CC BY
0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 111-124. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221206
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 37G35, 34C23, 34C28
Attractors of a Weakly Dissipative System Allowing Transition to the Stochastic Web in the Conservative Limit
A. V. Golokolenov, D. V. Savin
This article deals with the dynamics of a pulse-driven self-oscillating system — the Van der Pol oscillator — with the pulse amplitude depending on the oscillator coordinate. In the conservative limit the "stochastic web" can be obtained in the phase space when the function defining this dependence is a harmonic one. The paper focuses on the case where the frequency of external pulses is four times greater than the frequency of the autonomous system. The results of a numerical study of the structure of both parameter and phase planes are presented for systems with different forms of external pulses: the harmonic amplitude function and its power series expansions. Complication of the pulse amplitude function results in the complication of the parameter plane structure, while typical scenarios of transition to chaos visible in the parameter plane remain the same in different cases. In all cases the structure of bifurcation lines near the border of chaos is typical of the existence of the Hamiltonian type critical point. Changes in the number and the relative position of coexisting attractors are investigated while the system approaches the conservative limit. A typical scenario of destruction of attractors with a decrease in nonlinear dissipation is revealed, and it is shown to be in good agreement with the theory of 1:4 resonance. The number of attractors of period 4 seems to grow infinitely with the decrease of dissipation when the pulse amplitude function is harmonic, while in other cases all attractors undergo destruction at certain values of dissipation parameters after the birth of high-period periodic attractors.
Keywords: nonlinear dynamics, saddle-node bifurcation, stochastic web, Lyapunov exponent, multistability
Received October 21, 2021 Accepted October 05, 2022
Alexander V. Golokolenov golokolenovav@gmail.com Dmitry V. Savin savin.dmitry.v@gmail.com
Saratov State University
ul. Astrakhanskaya 83, Saratov, 410012 Russia
1. Introduction
One of the directions of research on the dynamics of systems with a low level of dissipation is the study of strong multistability that occurs in such systems when the level of dissipation approaches zero. The coexistence of a large number of low-period attractors was first shown for a rather simple weakly dissipative model, the so-called "rotor map" [12]. Over the past few decades there has been a large body of research devoted to the study of multistability and regularities of coexistence of attractors in various nonlinear systems with low dissipation, both physically motivated and model systems (see, for example, [4, 5, 10-13, 15, 19-22, 28]). At the same time it should be noted that in all these works the subject of investigation is the dynamics of systems obtained by adding a constant dissipation (i. e., one independent of the dynamical variables of the system) in conservative models that satisfy the condition of the KAM theorem [1, 16, 27]. A logical development of this line of research is to extend the consideration to systems with conservative version being degenerate from the point of view of KAM theory. One of the simplest and widely known models of this kind is the "stochastic web generator" system proposed by Zaslavsky [34]. This system is a pulse-driven conservative linear oscillator with the amplitude of external pulses depending on the generalized coordinate of the oscillator as a harmonic function. The dynamics of such a system in the weakly dissipative case was considered earlier both for the cases of introducing a linear small dissipation [30] and a small dissipation of the "self-oscillating" type, i.e., when the dissipative term looks like (7 — ¡ix2) x [8, 9]. In the latter case the system under investigation is actually a pulse-driven Van der Pol oscillator with small values of the parameters responsible for linear and nonlinear dissipation. The structures that arise in this case in the phase space retain the symmetry inherent in the conservative version of the system and, in general, the structure of the phase space differs from that for a typical weakly dissipative system with a constant level of dissipation.
