Hidden oscillations in dynamical systems · oscillations had been stated by D. Hilbert in 1900 (Hilbert’s 16th problem) for two-dimensional polyno-mial systems. For a more than

Hidden oscillations in dynamical systems

G.A. LEONOVa, N.V. KUZNETSOVb, O.A. KUZNETSOVAb, S.M. SELEDZHIa,V.I.VAGAITSEVb

aSt.Petersburg State University, Universitetsky pr. 28, St.Petersburg, 198504,RUSSIA

bUniversity of Jyvaskyla, P.O. Box 35 (Agora), FIN-40014,FINLAND

Abstract:- The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors arethose excited from unstable equilibria. From computational point of view this allows one to use standardnumerical method, in which after transient process a trajectory, started from a point of unstable manifoldin the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors ofanother type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilib-ria. Study of hidden oscillations and attractors requires the development of new analytical and numericalmethods which will be considered in this paper.

Key-Words:- Hidden oscillation, attractor localization, hidden attractor, harmonic balance, describing func-tion method, Aizerman conjecture, Kalman conjecture, Hilbert 16th problem

1 IntroductionIn the initial period of development of the theoryof nonlinear oscillations in the first half of last cen-tury [1, 2, 3, 4], a main attention has been given toanalysis and synthesis of oscillating systems for whichsolving the problem of the existence of the oscilla-tion modes did not present any great difficulties. Thestructure of many mechanical, electromechanical andelectronic systems was such that there were oscil-lation modes in them, the existence of which wasalmost obvious — oscillations are excited from un-stable equilibria. From computational point of viewthis allows one to use standard numerical method,in which after transient process a trajectory, startedfrom a point of unstable manifold in the neighborhoodof equilibrium, reaches an attractor and identifies it.

Consider corresponding classical examples.

Example 1 Van der Pol oscillator

Consider an oscillations arising in the electrical cir-cuit — the van der Pol oscillator [5]

x+ µ(x2 − 1)x+ x = 0 (1)

and carry out its simulation for the parameter µ = 2.

−3 −2 −1 0 1 2 3−5

−4

−3

−2

−1

0

1

2

3

4

5

x

y

Figure 1: Numerical localization of limit cycle in Vander Pol oscillator

Example 2 Belousov-Zhabotinsky (BZ) reaction

In 1951 B.P. Belousov first discovered oscillations inthe chemical reactions in liquid phase [6]. Consider

WSEAS TRANSACTIONS on SYSTEMS and CONTROLG. A. Leonov, N. V. Kuznetsov, O. A. Kuznetsova, S. M. Seledzhi, V. I. Vagaitsev

ISSN: 1991-8763 54 Issue 2, Volume 6, February 2011

http://www.math.spbu.ru/user/nk/

one of the Belousov-Zhabotinsky dynamic model

εx = x(1 − x) +f(q − x)

q + xz,

z = x− z

(2)

and carry out its simulation with standard parame-ters f = 2/3, q = 8 × 10−4, ε = 4 × 10−2.

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

y

Figure 2: Numerical localization of limit cycle inBelousov-Zhabotinsky (BZ) reaction

Now consider three-dimensional dynamic models.

Example 3 Lorenz system

Consider Lorenz system [7]

x = σ(y − x),

y = x(ρ− z) − y,

z = xy − βz

(3)

and carry out its simulation with standard parame-ters σ = 10, β = 8/3, ρ = 28.

Example 4 Chua system

Consider the behavior of the classical Chua circuit [8].Consider its dynamic model in dimensionless coordi-nates

x = α(y − x) − αf(x),

y = x− y + z,

z = −(βy + γz).

(4)

Here the function

f(x) = m1x+ (m0 −m1)sat(x) (5)

−20−15

−10−5

05

1015

20 −30

−20

−10

0

10

20

30

0

5

10

15

20

25

30

35

40

45

50

yx

z

Figure 3: Numerical localization of chaotic attractorin Lorenz system

characterizes a nonlinear element, of the system, calledChuas diode; α, β, γ,m0,m1 — are parameters of thesystem. In this system it was discovered the strangeattractors [9] called then Chuas attractors. To dateall known classical Chuas attractors are the attrac-tors that are excited from unstable equilibria. Thismakes it possible to compute different Chuas attrac-tors with relative easy [10, 11, 12, 13, 14, 15]. Herewe simulate this system with parameters α = 9.35,β = 14.79, γ = 0.016, m0 = −1.1384, m1 = 0.7225.

