Appearance of chaos and hyperchaos in evolving pendulum network
V. O. Munyaev, D. S. Khorkin, M. I. Bolotov, L. A. Smirnov, G. V. Osipov
AAppearance of chaos and hyperchaos in evolving pendulum network
Vyacheslav O. Munyaev, Dmitry S. Khorkin, Maxim I. Bolotov, Lev A. Smirnov,
1, 2 and Grigory V. Osipov Department of Control Theory, Scientific and EducationalMathematical Center “Mathematics of Future Technologies”,Nizhny Novgorod State University, Gagarin Av. 23, Nizhny Novgorod,603950 Russia Institute of Applied Physics, Russian Academy of Sciences, Ul’yanova Str. 46,Nizhny Novgorod, 603950 Russia (Dated: 14 December 2020)
The study of deterministic chaos continues to be one of the important problems inthe field of nonlinear dynamics. Interest in the study of chaos exists both in low-dimensional dynamical systems and in large ensembles of coupled oscillators. Inthis paper, we study the emergence of spatio-temporal chaos in chains of locallycoupled identical pendulums with constant torque. The study of the scenarios ofthe emergence (disappearance) and properties of chaos is done as a result of changesin: (i) the individual properties of elements due to the influence of dissipation inthis problem, and (ii) the properties of the entire ensemble under consideration,determined by the number of interacting elements and the strength of the connectionbetween them. It is shown that an increase of dissipation in an ensemble with afixed coupling force and elements number can lead to the appearance of chaos asa result of a cascade of period doubling bifurcations of periodic rotational motionsor as a result of invariant tori destruction bifurcation. Chaos and hyperchaos canoccur in an ensemble by adding or excluding one or more elements. Moreover, chaosarises hard, since in this case the control parameter is discrete. The influence of thecoupling strength on the occurrence of chaos is specific. The appearance of chaosoccurs with small and intermediate coupling and is caused by the overlap of thevarious out-of-phase rotational modes regions existence. The boundaries of theseareas are determined analytically and confirmed in a numerical experiment. Chaoticregimes in the chain do not exist if the coupling strength is strong enough.1 a r X i v : . [ n li n . C D ] D ec etworks of interacting nonlinear oscillators are encountered in various natu-ral and technical situations. They govern the behavior of coupled neurons andcardiac cells, many physical devices such as arrays of Josephson junctions andlasers, many engineering applications such as phase locked loops and electricpower machines. Because of many applications, the study of collective dynam-ics, especially synchronization spatio-temporal chaos are some of central sub-jects in nonlinear dynamics for the past three decades. Many significant andimportant results has been obtained. In our paper we present a theoretical andcomputational study of complex dynamics in the chains of coupled pendulums.The influence of individual (dissipation of pendulums) and collective (couplingstrength and number of elements) properties of population is analyzed and dis-cussed.I. INTRODUCTION The study of spatio-temporal dynamics in ensembles of nonlinear oscillators of variousnature is one of the most popular and interesting directions in modern nonlinear dynamics.The behavior of elements of such ensembles can be roughly divided into three types: (i) fullyorganized (consistent, coherent, synchronous), (ii) completely disorganized (inconsistent,incoherent, asynchronous), and (iii) intermediate between (i) and (ii) (partially consistent,partially coherent, partially synchronous, such as chimeric or cluster states). The mostdifficult variant of partially synchronous behavior is spatio-temporal chaos .Chaotic oscillations are one of the most common phenomena in nonlinear dynamical sys-tems of dimension three and higher. It can be assumed that the main mechanisms of theappearance