On the other hand, the same model with the quadratic function of dependence of the pulse amplitude on the coordinate — we will call it the amplitude function in what follows — was also studied in detail earlier [17]. This system shows a transition to chaos via a cascade of period-doubling bifurcations, which is typical for the dissipative case and is often called the Feigenbaum scenario [6, 7]. However, a certain choice of the path on the parameter plane allows one to observe on the border of chaos the scaling laws typical of the period-doubling cascade in conservative systems, with different scaling constants [29]. This type of critical behavior on the border of chaos is often called a Hamiltonian critical point, or a critical point of H-type, and can be found in a two-dimensional map as a phenomenon of codimension 2 [23, 24, 31], as it happens in the system under consideration despite the presence of the nonlinear dissipation [18, 32]. Let us remember here that the quadratic function can be considered as the simplest power series expansion of the harmonic function cos x. In this context it seems interesting to consider a set of systems obtained from a pulse-driven Van der Pol oscillator with a harmonic amplitude function by decomposing cos x into power series of various orders. Conservative versions of such systems will no longer be degenerate from the point of view of the KAM theorem, but it seems that the investigation of the evolution of their dynamics for amplitude functions of different orders will be useful to observe a transition between two classical phenomena of nonlinear dynamics of conservative systems: the stochastic web and the transition to chaos through a cascade of period-doubling bifurcations. Here we want to focus on two directions. First, we aim to investigate how the general structure of the parameter space of such a system, which was studied in detail earlier for the simplest case [17, 18, 32], changes as the amplitude function becomes more complex. Second, we aim to study the changes which occur in the structure of the phase space and, particularly,
of the coexisting attractors, with the change of the nonlinear dissipation for different amplitude functions. Attractors of such a system with the dissipation of the "self-oscillating" type were studied earlier in [8, 9], but those works were focused mainly on the dependence of the phase space structure on the amplitude of the external force at certain fixed values of dissipation parameters, and only for the harmonic amplitude function.
The article is organized as follows. Section 2 describes the set of systems under study and presents the results of a numerical study of their parameter space. In all cases transition to chaos both via a period-doubling cascade and via quasi-periodicity destruction can be obtained, and the structure of bifurcation lines on the border of chaos allows one to conclude that the critical behavior of H-type exists for all studied forms of an amplitude function. Section 3 presents results of the study of the structure of the phase plane and the transformation of attractors with the variation of the nonlinear dissipation. Numerical simulation shows that the number of coexisting low-period periodic attractors grows with the decrease of nonlinear dissipation, and a typical scenario exists for the birth of new attractors. For the harmonic amplitude function this growth continues in all intervals of nonlinear dissipation we have considered, while systems with a polynomial amplitude function demonstrate the birth of high-period periodic attractors at certain values of nonlinear dissipation, and further decrease of dissipation leads to destruction of all attractors.
2. Dissipative version of the stochastic web generator and its approximations
Following [6, 7], we consider the equations of the model system in the following form:
<x
x - (y - ix2) x + x = Y^ F- nT). (2.1)
n=-<x
Here y and i are parameters of linear and nonlinear dissipation, respectively. Negative values of y correspond to the presence of linear dissipation in the vicinity of the origin, which is in this case a stable fixed point for an autonomous system, while positive y values correspond to the presence of positive divergence, or pumping, around the origin, which becomes in this case an unstable fixed point. In the latter case a stable limit cycle appears for positive i values. To simplify the numerical analysis of the system (2.1), it is possible to approximately solve it in the intervals between pulses using the Van der Pol averaging method and obtain an approximate discrete stroboscopic map [23]
x = Bxn cos T + (F(xn) + yn) sin T = B -xn sin T + (F(x J + yn) cos T
'n+1 V^ + CK + iFixJ+yJ^ yn+1 Vl + C[xl + (F(xn)+yn)>] '
where the parameters B = exp (^7T) and C = [23] correspond now to linear and
nonlinear dissipation, respectively. The conservative case y = 0, i = 0 corresponds to C = = 0, B = 1 in the map (2.2). The autonomous system in (2.1) becomes in the conservative case a linear oscillator, its solution can be found analytically, and hence the stroboscopic map becomes in this case an exact one. The choice of the harmonic amplitude function, e.g., F (x) = A cos x, makes the system (2.1) in the conservative case a generator of the stochastic web [34]. In this paper we will further consider the frequency of external pulses ^ four times greater than the natural frequency of the autonomous system. In this case the structures formed in the phase
space have a crystal-type symmetry with the rotation angle | [34], and the discrete map (2.2) simplifies significantly:
x = B_F(xn) + yn___ ^^_x^_
VI + C[xl + (F(xn)+vn)*]' yn+l~ VI + C[xl + (F(xn)+yn)*]'
The introduction of dissipation into such a system leads to the appearance of symmetrically located coexisting periodic attractors, the number of which decreases with increasing dissipation [8, 9]. The "main" attractor, which continues to exist as the dissipation level increases, is located in the vicinity of the origin on the phase plane if F(x) = A cos x is chosen. Following the idea of considering a set of models approximating the original system, the conservative versions of which are no longer degenerate, we decompose F(x) into a Taylor series in the neighborhood of 0. Obviously, the decompositions are polynomials of even degrees. In the simplest case the function is quadratic, and we also analyze "intermediate" models with powers of the polynomial from 4 to 12:
n/2 x2i
Fn(x) = \Y,(~l Yj^- (2-4)
i=0 (2i)-
Let us start the investigation of the dynamics of these models by analyzing the structure of their parameter space in a wide range of parameters. To this end we obtain charts of the dynamical regimes for the system (2.1) in the stroboscopic section on the parameter plane (A, y) for various F(x) at the fixed value of the nonlinear dissipation parameter ¡ (Fig. 1). Let us also mention here that the structure of the chart of dynamical regimes for the discrete model (2.3) is qualitatively the same. The classification of nonperiodic regimes (quasi-periodic and chaotic) was made using the value of the larger Lyapunov exponent.