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−0.5

0

0.5

−4

−3

−2

−1

0

1

2

3

4

x

z

y

Figure 4: Numerical localization of chaotic attractorin Chua circuit

Here, in all the above examples, limit cycles andattractors are those excited from unstable equilibria.



2 Hidden oscillations and attractorsFurther there came to light so called hidden oscilla-tions - the oscillations, the existence itself of whichis not obvious (which are “small” and, therefore, aredifficult for numerical analysis or are not “connected”with equilibrium, i.e. the creation of numerical pro-cedure of integration of trajectories for the passagefrom equilibrium to periodic solution is impossible).In addition, in this case the integration of trajectorieswith random initial data is unlikely to furnish the de-sired result since a basin of attraction can be highlysmall and the considered system dimension can belarge.

For the first time the problem of finding hiddenoscillations had been stated by D. Hilbert in 1900(Hilbert’s 16th problem) for two-dimensional polyno-mial systems. For a more than century history, inthe framework of the solution of this problem thenumerous theoretical and numerical results were ob-tained. However the problem is still far from being re-solved even for the simple class of quadratic systems.In 40-50s of the 20th century A.N. Kolmogorov be-came the initiator of a few hundreds of computationalexperiments [16], in the result of which the limitcycles in two-dimensional quadratic systems wouldbeen found. The result was absolutely unexpected:in not a single experiment a limit cycle was found,though it is known that quadratic systems with limitcycles form open domains in the space of coefficientsand, therefore, for a random choice of polynomialcoefficients, the probability of hitting in these setsis positive. It should be noted also that small andnested cycles [17,18,16,19,20,21] are difficult to nu-merical analysis.

Example 5 Four limit cycles in quadratic system

Consider the following quadratic system

dx

dt= x2 + xy + y,

dy

dt= a2x

2 + b2xy + c2y2 + α2x+ β2y.

(6)

Application of special analytical methods [18,22] al-low us to visualize in this system four limit cycle. InFig. 5 for set of the coefficients b2 = 2.7, c2 = 0.4,a2 = −10, α2 = −437.5, β2 = 0.003 three “large”

Figure 5: Visualization of 4 limit cycles in quadraticsystem

limit cycles around zero point and 1 “large” limit cy-cle to the left of straight line x = −1 can be ob-served [23].

Further the problem of analysis of hidden oscil-lations arose in applied problems of automatic con-trol. In the process of investigation, connected withAizerman’s (1949) and Kalman’s (1957) conjectures,it was stated that the differential equations of sys-tems of automatic control, which satisfy generalizedRouth-Hurwitz stability criterion, can also have hid-den periodic regimes [24].

2.1 Analytical-numerical method forfinding hidden oscillations of multi-dimensional dynamical systemsConsider a system with one scalar1 nonlinearity

1vector nonlinearity can be considered similarly [25]



http://www.math.spbu.ru/user/nk/PDF/Limit_cycles_Focus_values.pdf

dx

dt= Px + qψ(r∗x), x ∈ R

n. (7)

Here P is a constant (n×n)-matrix, q, r are constantn-dimensional vectors, ∗ is a transposition operation,ψ(σ) is a continuous piecewise-differentiable2 scalarfunction, and ψ(0) = 0. Define a coefficient of har-monic linearization k in such a way that the matrix