of chaotic dynamics are currently well understood. . Chaos can be conserva-tive and dissipative. The mathematical image of dissipative chaos is a strange attractor –nontrivial stable closed invariant set with unstable behavior of trajectories on it. Below welist the main, most important, well-known scenarios of the appearance of strange attractorstypical for wide classes of dynamical systems:(i) bifurcation of the destruction of an invariant torus;(ii) intermittency; 2iii) infinite sequence of period doubling bifurcations of periodic motions;(iiii) internal crises of attractors.All these scenarios of the appearance (and disappearance) of dynamic chaos have beenstudied in detail for low-dimensional systems . From the point of view of bifurcationtheory, nothing unusual happens when chaos occurs in large distributed systems – discretenetworks and media . However, it is clear that the behavior of ensembles not only affects theindividual dynamics of its components, but also the different characteristics of the interactionof elements: the type and strength of connections, configuration and capacity of the networkand others.In this work, the complication of space-time behavior up to the onset of chaos is inves-tigated in a chain of pendulum-type systems depending on (i) the dissipation of a partialelement, (ii) the characteristics of the interaction between the elements – the strength ofconnections and (iii) the number of interacting elements.The work is structured as follows. Section II describes the studied model of a chainof pendulum elements. In the section III, in-phase rotational motion and the issue of itsstability are considered, asymptotic expressions are given for the boundaries of instabilityregions depending on the coupling strength of the elements in the chain. Section IV providesa detailed description of possible variants of the in-phase mode unstable regions intersection,depending on the change in the number of elements in the chain. Further, in section V, ascenario of the development of chaotic dynamics in a chain with an increase in the dissipationparameter with a fixed number of elements, as well as in the case of an increase in the numberof elements with a fixed large dissipation is described. In the Conclusion, the main findingson the presented results of the work are formulated. II. MODEL
We consider N coupled pendulum-type systems described by a system of ordinary differ-ential equations: ¨ ϕ + λ ˙ ϕ + sin ϕ = γ + K sin ( ϕ − ϕ ) , ¨ ϕ n + λ ˙ ϕ n + sin ϕ n = γ + K [sin ( ϕ n +1 − ϕ n ) + sin ( ϕ n − − ϕ n )] , n = 2 , . . . , N − , ¨ ϕ N + λ ˙ ϕ N + sin ϕ N = γ + K sin ( ϕ N − − ϕ N ) , (1)3here λ – damping parameter, responsible for dissipative processes in the system, γ –constant torque is the same for all N pendulums, K – parameter of coupling strengthbetween elements.The system (1) is used to describe the behavior of interacting pendulums (we presentour results in the interpretation of pendulums), connected Josephson junctions . Itis used to describe processes in superconductors , molecular biology , phase synchro-nization systems . Moreover, such a system can be considered as a generalization of theKuramoto model taking into account the inertia and intrinsic nonlinearity of the ensembleelements .Since the elements of the ensemble are identical, the following spatially homogeneousmodes exist for any values of the parameters:• equilibrium state with phase coherent elements ϕ = ϕ = . . . = ϕ N = ψ = const;• rotational motion of the elements with coherent phase ϕ ( t ) = ϕ ( t ) = . . . = ϕ N ( t ) ≡ ψ ( t ) .These modes satisfy the pendulum equation: ¨ ψ + λ ˙ ψ + sin ψ = γ. (2)The equation (2) has been carefully examined in .In this study, we are interested in regular and chaotic rotational modes, therefore, wefocus on the values of the parameters λ and γ , for which in the system (1) there is an in-phase rotation periodic motion – in-phase regime (IPR), which corresponds to the area onthe plane ( λ, γ ) , bounded below by the Tricomi bifurcation curve .In our previous works, we considered the features of the rotational dynamics of particularvariants of the system (1). In two coupled pendulums were investigated for the casesof symmetric and asymmetric coupling, respectively. In the article the case of a chain ofthree locally coupled elements, and in N globally coupled pendulums were considered. III. IN-PHASE REGIME AND ITS STABILITY