The charts show regions of periodic, quasi-periodic and chaotic dynamics, and it can be seen that two different scenarios of transition to chaos take place in a wide range of parameters: via the cascade of period-doubling bifurcations (Feigenbaum scenario) with an increase in the parameter A in the region of negative values of 7 as well as via the destruction of quasi-periodic dynamics in the vicinity of the line A = 0 in the region y > 0. In the latter case, transition to chaos via the Feigenbaum scenario also exists within the synchronisation tongues. Note that in the case of choosing a quadratic amplitude function the system can be transformed by rescaling variables and parameters to the model previously studied in [17, 18, 32], for which, as we mentioned above, the existence of a Hamiltonian critical point (H-type point) as a phenomenon of codimension 2 was shown for the approximate discrete model (2.3). Obviously, the rescaling of the amplitude does not change qualitatively the structure of the parameter space. The structure of the bifurcation lines which appears in the parameter plane in this case is typical of the presence of the H-type critical point — which can be found as an accumulation of the intersections of the period-doubling and Neimark-Sacker bifurcation lines for consecutive cycles of the period-doubling cascade; it was studied in detail for a similar system in [32] and can be seen in the chart of dynamical regimes obtained for the system (2.1) (Fig. 1a). Moreover, we note that a similar structure of bifurcation lines exists for higher expansion orders of F(x) (Figs. 1b and 1c) as well as for the harmonic amplitude function (Fig. 1d), corresponding points are marked by the red cross in Fig. 1. We also note that, when the function F(x) becomes more complicated, the parameter plane transforms: chaotic and quasi-periodic regions expand and the synchronization tongues become more narrow along the A = 0 axis. Areas of chaos emerged in two different ways: via the period-doubling cascade and via the quasi-periodicity destruction, which were separated in the parameter plane for quadratic F(x), are now connected. In particular, it can be seen that the
Fig. 1. Charts of the dynamical regimes for the system (2.1) in the stroboscopic section for different expansion orders Fn(x): F2 (x) (a), F4(x) (b), F8(x) (c) and F(x) = A cos x (d) on the parameter plane (A, y). I = 0.5 in all cases. Areas of different color correspond to regions of the existence of different regimes of the dynamics: chaos (Ch), quasi-periodic (q), dynamics unstable in the sense of Lagrange (infinity), other colors — periodic cycles (the numbers in the figure specify the period of the cycle). The red cross denotes the estimated location of the H-type critical point
vicinity of the H-type critical point which was previously located on the boundary between these two chaotic areas and the area of dynamics unstable in the sense of Lagrange is now located on the boundary of this unified chaotic area. It can also be noted that some synchronization tongues expand in the region of chaos that arose from the cascade of period-doubling bifurcations. For the harmonic F(x) the area of dynamics unstable in the sense of Lagrange disappears completely, and the form of synchronization tongues becomes sufficiently more complicated.
When the nonlinear dissipation parameter i decreases, obtaining charts of dynamical regimes becomes ineffective. The system in this case becomes highly multistable, which is typical of sys-
tems with a low level of dissipation, and a chart of dynamical regimes is useful for visual analysis of the behavior of only one attractor. Therefore, in order to study the behavior of the considered models at small values of the dissipation parameters, let us turn to the analysis of coexisting attractors and their location on the phase plane, using more suitable methods.