P0 = P + kqr∗ (8)

has a pair of purely imaginary eigenvalues ±iω0 (ω0 >0) and the rest of its eigenvalues have negative realparts. We assume that such k exists. Rewrite system(7) as

dx

dt= P0x + qϕ(r∗x), (9)

where ϕ(σ) = ψ(σ) − kσ.Introduce a finite sequence of functions ϕ0(σ),

ϕ1(σ), . . . , ϕm(σ) such that the graphs of neighboringfunctions ϕj(σ) and ϕj+1(σ) slightly differ from oneanother, the function ϕ0(σ) is small, and ϕm(σ) =ϕ(σ). Using a smallness of function ϕ0(σ), we canapply and mathematically strictly justify [26, 27, 28,?, 25, 29] the method of harmonic linearization (de-scribing function method) for the system

dx

dt= P0x + qϕ0(r∗x) (10)

and determine a stable nontrivial periodic solutionx0(t). For the localization of oscillating solution (orattractor) of original system (9), we shall follow nu-merically the transformation of this periodic solution(a starting oscillating attractor — an attractor, notincluding equilibria, denoted further by A0) with in-creasing j. Here two cases are possible: all the pointsof A0 are in an attraction domain of attractor A1, be-ing an oscillating attractor of the system

dx

dt= P0x + qϕj(r∗x) (11)

with j = 1, or in the change from system (10) tosystem (11) with j = 1 it is observed a loss of sta-bility (bifurcation) and the vanishing of A0. In thefirst case the solution x1(t) can be determined nu-merically by starting a trajectory of system (11) with

2This condition can be weakened if a piecewise-continuousfunction being Lipschitz on closed continuity intervals is con-sidered [?]

j = 1 from the initial point x0(0). If in the process ofcomputation the solution x1(t) has not fallen to anequilibrium and it is not increased indefinitely (here asufficiently large computational interval [0, T ] shouldalways be considered), then this solution reaches anattractor A1. Then it is possible to proceed to system(11) with j = 2 and to perform a similar procedureof computation of A2, by starting a trajectory of sys-tem (11) with j = 2 from the initial point x1(T ) andcomputing the trajectory x2(t).

Proceeding this procedure and sequentially in-creasing j and computing xj(t) (being a trajectoryof system (11) with initial data xj−1(T )) we eitherarrive at the computation of Am (being an attractorof system (11) with j = m, i.e. original system (9)),either, at a certain step, observe a loss of stability(bifurcation) and the vanishing of attractor.

To determine the initial data x0(0) of starting pe-riodic solution, system (10) with nonlinearity ϕ0(σ)is transformed by linear nonsingular transformationS to the form

y1 = −ω0y2 + b1ϕ0(y1 + c∗3y3),

y2 = ω0y1 + b2ϕ0(y1 + c∗3y3),

y3 = A3y3 + b3ϕ0(y1 + c∗3y3).

(12)

Here y1, y2 are scalar values, y3 is (n−2)-dimensionalvector; b3 and c3 are (n−2)-dimensional vectors, b1and b2 are real numbers; A3 is an ((n−2)× (n−2))-matrix, all eigenvalues of which have negative realparts. Without loss of generality, it can be assumedthat for the matrix A3 there exists a positive numberd > 0 such that

y∗

3(A3 + A∗

3)y3 ≤ −2d|y3|2, ∀y3 ∈ R

n−2. (13)

Introduce the describing function

Φ(a) =

2π/ω0∫

0

ϕ(

cos(ω0t)a)

cos(ω0t)dt.

In practice, to determine k and ω0 it is used thetransfer function W (p) of system (7):

W (p) = r∗(P − pI)−1q,

where p is a complex variable. The number ω0 isdetermined from the equation ImW (iω0) = 0 and kis computed then by formula k = −(ReW (iω0))

−1.



In 1957 R.E. Kalman formulated the followingconjecture [30]: Suppose that for all k ∈ (µ1, µ2) azero solution of system (9) with ϕ(σ) = kσ is asymp-totically stable in the large (i.e., a zero solution isLyapunov stable and any solution of system (9) tendsto zero as t → ∞. In other words, a zero solution isa global attractor of system (9) with ϕ(σ) = kσ).

If at the points of differentiability of ϕ(σ) the con-dition

µ1 < ϕ′(σ) < µ2 (14)

is satisfied, then system (9) is stable in the large?Consider a method for counterexamples construc-

tion Kalman’s conjecture. Let us assume first thatµ1 = 0, µ2 > 0 and consider system (12) with nonlin-earity ϕ0(σ) of special form

ϕ0(σ) =

{

µσ, ∀|σ| ≤ ε;

sign(σ)Mε3, ∀|σ| > ε.(15)

Here µ < µ2 and M are certain positive numbers,ε is a small positive parameter.

