilibrium, after transient process

reaches an self-excited oscillation and localizes it]allows to �nd self-excited but not hidden oscillations. −15 −10 −5 0 5 10 15

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y

x

z

How to predict existence and choose initial data in attraction domain?

Survey: Leonov G.A.,Kuznetsov N.V., Hidden attractors in dynamical systems. From hiddenoscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor inChua circuits, Int. J. Bif. andChaos, 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 di�erential equations

16th Hilbert problem (second part) 1900:number and mutual disposition of limit cycles forx = Pn(x, y) = a1x

2 + b1xy + c1y2 + α1x+ β1y + ...

y = Qn(x, y) = a2x2 + b2xy + c2y

2 + 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 oscillationsN.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+ βyc∈(1/3,1), α=−ε−1, bc<1, b>a+ c, 2c<b+1, 4a(c−1)>(b−1)2, β=0Theorem. For su�ciently small ε the system has three limit cycles:

one to the left of line{x = −1} and two to the right of it.

(Increase β and get four normal size limit cycles)

Kuznetsov N.V., Kuznetsova O.A., Leonov G.A., Visualization of four normal size limit cycles intwo-dimensional polynomial quadratic system, Di�erential 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 conjecturesif z=Az+bkc∗z, is asympt. stable ∀ k∈(k1, k2) : z(t, z0)→0 ∀z0, thenis x=Ax+bϕ(σ), σ=c∗x, ϕ(0)=0, k1<ϕ(σ)/σ<k2: x(t,x0)→0 ∀x0?

ϕ(σ)

k

1949 : k1 < ϕ(σ)/σ < k2 1957 : k1 < ϕ′(σ) < k2In general, conjectures are not true (Aizerman's in R2 , Kalman's in R4)Periodic solution can coexist with the only stable equilibria fornonlinearity from the sector of linear stability � hidden oscillationAizerman's conj.: N.Krasovsky 1952:R2, x(t)→∞; V.Pliss 1958:R3, periodicx(t)

Why such conjectures: by describing function method (DFM) there areno periodic solutions in the case of Aizerman's and Kalman's conditionsN.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 7/26

Kalman conjecture (Kalman problem) 1957x=Ax+bϕ(σ), σ=c∗x, ϕ(0)=0, 0<ϕ′(σ)<k2: x(t,x0)→0 ∀x0?

I Fitts R. 1966: series of counterexamples in R4, nonlinearity ϕ(σ) = σ3

I Barabanov N. 1979-1988: some of Fitts counterexamples are false;

analytical `counterex.' construction in R4, ϕ(σ) `close' to sign(σ) (0≤ϕ′(σ))`gaps' were reported by Glutsyuk, Meisters, Bernat & Llibre

I Yakubovich V. 1965 freq. thm. with ϕ′(σ)⇒:Kalman conj. is true in R3

(Barabanov 1988; Leonov 1996)I Bernat J. & Llibre J. 1996: analytical-numerical `counterexamples'

construction in R4, ϕ(σ) `close' to sat(σ) (0≤ϕ′(σ))I Leonov G., Kuznetsov N.: some of Fitts counterexamples are true;

smooth counterex. in R4: ϕ(σ) = tanh(σ) : 0<tanh′(σ)61<k2 = 9.9;

analytical-numerical counterexamples construction for any type of ϕ(σ);

I Carrasco J. et al. 2014: Discrete-time Kalman conjecture is false in R2

Survey: V.O. Bragin, V.I. Vagaitsev, N.V. Kuznetsov, G.A. Leonov (2011) Algorithms for �ndinghidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua's circuits,J. of Computer and Systems Sciences Int., 50(4), 511-544 (doi:10.1134/S106423071104006X)

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) x1=Px1+ϕ1(x1)(2) xε=Pxε+εϕε(xε) (3) xεj=Pxεj+εjϕεj(xεj)

small ε allows one to determine analytically a stable nontrivial periodicsolution 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 j = 1), then solution xε1(t) can be determined numerically by starting atrajectory of (3) with j=1 from initial point xε(0). If in computational processxε1(t) is not fallen to equilibria and is not →∞ (on su�. large [0, T ]), then xε1(t)computes attractor A1. Then perform similar procedure for (3) with j=2: bystarting trajectory xε2(t) of (3) with j = 2 from init. point xε1(T ) (last point onprevious step) A2 can be computed. And so on.