3. Attractors in the case of weak dissipation
3.1. Methods of numerical research
An analysis of systems with a low level of dissipation requires a significant increase in the time of transient process up to hundreds of thousands or millions of iterations of the stroboscopic map. For analyzing the dynamics of discrete maps that are explicitly defined analytically such changes do not add any additional difficulties, except for an increase in the calculation time. When it comes to modeling the dynamics of an ODE system, the question arises about the accumulation of calculation errors of approximate numerical methods, especially taking into account the fact that integration of conservative systems requires special symplectic methods that preserve the value of the Hamiltonian of the system, and classical methods, for example, the Runge-Kutta methods, for integrating such systems over long-time intervals are not applicable [2]. In this regard the choice of an integration method which should be used to obtain correct results for the system (2.1) at a small dissipation level is nontrivial. We have compared results obtained by two methods: the classical Runge-Kutta method of fourth order and the symplectic Forest-Stremer-Verlet method [14] at integration times typical of our analysis. We integrated the system (2.1) in the conservative case and in the case of small dissipation, using values of the dissipation parameters typical of further investigation, and compare the phase portraits and attractors obtained in both ways visually. In the dissipative case we modified the fourth-order Forest-Stremer-Verlet method for systems with small non-Hamiltonian perturbation. The modification includes adding a dissipative term to the derivative of the Hamiltonian for the corresponding variable: instead of
dy dx X ^
we use
dH dH , 2, . .
-^=x-(r,-nx)v. (3-2)
The structures of coexisting attractors obtained with computations via both this method and the Runge-Kutta method of fourth order at the same values of dissipation parameters do not have any differences which could be found by visual analysis, which indicates the possibility of using the Runge-Kutta method of fourth order in further calculations.
For analysis of coexisting attractors and its evolution we will use phase portraits, bifurcation trees and bifurcation diagrams. Phase portraits of weakly dissipative systems provide information on the structure of transient trajectories in the phase plane, as well as on the number and location of attractors for fixed values of parameters. To study the transformations that occur with the change of parameters, we use bifurcation trees and bifurcation diagrams.
We obtain bifurcation trees for a set of initial conditions using direct modeling of the system dynamics. To do this we model the dynamics of the system in direct time for each initial condition from that set and, after skipping the transient process, put points on the parameter-variable plane for several iterations of the map, the described process is repeated for all values of the parameter in some range with a certain step. Such bifurcation trees allow us to show in one figure the
dependence of a large number of coexisting attractors on the values of the parameter except only attractors with basins too small to include any of the initial conditions from the selected set. We use this method for both the original system of ODE (2.1) and the approximate discrete map (2.3).
For the discrete map (2.3) we also obtain bifurcation diagrams using specialized software Content [26]. The process includes a search for the coordinates of a fixed point or a cycle of a certain period using the Newton method with only one initial condition which allows one to find a cycle or a fixed point, regardless of their stability, but only one regardless of their number. The functionality available in the program also allows one to determine the type of stability of the selected cycle, track changes in its coordinates with the change of parameter, perform bifurcation analysis and continue the lines of codimension 1 bifurcations on the parameter plane.
Thus, the combination of two methods described above makes it possible to combine a simple and efficient analysis of the dynamics of attractors with a more thorough analysis of the behavior of a certain fixed point of the cycle of the discrete map.
3.2. Simulation results
Adding dissipation to the system (2.1) destroys the stochastic web existing in the phase space of the conservative version of the system. Let us consider step-by-step the process of transformation of the phase space structure with the change of the nonlinear dissipation parameter ¡ . We start with a sufficiently large dissipation value corresponding to the existence of a single "main" attractor in the system — a stable fixed point — and decrease ¡ while leaving the linear dissipation parameter 7 fixed and small. We consider not only the system generating a stochastic web, which corresponds to the choice F(x) = A cos x in system (2.1), but also the set of systems with different orders of polynomial expansion Fn(x), following (2.4). The algorithm of changes which the "main" attractor undergoes is the same for various choices of the external forcing F(x), both a harmonic function and its polynomial expansions. Figure 2 shows a series of phase portraits illustrating these changes.