The possibility of realizing rotational regimes of varying degrees of complexity in thesystem (1) is directly related to the issue of stability of the in-phase regime, the development4f instability of which leads to the appearance of out-of-phase regimes (OPR). For thisreason, let us briefly consider the issue of stability of the in-phase mode. For this, welinearize the system (1) in the neighborhood of φ ( t ) , representing ϕ n ( t ) in the form ϕ n ( t ) = φ ( t ) + δϕ n ( t ) . The linearized system of equations for perturbations δϕ n ( t ) has the form: δ ¨ ϕ + λδ ˙ ϕ + cos φ ( t ) δϕ = K ( δϕ − δϕ ) ,δ ¨ ϕ n + λδ ˙ ϕ n + cos φ ( t ) δϕ n = K ( δϕ n − − δϕ n + δϕ n +1 ) , n = 2 , . . . , N − ,δ ¨ ϕ N + λδ ˙ ϕ N + cos φ ( t ) δϕ N = K ( δϕ N − − δϕ N ) . (3)We pass in the system (3) to the normal coordinates ψ , ψ , . . . , ψ N and obtain the followingsystem of independent equations : ¨ ψ n + λ ˙ ψ n + [cos φ ( t ) − Kµ n ] ψ n = 0 , (4)where µ n = − nπ/N )] . In our previous works it was shown that themodes ψ , ψ , . . . , ψ N − can become unstable, which leads to the appearance of out-of-phaserotational modes, while the boundaries of the instability interval ( K ( n )1 , K ( n )2 ) are determinedby the expressions K ( n )1 , = K ∗ , / | µ n | , (5)where K ∗ , ( λ, γ ) are, respectively, the left and right boundaries of the region of instabilityof the in-phase solution of the equation ¨ ψ + λ ˙ ψ + [cos φ ( t ) + K ∗ ] ψ = 0 , which can bedetermined by the asymptotic expressions K ∗ , = 14 (cid:20) γ λ ∓ (cid:112) − γ + 12 λ γ (cid:21) + O (cid:18) λ γ (cid:19) . (6)Thus, in the system (1), for certain values of the control parameters λ , γ , an N − regionof instability of the in-phase mode φ ( t ) can exist. Moreover, these instability regions do notoverlap at low dissipation ( λ (cid:28) ). IV. INTERSECTION OF IN-PHASE REGIME INSTABILITY REGIONS
In this section, we show that the mutual arrangement of the regions of instability of thein-phase regime significantly affects the complexity of the dynamic regimes being realized.As a result of direct numerical simulation of the system (1) in the parameter area ( λ, γ ) ,maps of rotational modes were built, shown in Fig. 1, analyzing which we see that there is5 complication of structures with an increase of dissipation parameter. There is a conver-gence and overlap of the zones of various structures existence, as well as the emergence ofmultistability, which ultimately leads to the emergence of a regime of dynamic chaos. FIG. 1. Maps of rotational modes are realized in the system (1) for N = 7 , γ = 0 . for differentvalues of the λ and K . Black lines – boundaries of instability regions of in-phase rotational motion φ ( t ) , obtained numerically. The color indicates the types of realized rotational modes: green – (2 : 2 : 2 : 1) regimes, yellow – (1 : 1 : . . . : 1) regimes. Red shading indicates areas of chaos. Mapsare obtained by inheriting the initial conditions as a result of increasing (a) and decreasing (b) theparameter K . Let us analyze how the emergence, evolution and overlapping of the zones of instability ofthe IPR occurs. We show that for fixed values of the parameters λ and γ increased numberof the chain elements N leads to the following scenarios of overlap regions common-modeinstability:(i) one of the instability regions is separated from the rest when new elements are added;(ii) there is such a critical number of elements N ∗ , when an exceeding instability region