Then the following result is valid.

Theorem 1 [31] If the inequalities

b1 < 0,

0 < µb2ω0(c3∗b3 + b1) + b1ω

20

are satisfied, then for small enough ε system (12) withnonlinearity (15) has orbitally stable periodic solu-tion, satisfying the following relations

y1(t) = − sin(ω0t)x2(0) +O(ε),

y2(t) = cos(ω0t)x2(0) +O(ε),

y3(t) = On−2(ε),

y1(0) = O(ε2),

y2(0) = −

√

µ(µb2ω0(c∗b+ b1) + b1ω

20)

−3ω20Mb1

+O(ε),

y3(0) = On−2(ε2).

(16)

The methods for the proof of this theorem aredeveloped in [31,16,32,33].

Based on this theorem, it is possible to apply de-scribed above multi-step procedure for the localiza-tion of hidden oscillations: initial data obtained in

this theorem allow to step aside from stable zero equi-librium and to start numerical localization of possibleoscillations.

For that we consider a finite sequence of piecewise-linear functions

ϕj(σ) =

{

µσ, ∀|σ| ≤ εj ;

sign(σ)Mε3j , ∀|σ| > εj ., εj = j

m

√

µM

j = 1, . . . ,m.(17)

Here function ϕm(σ) is monotone continuous piecewise-linear function sat(σ) (“saturation”). We choosem insuch a way that the graphs of functions ϕj and ϕj+1

are slightly distinct from each other outside smallneighborhoods of points of discontinuity.

Suppose that the periodic solution xm(t) of sys-tem (9) with monotone and continuous function ϕm(σ)(“saturation”) is computed. In this case we organizea similar computational procedure for the sequenceof systems

dx

dt= Px+ qψi(r∗x). (18)

Here i = 0, . . . , h, ψ0(σ) = ϕm(σ) and

ψi(σ) = ϕm(σ) +

{

0, ∀|σ| ≤ εm;

i(σ − sign(σ)εm)N, ∀|σ| > εm,

where N is a certain positive parameter such thathN < µ2 (using the technique of small changes, it isalso possible to approach other continuous monotonicincreasing functions [25]).

The finding of periodic solutions xi(t) of system(18) gives a certain counterexample to Kalman’s con-jecture for each i = 1, . . . , h.

Consider a system

x1 = −x2 − 10ϕ(x1 − 10.1x3 − 0.1x4),

x2 = x1 − 10.1ϕ(x1 − 10.1x3 − 0.1x4),

x3 = x4,

x4 = −x3 − x4 + ϕ(x1 − 10.1x3 − 0.1x4).

(19)

Here for ϕ(σ) = kσ linear system (19) is stable fork ∈ (0, 9.9) and by the above-mentioned theorem forpiecewise-continuous nonlinearity ϕ(σ) = ϕ0(σ) withsufficiently small ε there exists periodic solution.

Here for ϕ(σ) = kσ linear system (19) is stable fork ∈ (0, 9.9) and by the above-mentioned theorem forpiecewise-continuous nonlinearity ϕ(σ) = ϕ0(σ) withsufficiently small ε there exists periodic solution.



Now we make use of the algorithm of construct-ing of periodic solutions. Suppose µ = M = 1,ε1 = 0.1, ε2 = 0.2, ..., ε10 = 1. For j = 1, ..., 10,we construct sequentially solutions of system (19),assuming that by (17) the nonlinearity ϕ(σ) is equalto ϕj(σ). Here for all εj , j = 1, ..., 10 there existsperiodic solution.

At the first step for j = 0 by the theorem the ini-tial data of stable periodic oscillation take the form:

x1(0) = O(ε), x3(0) = O(ε), x4(0) = O(ε),x2(0) = −1.7513 +O(ε).

Therefore for j = 1 a trajectory starts from the pointx1(0) = x3(0) = x4(0) = 0, x2(0) = −1.7513. Theprojection of this trajectory on the plane (x1, x2) andthe output of system r∗x(t) = x1(t) − 10.1x3(t) −0.1x4(t) are shown in Fig. 6.