The decrease in the nonlinear dissipation results in the following sequence of bifurcations: the supercritical Neimark-Sacker bifurcation, after which the fixed point loses its stability and a stable invariant curve appears (Figs. 2a, 2d, 2g, 2j); the saddle-node bifurcation, leading to the birth of a pair of stable and saddle cycles of period 4 (Figs. 2b, 2e, 2h, 2k); the destruction of the invariant curve (Figs. 2c, 2f, 2i, 2l). Note that the structures of the phase portraits of the system of ODE (2.1) and the map (2.3) correspond well to each other. Also, it is worth mentioning that the sequence of bifurcations obtained here is in good correspondence with the organization of phase space around the resonance 1:4 point [3, 25, 33]. Thereby, at the end of the described bifurcation process, instead of the "main" attractor the newly formed attractor of period 4 exists. This new attractor undergoes a similar bifurcation sequence with a decrease in nonlinear dissipation, and so on. To study the dependence of the coordinates of attractors and their number on the parameter, we obtain bifurcation trees and bifurcation diagrams.
Figure 3 shows bifurcation trees for a set of initial conditions constructed for the system of ODE (2.1) for different choices of F(x) as well as their enlarged fragments. It can be seen that, as the dissipation decreases, the number of coexisting attractors increases for both the harmonic function F(x) (Figs. 3d-3f) and its polynomial expansion (Figs. 3a-3c). The enlarged fragments show clearly that the "new" periodic attractors appearing with a decrease of dissipation lose stability with further decrease in dissipation. The type of the bifurcation tree structure suggests that it happens as a result of the Neimark-Sacker bifurcation. Phase portraits confirming the
fi = 0.01
(a)
/x = 0.0018
At = 0.00117
C = 0.01
(d)
C = 0.001
(e)
C = 0.0005
f^1
1
(1 = 0.01
(g)
(1 = 0.0018,
00 /
(i = 0.0009^
C = 0.01
(j)
C = 0.001 .
00
c=o.ooo^. ' . (i)
Fig. 2. Phase portraits for F(x) = A cos x: 1st row (a)-(c) for the ODE (2.1), 2nd row (d)-(f) for the map (2.3), and for F(x) = F4(x): 3rd row (g)-(i) for the system (2.1), 4th row (j)-(l) for map (2.3). A = 0.3 in all cases, y = 0.001 (a)-(c), (g)-(i), B = 1.001 (d)-(f), (j)-(l). Coordinate range on the plane (x, x) (—5 : 5; —5 : 5) for all portraits, values of parameters of nonlinear dissipation are shown in corresponding figures. The transient trajectories are shown in black, attractors in red, and saddle points (in (e) and (k)) as yellow stars
0 M 0.005
, i. flr^— (d)
..... -
Mp|H*5 m
iL^ ^—""
M 0.005 0
0
M
-15
0.007
(ë)j 25
0 A» 0.0005
(f)
».11
»————
-25
0.0005
o
M 0.00014
Fig. 3. Bifurcation trees for the system (2.1): F(x) = F4(x) (a), F(x) = F8(x) (b) and (c), (c) is an enlarged fragment of (b), F (x) = A cos x (d)-(f) with different scale at the ordinate axis. A = 0.3, 7 = = 0.001 in all cases. The red vertical lines in (c) and (e) correspond to the n values for which phase portraits are shown later in Fig. 4
• O
0 0 '
. 0
Fig. 4. Phase portraits for the system (2.1): F (x) = F4(x), ¡1 = 0.00021 (a), F (x) = A cos x, ¡1 = = 0.0002 (b). The values of ¡1 correspond to the red vertical lines in Figs. 3c and 3e. A = 0.3, 7 = 0.001 in all cases. The coordinate range on the plane (x, X) (-7 : 7; —7 : 7)
presence of the stable invariant curves at the ¡1 values slightly smaller than the bifurcation values are shown in Fig. 4.
To study the transformations of the attractors in more detail, we proceed to analyze the dynamics of the discrete map (2.3). The bifurcation trees for the map (2.3) presented in Figs. 5a-5c
demonstrate a qualitative correspondence with similar trees for the system (2.1). Figures 5d-5f show enlarged fragments that demonstrate typical features of bifurcation trees combined with bifurcation diagrams with bifurcation points marked on them. It can be seen that attractors appear due to saddle-node bifurcations and lose stability due to the supercritical Neimark-Sacker bifurcation.