isseparated from the rest;(iii) for any number of elements N , all regions of instability have intersections with others.From the expression (5) and the monotonic increase of the expressions µ , µ , . . . , µ N − itfollows that the sequences of the left K (1)1 , K (2)1 , . . . , K ( N − and right K (1)2 , K (2)2 , . . . , K ( N − boundaries of instability regions increase monotonically (the width of instability regions K ( n )2 − K ( n )1 also increases monotonically). Thus, two near regions of instability can partiallyoverlap ( K ( n +1)1 ≤ K ( n )2 ), or be separated by a stability window ( K ( n +1)1 > K ( n )2 ). Thus, thezone of instability cannot entirely enter into another zone of instability. Let us introduce6he number n ∗ ( N ) , which determines the number of the first overlapping zones of instabilitywith increasing bond strength parameter K . For example, if n ∗ (8) = 3 , then the first threezones of instability overlap, then the stability window follows (the location of the other fourzones does not interest us, it is important that they are separated from each other). Theexpression n ∗ (8) = 7 means that all zones of instability have overlapped and merged intoone. It’s obvious that ≤ n ∗ ( N ) ≤ N − . Formally, n ∗ ( N ) is defined by the expression n ∗ ( N ) = min (cid:16)(cid:110) n ∈ { , , . . . , N − } | K ( n +1)1 − K ( n )2 > (cid:111) ∪ { N − } (cid:17) . (7)To find an explicit expression for the number n ∗ ( N ) , consider the expression continualizedby n K ( n +1)1 − K ( n )2 , introduce the distance function between instability regions L ( x ) = K ( x +1)1 − K ( x )2 . Obviously, the function L ( x ) is periodic: L ( x + 2 N ) = L ( x ) . In addition,there are gaps at the points x = (2 k + 1) N and x = (2 k + 1) N − . It can be verified that lim x → N L ( x ) = −∞ and lim x → N − L ( x ) = + ∞ . Thus, the intervals [ − N, N − and [ N − , N ] contain at least one root of the equation L ( x ) = 0 . The direct solution of the equation L ( x ) = 0 leads to the roots x ± k = 2 Nπ arctan (cid:32) cos (cid:0) π N (cid:1) ± (cid:112) K ∗ /K ∗ sin (cid:0) π N (cid:1) (cid:33) + 2 N k. (8)The roots x − and x +0 are on the interval [ − N, N ] . We conclude that each interval [ − N, N − and [ N − , N ] contains one root, and the smaller root x − lies in the interval [ − N, N − .Thus, for x > x − we have L ( x ) > : the length of the gap between the regions of instabilitybecomes positive, those zones no longer overlap. Then n ∗ ( N ) = max (cid:40) , (cid:38) Nπ arctan (cid:32) cos (cid:0) π N (cid:1) − (cid:112) K ∗ /K ∗ sin (cid:0) π N (cid:1) (cid:33) (cid:39)(cid:41) , (9)where (cid:100) . . . (cid:101) – round up. Thus, as the parameter K increases, only the first n ∗ ( N ) instabilityregions can overlap. All other unstable regions (if they exist) are separated from each otherby stability windows. It is also useful to note that due to the monotonic growth of thefunction under the round-up brackets in (9), as the number N grows, the sequence n ∗ ( N ) isnon-decreasing. In other words, an increase of the chain elements number N can only leadto an increase in the number of intersecting regions of instability of IPR.Let us define the right boundary K ( n ∗ )2 of the instability zone formed by the overlappingof the instability areas: K ( n ∗ )2 = K ∗ (cid:16) n ∗ ( N ) N π (cid:17) . (10)7hen N → ∞ , there is a limit lim N →∞ n ∗ ( N ) /N = 1 . In fact, for sufficiently large N (when cos (cid:0) π N (cid:1) > (cid:112) K ∗ /K ∗ ) n ∗ ( N ) = (cid:38) Nπ arctan (cid:32) cos (cid:0) π N (cid:1) − (cid:112) K ∗ /K ∗ sin (cid:0) π N (cid:1) (cid:33) (cid:39) , (11)the following inequalities are satisfied π arctan (cid:32) cos (cid:0) π N (cid:1) − (cid:112) K ∗ /K ∗ sin (cid:0) π N (cid:1) (cid:33) ≤ n ∗ ( N ) N ≤ − N . (12)Then, calculating the limits on both sides, by the two attendant theorem we arrive at theindicated limit. As a result, we arrive at lim N →∞ K ( n ∗ )2 = + ∞ . (13)The