Figure 6: ε = 0.1: trajectory projection on the plane(x1, x2)

From the figure it follows that after transient pro-cess stable periodic solution is reached. At the firststep the computational procedure is ended at thepoint x1(T ) = 0.7945, x2(T ) = 1.7846, x3(T ) =0.0018, x4(T ) = −0.0002, where T = 1000π.

Further, for j = 2 we take the following initialdata: x1(0) = 0.7945, x2(0) = 1.7846, x3(0) =0.0018, x4(0) = −0.0002, and obtain next periodicsolutions.

Proceeding this procedure for j = 3, ...10, we se-quentially approximate (Fig. 8-14) a periodic solutionof system (19) (Fig. 15).

Note that for εj = 1 the nonlinearity ϕj(σ) ismonotone. The computational process is ended atthe point x1(T ) = 1.6193, x2(T ) = −29.7162, x3(T ) =−0.2529, x4(T ) = 1.2179, where T = 1000π.




We also remark that here if instead of sequentialincreasing of εj , we compute a solution with initialdata according to (16) for ε = 1, then the solutionwill “falls down” to zero.

Change the nonlinearity ϕ(σ) to the strictly in-creasing function ψi(σ), where µ = 1, εm = 1, N =0.01, for i=1,...,5, and, continue the sequential con-



Figure 10: ε = 0.5: trajectory projection on theplane (x1, x2)



struction of periodic solutions for system (19). Thegraph of such nonlinearity is shown in Fig. 16.

The periodic solutions obtained are shown in Fig. 17–21.

In the case of the computation of solution for



Figure 15: ε = 1: trajectory projection on the plane(x1, x2)

i = 6 there occurs the vanishing of periodic solution(Fig. 22).

For system (19) with smooth strictly increasingnonlinearity

ϕ(σ) = tanh(σ) =eσ − e−σ

eσ + e−σ(20)



Figure 16: The graph of ψi(σ) for i=5 and stabilitysector

Figure 17: ı = 1: trajectory projection on the plane(x1, x2)


there exists a periodic solution (Fig. 23). Here

0 <d

dσtanh(σ) ≤ 1, ∀σ.




Here on the first step it is possible to apply describedabove method to reach saturation function; on thesecond — “slightly” by small steps transform satura-tion to tanh.

Further, the issues analysis of hidden oscillationsarose in the study of dynamical phase locked loops




Figure 23: The projection of trajectory with theinitial data x1(0) = x3(0) = x4(0) = 0, x2(0) = −20of system (20) on the plane (x1, x2)

[34,35,36]. In 1961 semi-stable limit cycle were foundedin two-dimensional model of PLL [37] (which also cannot be detected by numerical simulations).

Example 6 Hidden attractor in Chua’s circuit

Similar situation arises in attractors localization.The classical attractors of Lorenz, Rossler, Chua,Chen, and other widely-known attractors are thoseexcited from unstable equilibria. However there areattractors of another type [38]: hidden attractors, abasin of attraction of which does not contain neigh-borhoods of equilibria. Numerical localization, com-putation, and analytical investigation of such attrac-tors are much more difficult problems.

Recently such hidden attractors were discovered[38,39] in classical Chua’s circuit.

Here we consider system (10) with ϕ0(σ) = εϕ(σ)were ε is a small positive parameter and introduce

class of functions ϕj : ϕ1 = ε1ϕ(σ), . . ., ϕm−1 =εm−1ϕ(σ), ϕm = εmϕ(σ), where εj = j/m, j =1, . . . ,m.

For such class on nonlinearities ϕj the followingtheorem was proved in [16]

Theorem 2 If it can be found a positive a0 such that

Φ(a0) = 0, (21)

then for the initial data of periodic solution x0(0) =S(y1(0), y2(0),y3(0))

∗ at the first step of algorithmwe have

y1(0) = a0 +O(ε), y2(0) = 0, y3(0) = On−2(ε),(22)

where On−2(ε) is an (n−2)-dimensional vector suchthat all its components are O(ε).