Localization of attractorA in (1): numerically follow transformation ofAj withincreasing 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

−30 −20 −10 0 10 20 30

−30

−20

−10

0

10

20

30

x1

x2

ε = 0.3

−0.4 −0.2 0 0.2 0.4

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

�3 (

)

σ

σ

−30 −20 −10 0 10 20 30

−30

−20

−10

0

10

20

30

x1

x2

ε = 0.7

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

�7 ()

σ

σ

x1=−x2−10ϕ(σ), x2=x1−10.1ϕ(σ)x3=x4, x4=−x3−x4+ϕ(σ)σ=x1−10.1x3−0.1x4

ϕj(σ)=

{σ, |σ| ≤ εj ;sign(σ)ε3j , |σ| > εj

ε0 = ε << 1, εj = 0.1, ..., 1

Thm: For su�. small ε ∃ periodic solution:x1(0)=x3(0)=x4(0)=0, x2(0)=−1.7513

−30 −20 −10 0 10 20 30

−30

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30

x1

x2

ε = 1

0 20 40 60 80 100−20

−15

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−5

0

5

10

15

20

t

−1 −0.5 0 0.5 1 1.5

−1

−0.5

0

0.5

1

1.5

�10

σ

σ ()

σ

N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 10/26

Smooth counterexample to Kalman conjecture

−30 −20 −10 0 10 20 30

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x1

x2

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t

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

θ1(σ

(

σ

σ

x1 = −x2 − 10ϕ(σ)x2 = x1 − 10.1ϕ(σ)x3 = x4x4 = −x3 − x4 + ϕ(σ)σ = x1 − 10.1x3 − 0.1x4

−30 −20 −10 0 10 20 30

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x2

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t

−2 −1 0 1 2−1.5

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1.5

θ5(σ

(σ

σ

ϕ(σ) = θi(σ)=sat(σ)+i(tanh(σ)−sat(σ))/10i = 1, .., 10tanh(σ) = eσ−e−σ

eσ+e−σ

−30 −20 −10 0 10 20 30

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x2

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t

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1.5

θ10(

σ

σ

σ

(

Smooth counterexampleto Kalman conj-re (i=10):(0< d

dσtanh(σ) ≤ 1),

periodic solution exists,linear system is stable.

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),θ = kx+ kb(sin θ − γ).lead-lag �lter tr.f.(s)=αs+β

s+λ;

x(t) - describes �lter state;k-vco gain; ωvcofree, ω

ref -freq.θ(t) - phase di�erence

γ =ωvcofree−ω

ref

kα+k 1λ(β−aλ)

hidden oscillation:stable & unstable or semistable regimes

σ1σ2

130 135 140 145 150 155

1.4

1.2

1

1.2

0

-0.2

0.6

0.4

x

0.8

σ1σ2σ1σ2σ1σ2

σ1σ2

130 135 140 145 150 155

1.4

1.2

1

1.2

0

-0.2

0.6

0.4

0.8

σ1 σ

2

σ1 σ

2

σ1 σ

2

σ1 σ

2σ1σ2

x

in simulation all trajectories tend to equilibria (global stability - in�nitepull-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 systemsLauvdal, 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 thepresence 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 controlsystem 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)

Li1 +Ri1=nBSθu cos(θu +

π2

),

Li2 +Ri2=nBSθu cos(θu − π

6

),

Li3 +Ri3=nBSθu cos(θu − 5π

6

),

Juθu + k(θu − θl) + b(θu − θl)

+nBS∑3k=1 ik cos

(θu+

(7−4k)π6

)=0,

Jlθl − k(θu − θl)− b(θu − θl)− Tfl(ω − θl) = 0.

Friction torque

θl

θu

Drive part

Upper disc

Lower disc

−5 0 5 10 15

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46

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4

6

8

10

12

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16

θ

ωu

ωl

θ = 0ωu=8ωl =8

θ = 0.5ωu=1ωl =1

Stable limit cycle Stable equilibrium

θ ≈ 3.7ωu≈6.65ωl ≈6.65

θu, θl � angular displacements relative to ω (speed of rotation of bothdiscs in idle mode); i1, i2, i3 � currents in the rotor coils;Tfl � friction between lower disk and the bedrock;

ωu = (ω − θu), ωl = (ω − θl) � 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 angulardisplacement θ may lead to breakdownLeonov G.A., Kuznetsov N.V., Kiseleva M.A., Solovyeva E.P., Zaretskiy A.M., Hidden oscillationsin 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

L.Chua (1983)

x = α(y − x− f(x)

),

y = x− y + z,

z = −(βy + γz),

f(x)=m1x+sat(x)=m1x+1

2(m0−m1)(|x+1|+|x−1|)