left boundary of the instability zone formed by the overlap of individual instabilityregions is defined as K (1)1 . When N → ∞ , lim N →∞ K (1)1 = K ∗ / . (14)For a more detailed study of the behavior of the instability regions we consider the number n ∗ ( N ) = N − − n ∗ ( N ) ( ≤ n ∗ ( N ) ≤ N − ), that determines the number of isolated (bystability windows) unstable regions. From the expression (9) we get that n ∗ ( N ) = min (cid:40) N − , (cid:36) Nπ (cid:32) π − arctan (cid:32) cos (cid:0) π N (cid:1) − (cid:112) K ∗ /K ∗ sin (cid:0) π N (cid:1) (cid:33)(cid:33) (cid:37)(cid:41) − . (15)Let’s examine the behavior of the function under the round-down brackets. Direct computa-tion can verify that its second derivative with respect to N is always negative for ∀ N ≥ (weare interested in the values N ≥ ). Thus, for N ≥ , the first derivative decreases monoton-ically with increasing N . At the point N = 1 , the derivative is K ∗ K ∗ + K ∗ + π arctan (cid:113) K ∗ K ∗ > ;as N → + ∞ , the value of the derivative tends to +0 . Therefore it can be argued that thederivative all N ≥ is positive and of the function increases monotonically with an increaseof the number N ≥ . Then the sequence n ∗ ( N ) , like the sequence n ∗ ( N ) , is non-decreasing. Scenario A.
Let us find a condition under which an isolated region of instability of IPRexists for any N . From the non-decreasing sequence n ∗ ( N ) it follows that a necessaryand sufficient condition is the fulfillment of the equality n ∗ (3) = 1 . From (15) we find K ∗ /K ∗ < . The example of this scenario is demonstrated in Fig. 2 (a).8 cenario B. Similarly, one can find a condition under which all regions of instability overlapfor any N . A necessary and sufficient condition is the equality lim N → + ∞ n ∗ ( N ) = 0 . From (15)it follows lim N → + ∞ n ∗ ( N ) = (cid:22) (cid:16) − (cid:112) K ∗ /K ∗ (cid:17) − (cid:23) − . Then we get K ∗ /K ∗ > (Fig. 2(b)). Scenario C.
When < K ∗ /K ∗ < for small N all instability regions overlap, but startingfrom some number N isolated regions appear separated by stability windows (Fig. 2(c)).Another conclusion from the non-decreasing sequences n ∗ ( N ) and n ∗ ( N ) is the fact that K (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) K K -1 -1 N (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) (cid:1) (cid:2) N =14 N =10 N =12 (a) (b) (c) FIG. 2. Lower panel. The regions of instability of the ψ n modes (orange solid lines), and thecorresponding regions of instability of the IPR (red dashed lines), found from the expression (5)and the numerical calculation of the boundaries K ∗ , , for γ = 0 . and different N and λ dependingon the bond strength K . Blue solid lines represent stable IPR. Over panel. Instability regions of ψ n modes and the corresponding types of OPR for indicated by arrow number N . The numberin square brackets indicates the number of in-phase clusters. (a) λ = 0 . For any N , there is atleast one region of in-phase instability, isolated from the others by stability windows. (b) λ = 0 . .Since some N (here N = 6 ) appears common mode instability region isolated from other stabilitywindows. (c) λ = 0 . . For any number of elements N , overlap of all instability regions of the ψ n modes is observed with the formation of instability area of the in-phase mode without stablewindows. Over panel. Instability regions of ψ n modes and the corresponding types of out-of-phaserotation at N = 15 . The number in square brackets indicates the number of in-phase clusters. N increases by one, either one new isolated region ofinstability appears and the number of overlapping regions does not change, or the numberof overlapping regions increases by one, but the number of isolated regions does not change.10 . THE CHAOTIC DYNAMICS DEVELOPMENT Here we analyze in detail the complex periodic and chaotic regimes occurrence mech-anisms, and the various OPRs existence regions overlapping role. Below we present theresults of computations. First, we introduce the synchronicity parameter