For the stability of x0(t) (if the stability is re-garded in the sense that for all solutions with theinitial data sufficiently close to x0(0) the modulus oftheir difference with x0(t) is uniformly bounded forall t > 0), it is sufficient to require the satisfaction ofthe following condition

b1dΦ(a)

da

∣

∣

∣

∣

a=a0

< 0.

Consider Chua system (4) with the parameters

α = 8.4562, β = 12.0732, γ = 0.0052,

m0 = −0.1768, m1 = −1.1468.(23)

Note that for the considered values of parametersthere are three equilibria in the system: a locallystable zero equilibrium and two saddle equilibria.

Modeling of this system was carried out in 10steps increasing parameter εj from the start valueε1 = 0.1 to the finish value ε10 = 1 with step 0.1.Projections of trajectories of the system into (x, y)plane at the each step of the multistage numericalprocedure described above are shown in Figs. 24—33.

Here application of special analytical-numericalalgorithm described above [25] allow us to find hiddenattractor — Ahidden (see Fig. 34).



http://www.math.spbu.ru/user/nk/PDF/Hidden_Chua_Attractor_Localization.pdf

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.1

x

y

Figure 24: Projections of trajectory into (x, y) planefor ε1 = 0.1

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.2

x

y

Figure 25: ε2 = 0.2

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.3

x

y

Figure 26: ε3 = 0.3

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.4

x

y

Figure 27: ε4 = 0.4

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.5

x

y

Figure 28: ε5 = 0.5

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.6

x

y

Figure 29: ε6 = 0.6



−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.7

x

y

Figure 30: ε7 = 0.7

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.8

x

y

Figure 31: ε8 = 0.8

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=0.9

x

y

Figure 32: ε9 = 0.9

−8 −6 −4 −2 0 2 4 6 8−2

−1.5

−1

−0.5

0

0.5

1

1.5

2ε=1

x

y

Figure 33: Projections of trajectory into (x, y) planefor ε10 = 1

It should be noted that the decreasing of inte-gration step, the increasing of integration time, andthe computation of different trajectories of originalsystem with initial data from a small neighborhoodof Ahidden lead to the localization of the same setAhidden (all the computed trajectories densely tracethe set Ahidden). Note also that for the computedtrajectories it is observed Zhukovsky instability andthe positiveness of Lyapunov exponent [40,41].

−15−10

−50

510

15−5

0

5

−15

−10

−5

0

5

10

15

M2

unst

M1

unst

Ahidden

F0

S1

S2

M2

st

M1

st

x

y

z



−15

−10

−5

0

5

10

15

−5

−4

−3

−2

−1

0

1

2

3

4

5

−15

−10

−5

0

5

10

15

x

y

z

M2unst

M1

unst

Ahidden

S2

S1

M2

stM1

st

F0

Figure 34: Equilibrium, stable manifolds of saddles,and localization of hidden attractor.

The behavior of system trajectories in the neigh-borhood of equilibria is shown in Fig. 34. Here Munst

1,2

are unstable manifolds, M st1,2 are stable manifolds.

Thus, in a phase space of system there are stableseparating manifolds of saddles.

The above and the remark on the existence, insystem, of locally stable zero equilibrium F0 attractedthe stable manifolds M st

1,2 of two symmetric saddlesS1 and S2 led to the conclusion that in Ahidden thereis computed a hidden strange attractor.

3 ConclusionStudy of hidden oscillations and attractors requiresthe development of new analytical and numerical meth-ods. At this invited lecture there are discussed thenew analytic-numerical approaches to investigationof hidden oscillations in dynamical systems, basedon the development of numerical methods, comput-ers, and applied bifurcation theory, which suggestsrevisiting and revising early ideas on the applicationof the small parameter method and the harmonic lin-earization [16,25,29,38].

AcknowledgementsThis work was supported by the Academy of Fin-

land, Russian Ministry of Education and Science, andSaint-Petersburg State University.

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[3] A. Andronov, A. Vitt, and S. Khaikin, Theoryof Oscillators. Oxford: Pergamon, 1966.

[4] J. Stoker, Nonlinear Vibrations in Mechanicaland Electrical Systems. N.Y: L.: Interscience,1950.

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