0 C 0.003 0 C 0.003 0 C 0.0005
Fig. 5. Bifurcation trees for the map (2.3): F(x) = F4(x) (a), F(x) = F8(x) (b) and (c), (c) is an enlarged fragment of (b), F(x) = F4(x) (d) and (e), (e) is an enlarged fragment of (d), F(x) = A cos x (f). In (d)-(f) bifurcation diagrams are superimposed on top of the bifurcation trees, green curves correspond to the stable points/cycles, purple curves, to saddle points/cycles, and blue curves, to unstable ones. The Neimark-Sacker bifurcation points are shown by triangles, and the saddle-node bifurcation points are shown by dots. A = 0.3, B = 1.001 in all cases
The invariant curve born as a result of this Neimark-Sacker bifurcation subsequently disappears via interaction with saddle points. This process is illustrated in Fig. 6.
Further decrease of dissipation leads to sufficient differences in the dynamics of systems with harmonic F(x) and its polynomial expansions. In the first case the number of coexisting attractors is constantly increasing, the newly born attractors are located farther from the origin, and the hierarchical structure of the bifurcation tree is clearly visible (see, e.g., Figs. 4d-4f), while in the latter case the number of attractors appearing with the decrease of i is significantly smaller, and the structure of the bifurcation tree is simpler. Phase portraits illustrating the main features of attractor structure in both of these cases are shown in Fig. 7. In the case of a harmonic function, attractors on the phase plane form an annular structure (Fig. 7a), which has similar form in a wide range of the dissipation parameter, the only difference being that the radius of this structure increases with the decrease of dissipation. Is it worth mentioning here that all attractors appearing in this case are of the same period 4. For polynomial expansions F(x), in contrast to the previous case, a decrease in dissipation leads at some point to the appearance of highly periodic attractors which later also undergo the Neimark-Sacker
Fig. 6. Phase portraits of the map (2.3) at the values of parameters close to the points of destruction of the invariant curves. F(x) = A cos x (a), F(x) = F4(x) (b), (c). Attractors are designated by dots with numbers, saddle points, by yellows stars and the letter "S". The numbers indicate the order in which the attractors appear. The coordinate range on the plane (x, x) (—5 : 5; —5 : 5) for all portraits. A = 0.3, B = 1.001 in all cases. (a) C = 0.000582698, (b) C = 0.00059909, (c) C = 0.00015827
Fig. 7. Phase portraits of the map (2.3): F (x) = A cos x (a), F (x) = F4 (x) (b), (c) is an enlargement of the red rectangle from (b), (d) is an enlargement of the red rectangle from (c). A = 0.3, B = 1.001, C = 0.00004 in all cases
bifurcation (Figs. 7b-7d). These attractors seem to have rather small basins of attraction and undergo destruction rather fast as dissipation decreases further.
4. Summary
In this paper we have studied the structure of the parameter plane and phase space for a set of the pulse-driven autooscillating systems with weak dissipation, including the system allowing transition to the stochastic web in the conservative limit, the system possessing the critical behavior of H-type in the dissipative case, and several intermediate models with different functions of the external force. Concerning the structure of the parameter plane we have found that the structure of bifurcation lines typical of the existence of the H-type critical point exists for all considered models, while the chaotic area widens as the external force amplitude function becomes more complex, and the structure of this area also becomes more complex.
A detailed study of attractors and their evolution on the phase plane and in bifurcation diagrams was also carried out. The following scenario of their evolution was shown: the birth of a low-periodic stable cycle as a consequence of the saddle-node bifurcation, the loss of stability of this cycle due to the supercritical Neimark-Sacker bifurcation, and the destruction of the invariant curve born via this Neimark-Sacker bifurcation due to interaction with the saddle cycle. The above-mentioned scenario repeats for different cycles which exist at different values of dissipation parameter. The period of the cycles was equal to the ratio of the frequency of the external pulses and the frequency of the autonomous system. We have shown that this algorithm takes place both for the original system of ODE and for the approximate discrete mapping with different functions of external pulse amplitude. When dissipation approaches zero, the scenarios of attractor evolution differ for the system with a harmonic amplitude function, which has a stochastic web as a conservative limit, and with its polynomial expansions. In the first case the number of periodic cycles demonstrating the described scenario of evolution seems to grow infinitely with the decrease of dissipation, while in other cases, after several iterations, all attractors undergo destruction. This process is associated with the birth of high-period periodic attractors with small basins of attraction and their further destruction.