Chua circuit is used in chaotic communications

1983�now: computations of Chua self-excited attractorsby standard procedure: trajectory from neighborhood ofunstable 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:[LeonovG.A.,KuznetsovN.V., Vagaitsev V.I, Localization of hidden Chua's attractors,

Physics Letters A, 375(23), 2011, 2230-2233]

x=α(y−x−m1x−ψ(x))y=x−y+z, z=−(βy+γz)ψ(x)=(m0−m1)sat(x)

α=8.4562, β=12.0732γ=0.0052m0=−0.1768,m1=−1.1468equilibria: stable zeroF0 &2 saddlesS1,2

trajectories: 'from'S1,2 tend (black)to zero F0 or tend (red) to in�nity;

Hidden chaotic attractor (in green)

with positive Lyapunov exponent0

5

10

15

−3

−2

−1

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1

2

3

4

5

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−10

−5

0

5

10

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x

y

z

M2unst

M1unst

Ahidden

S2

S1

M2st

M1st

F0

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

−10−5

05

10

−5

0

5

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−5

0

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15

x

ε=0.1

y

z

−10−5

05

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5

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x

ε=0.2

y

z

−10−5

05

10

−5

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5

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−5

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5

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x

ε=0.3

y

z

−10−5

05

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5

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−5

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x

ε=0.5

y

z

−10−5

05

10

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5

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x

ε=0.7

y

z

−10−5

05

10

−5

0

5

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−5

0

5

10

15

x

ε=0.9

y

z

N.V.Kuznetsov, G.A.Leonov Hidden attractors: systems without equilibria, attractors coexistence & multistability 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)x = α(y − x− ψ(x))y = x− y + zz = −(βy + γz)ψ(x) = m1x+ (m0 −m1)sat(x)

Receiver (Chua circuit 2)˙x = α(y − x− ψ(x))˙y = x− y + z +K(y − y)˙z = −(βy + γz)ψ(x) = m1x+ (m0 −m1)sat(x).

Synchronization on self-excited attractor No synchronization on hidden attracton

−5

0

5

−0.4

−0.2

0

0.2

0.4−4

−2

0

2

4

Circuit 1Circuit 2

z(t

) a

nd

z(t

)~

x(t) and x(t)~

y(t) and y(t)~

K = 1.8

x(0) = 2.0848, y(0) = 0.0868, z(0) = −2.819

x(0) = 0.01, y(0) = 0, z(0) = 0~ ~ ~

0 50 100 150 200 250 300−5

0

5

t

0 50 100 150 200 250 300−0.5

0

0.5

t

0 50 100 150 200 250 300−5

0

5

t

z(t) − z(t)

~y(t) − y(t)

~x(t) − x(t)

~

−10

0

10

−10

−5

0

5

10−10

−5

0

5

10

Circuit 1Circuit 2

z(t

) a

nd

z(t

)~

x(t) and x(t)~

y(t) and y(t)~

K = 1.8

x(0) = -3.7727, y(0) = -1.3511, z(0) = 4.6657

x(0) = 0.01, y(0) = 0, z(0) = 0~ ~ ~

0 50 100 150 200 250 300−10

0

10

t

0 50 100 150 200 250 300−2

0

2

t

0 50 100 150 200 250 300−10

0

10

t

z(t) − z(t)

~y(t) − y(t)