The authors' contributions
The author G.A.V. obtained all numerical results. Both authors were involved in writing the text of the paper and participated in discussing the results.
Conflict of interest
The authors declare that they have no conflict of interest.
References
[1] Arnol'd, V. I., Proof of a Theorem of A. N. Kolmogorov on the Invariance of Quasi-Periodic Motions under Small Perturbations of the Hamiltonian, Russian Math. Surveys, 1963, vol. 18, no. 5, pp. 9-36; see also: Uspekhi Mat. Nauk, 1963, vol. 18, no. 5, pp. 13-40.
[2] Ascher, U. M., Numerical Methods for Evolutionary Differential Equations, Comput. Sci. Eng., vol. 5, Philadelphia, Penn.: SIAM, 2008.
[3] Biragov, V. S., Bifurcations in a Two-Parameter Family of Conservative Mappings That Are Close to the Henon Mapping, Selecta Math. Soviet, 1990, vol. 9, no. 3, pp. 273-282; see also: Methods of the Qualitative Theory of Differential Equations, Gorki: GGU, 1987, pp. 10-24.
[4] Blazejczyk-Okolewska, B. and Kapitaniak, T., Coexisting Attractors of Impact Oscillator, Chaos Solitons Fractals, 1998, vol. 9, no. 8, pp. 1439-1443.
[5] de Freitas, M.S.T., Viana, R. L., and Grebogi, C., Multistability, Basin Boundary Structure, and Chaotic Behavior in a Suspension Bridge Model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2004, vol. 14, no. 3, pp. 927-950.
[6] Feigenbaum, M. J., Quantitative Universality for a Class of Nonlinear Transformations, J. Stat. Phys, 1978, vol. 19, no. 1, pp. 25-52.
[7] Feigenbaum, M. J., The Universal Metric Properties of Nonlinear Transformations, J. Statist. Phys., 1979, vol. 21, no. 6, pp. 669-706.
[8] Felk, E. V., The Effect of Weak Nonlinear Dissipation on the Stochastic Web, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2013, vol. 21, no. 3, pp. 72-79 (Russian).
[9] Felk, E. V., Kuznetsov, A. P., and Savin, A. V., Multistability and Transition to Chaos in the Degenerate Hamiltonian System with Weak Nonlinear Dissipative Perturbation, Phys. A, 2014, vol. 410, pp. 561-572.
[10] Feudel, U., Complex Dynamics in Multistable Systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2008, vol. 18, no. 6, pp. 1607-1626.
[11] Feudel, U. and Grebogi, C., Why Are Chaotic Attractors Rare in Multistable Systems?, Phys. Rev. Lett., 2003, vol. 91, no. 13, 134102, 4 pp.
[12] Feudel, U., Grebogi, C., Hunt, B.R., and Yorke, J. A., Map with More Than 100 Coexisting Low-Period Periodic Attractors, Phys. Rev. E, 1996, vol. 54, no. 1, pp. 71-81.
[13] Feudel, U., Grebogi, C., Poon, L., and Yorke, J. A., Dynamical Properties of a Simple Mechanical System with a Large Number of Coexisting Periodic Attractors, Chaos Solitons Fractals, 1998, vol. 9, nos. 1-2, pp. 171-180.
[14] Forest, E. and Ruth, R. D., Fourth-Order Symplectic Integration, Phys. D, 1990, vol. 43, no. 1, pp.105-117.
[15] Kolesov, A. Yu. and Rozov, N. K., The Nature of the Bufferness Phenomenon in Weakly Dissipative Systems, Theor. Math. Phys., 2006, vol. 146, no. 3, pp. 376-392; see also: Teoret. Mat. Fiz., 2006, vol. 146, no. 3, pp. 447-466.
[16] Kolmogorov, A. N., Preservation of Conditionally Periodic Movements with Small Change in the Hamilton Function, in Stochastic Behaviour in Classical and Quantum Hamiltonian Systems (Volta Memorial Conference, Como, 1977), G.Casati, J.Ford (Eds.), Lect. Notes Phys. Monogr., vol. 93, Berlin: Springer, 1979, pp. 51-56; see also: Dokl. Akad. Nauk SSSR (N. S.), 1954, vol. 98, pp. 527-530.
[17] Kuznetsov, A. P., Kuznetsov, S. P., Savin, A. V., and Savin, D. V., Autooscillating System with Compensated Dissipation: Dynamics of Approximated Discrete Map, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2008, vol. 16, no. 5, pp. 127-138 (Russian).
[18] Kuznetsov, A. P., Kuznetsov, S. P., Savin, A. V., and Savin, D. V., On the Possibility for an Autooscil-latory System under External Periodic Drive Action to Exhibit Universal Behavior Characteristic of the Transition to Chaos via Period-Doubling Bifurcations in Conservative Systems, Tech. Phys. Lett., 2008, vol. 34, no. 11, pp. 985-988; see also: Pis'ma Zh. Tekh. Fiz, 2008, vol. 34, no. 22, pp. 72-80.
[19] Kuznetsov, A. P., Savin, A.V., and Savin, D.V., Conservative and Dissipative Dynamics of Ikeda Map, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 2006, vol. 14, no. 2, pp. 94-106 (Russian).
[20] Kuznetsov, A. P., Savin, A.V., and Savin, D. V., Features in Dynamics of an Almost Conservative Ikeda Map, Tech. Phys. Lett., 2007, vol. 33, no. 2, pp. 122-124; see also: Pis'ma Zh. Tekh. Fiz., 2007, vol. 33, no. 3, pp. 57-63.
[21] Kuznetsov, A. P., Savin, A.V., and Savin, D. V., On Some Properties of Nearly Conservative Dynamics of Ikeda Map, Nonlinear Phenom. Complex Syst., 2007, vol. 10, no. 4, pp. 393-400.
[22] Kuznetsov, A. P., Savin, A.V., and Savin, D. V., On Some Properties of Nearly Conservative Dynamics of Ikeda Map and Its Relation with the Conservative Case, Phys. A, 2008, vol. 387, no. 7, pp. 1464-1474.
[23] Kuznetsov, S.P., Kuznetsov, A. P., and Sataev, I.R., Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps, J. Stat. Phys., 2005, vol. 121, nos. 5-6, pp. 697-748.
[24] Kuznetsov, S.P. and Sataev, I.R., New Types of Critical Dynamics for Two-Dimensional Maps, Phys. Lett. A, 1992, vol. 162, no. 3, pp. 236-242.
[25] Kuznetsov, Yu. A., Elements of Applied Bifurcation Theory, 3rd ed., Appl. Math. Sci., vol. 112, New York: Springer, 2004.
[26] Kuznetsov, Yu. A. and Levitin, V. V., CONTENT: A Multiplatform Environment for Analyzing Dynamical Systems, Amsterdam: Dynamical Systems Laboratory, Centrum voor Wiskunde en Informatica, 1997 (available at http://www.math.uu.nl/people/kuznet/CONTENT).
[27] Moser, J., On Invariant Curves of Area-Preserving Mappings of an Annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962, vol. 1962, pp. 1-20.
[28] Rech, P. C., Beims, M. W., and Gallas, J. A. C., Basin Size Evolution between Dissipative and Conservative Limits, Phys. Rev. E, 2005, vol. 71, no. 1, 017202, 4 pp.
[29] Reichl, L.E., The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations, 2nd ed., New York: Springer, 2004.
[30] Savin, A.V. and Savin, D.V., The Basins of Attractors in the Web Map with Weak Dissipation, Nelin. Mir, 2010, vol. 8, no. 2, pp. 70-71 (Russian).
[31] Savin, D.V., Kuznetsov, A. P., Savin, A.V., and Feudel, U., Different Types of Critical Behavior in Conservatively Coupled Hénon Maps, Phys. Rev. E, 2015, vol. 91, no. 6, 062905, 9 pp.
[32] Savin, D.V., Savin, A.V., Kuznetsov, A. P., Kuznetsov, S.P., and Feudel, U., The Self-Oscillating System with Compensated Dissipation: The Dynamics of the Approximate Discrete Map, Dynam. Syst., 2012, vol. 27, no. 1, pp. 117-129.
[33] Simô, C. and Vieiro, A., Resonant Zones, Inner and Outer Splittings in Generic and Low Order Resonances of Area Preserving Maps, Nonlinearity, 2009, vol. 22, no. 5, pp. 1191-1245.
[34] Zaslavsky, G. M., The Physics of Chaos in Hamiltonian Systems, 2nd ed., London: Imperial College Press, 2007.