~x(t) − x(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, 2014V.-T. Pham, F. Rahma, M. Frasca, L. Fortuna, Dynamics and synchronization of a novelhyperchaotic system without equilibrium, Int. J. of Bif.&Chaos, 24(6), 2014Q. Li, H. Zeng, X.-S. Yang, On hidden twin attractors and bifurcation in the Chua's circuit,Nonlinear Dynamics, 2014Wei Z., A new �nding of the existence of hyperchaotic attractors with no equilibria, Mathematicsand Computers in Simulation, 100, 2014C. Li, J.C. Sprott, Coexisting hidden attractors in a 4-d simpli�ed Lorenz system, IJBC, 2014C. Li, J.C. Sprott, Chaotic �ows with a single nonquadratic term, Physics Letters A, 378, 2014Z. Wei, Hidden hyperchaotic attractors in a modi�ed Lorenz-Sten�o system with only one stableequilibrium, Int. J. of Bif.&Chaos, 2014Q. Li, H. Zeng, X.-S. Yang, On hidden twin attractors and bifurcation in the Chua's circuit,Nonlinear Dynamics, 2014H. Zhao, Y. Lin, Y. Dai, Hidden attractors and dynamics of a general autonomous van derPol-Du�ng oscillator, Int. J. of Bif.&Chaos, 2014S.-K. Lao, Y. Shekofteh, S. Jafari, J.C. Sprott, Cost function based on Gaussian mixture model forparameter estimation of a chaotic circuit with a hidden attractor, Int. J. of Bif.&Chaos, 24(1), 2014U. Chaudhuri, A. Prasad , Complicated basins and the phenomenon of amplitude death in coupledhidden attractors, Physics Letters A, 378, 2014M. Molaie, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Simple chaotic �ows with one stableequilibrium, Int. J. of Bif.&Chaos, 23(11), 2013Jafari S., Sprott J., Simple chaotic �ows with a line equilibrium, Chaos, Sol & Fra, 57, 2013C. 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 oscillationsin mathematical model of drilling system actuated by induction motor with a wound rotor, Nonlineardynamics, 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 periodicoscillations 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 classicalphase-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: 2013X Leonov G.A., Kuznetsov N.V. Hidden attractors in dynamical systems. From hidden oscillations inHilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits,Int. J. of Bifurcation and Chaos, 23(1), 2013, 1�69 (doi:10.1142/S0218127413300024)X Leonov G.A., Kuznetsov N.V. Prediction of hidden oscillations existence in nonlinear dynamicalsystems: analytics and simulation, Advances in Intelligent Systems and Computing, Volume 210AISC, 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 ofhidden attractor in electrical Chua's circuit, Lecture Notes in Electrical Engineering, Volume 174LNEE, 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 Drillingsystems: 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 intwo-dimensional polynomial quadratic system, Di�erential 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 drillingsystem 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 AircraftFlight 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 stabilizationsystem 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 withanti-windup compensation, Di�erential 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 BasedAnti-windup Design Method and Its Application to Flight Control, 2012 4th International Congresson 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 controlsystem 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 forcomputation 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 andHidden 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 dynamicalsystems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hiddenchaotic attractor in Chua circuits, Chaos-Fractals Theories and Applications (IWCFTA), 2012 FifthInternational 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 hiddenoscillations in nonlinear systems. The Aizerman and Kalman Conjectures and Chua's Circuits, J. ofComputer 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 , PhysicsLetters 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 oscillationsin dynamical systems, Transaction on Systems and Control, 6(2), 2011, 54�67 (survey)X G.A. Leonov, N.V. Kuznetsov, Algorithms for searching for hidden oscillations in theAizerman andKalman 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 LyapunovQuantities of Two-Dimensional Dynamical Systems, Proceedings of the Steklov Institute ofMathematics, 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. ofBifurcation and Chaos, 21(2), 2011, 425-429X Leonov G.A., Kuznetsov N.V., Analytical-numerical methods for investigation of hidden oscillationsin 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'scircuits, ICINCO 2011 - Proc. of the 8th Int. Conf. on Informatics in Control, Automation andRobotics, 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 theKalman 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 orderweak 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 oforder 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 onthe 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 dynamicalsystems. Analytical methods and symbolic computation, R&C dynamics, 15(2-3), 2010, 354-377X Leonov G.A., E�cient methods for search of periodical oscillations in dynamic systems. Appl.mathematics and mechanics, 1, 2010, 37-73X 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 attractorlocalization 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: computercalculations, 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 oftrajectories 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 e�ects, chaos, linearization{x = F (x), x ∈ Rn, F (x0) = 0

x(t) ≡ x0, A = dF (x)dx

∣∣x=x0

{y = Ay + o(y)y(t) ≡ 0, (y=x−x0)

{z=Azz(t)≡0

Xstationary: z(t) = 0 is exp. stable ⇒ y(t) = 0 is asympt. stablex = F (x), x(t) = F (x(t)) 6≡ 0

x(t) 6≡ x0, A(t) =dF (x)

dx

∣∣∣∣x=x(t)

{y=A(t)y + o(y)y(t) ≡ 0, (y=x−x(t))

{z=A(t)zz(t)≡0

? nonstationary: z(t) = 0 is exp. stable ⇒? y(t) = 0 is asympt. stable

! Perron e�ects:z(t)=0 is exp. stable(unst), y(t)=0 is exp. unstable(st)

Positive largest Lyapunov exponentdoesn't, in general, indicate chaos

Survey: G.A. Leonov, N.V. Kuznetsov, Time-Varying Linearization and the Perron e�ects,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, 2005Int. 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

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