Ξ = 1 N ( N − N (cid:88) n ,n =1 max ≤ t ≤ T | ˙ ϕ n ( t ) − ˙ ϕ n ( t ) | , (16)which characterizes the rotational motion phase synchronization degree. The value Ξ = 0 shows that the rotational regime under consideration is in-phase, the values Ξ > indicatethe realization of OPR. For a more detailed analysis, we present bifurcation diagrams, as wellas graphs that show the local maxima of the oscillator frequencies and the largest Lyapunovexponent, which positive value indicates the dynamic chaos regime presence in the system.Let us investigate the rotational regimes dynamics in the chain depending on the param-eter K for different dissipation parameter λ values and different elements number N . Ourcomputational experiments show that with a parameter K value increase the IPR loses itsstability softly, but the corresponding OPR stability loss with a further increase in K occursin a hard manner. Thus, hysteresis occurs when there is a range of K values where in-phasemode coexists with out-of-phase mode. We will call such an interval of in-phase regime in-stability as “right”. If the in-phase mode loses its stability hardly and the out-of-phase modeloses its stability softly with K increasing then we will call such an instability range “left”.The scenario for the appearance and existence of “left” and “right” regions for a differentchain elements number N is as follows: N = 2 – one “left” (1 : 1) region; N = 3 – one “left” (2 : 1) , one “right” (1 : 1 : 1) ; N = 4 – two “left” (1 : 1 : 1 : 1) and (2 : 2) , one “right” (1 : 1 : 1 : 1) ; N = 5 – two “left” (2 : 2 : 1) and (1 : 1 : 1 : 1 : 1) , two “right” (2 : 2 : 1) and (1 : 1 : 1 : 1 : 1) ;etc.Thus, the transition from an even number of elements to an odd number is accompaniedby the addition of one “right” area; similarly, from an odd number to an even number one“left” area is added.Fixed number of elements N . Consider the chain dynamics for N = 7 , γ = 0 . , λ = 0 . .In this case, with the parameter K value increase the out-of-phase (2 : 2 : 2 : 1) and11 . . . : 1) regimes associated with the in-phase rotation instability development andcorresponding to the ψ n ( n = 1 , . . . , ) modes are sequentially realized in the system (1).Note that modes with the same designation can differ from each other and represent differentobjects in the system (1) phase space. Consider the range of the parameter K values whenthe rotational motion (2 : 2 : 2 : 1) is realized for the first time. Here, as the parameter K increases, the in-phase periodic rotation φ ( t ) undergoes a period-doubling bifurcation at K ≈ . . In this case, stable in-phase π -periodic motion gives rise to stable π -periodic (2 : 2 : 2 : 1) motion, but φ ( t ) loses its stability. The bifurcation diagram (Fig. 3) showsthat there is also an unstable motion (2 : 2 : 2 : 1) , which at K ≈ . arises froman unstable in-phase motion as the period doubling bifurcation result, while φ ( t ) becomesstable again. Further, as the parameter K increases, stable and unstable (2 : 2 : 2 : 1) motions merge and disappear as a result of saddle-node bifurcation. With a further increasein the coupling parameter K the regimes (1 : 1 : . . . : 1) and (2 : 2 : 2 : 1) appear in asoft manner, and then regimes (1 : 1 : . . . : 1) , (2 : 2 : 2 : 1) , (1 : 1 : . . . : 1) appear in ahard manner. Thus, when the parameter K changes there are three “right” and three “left”in-phase regime φ ( t ) instability regions in the system (1) with small dissipation λ . Notealso that for λ = 0 . π -periodic rotational regimes (2 : 2 : 2 : 1) and (1 : 1 : . . . : 1) ,referring to “right” instability intervals, can coexist on the interval of the coupling strength . ≤ K ≤ . , although the corresponding instability regions do not intersect.Let us analyze further the dissipation parameter λ influence on the space-time dynam-ics complication in the system under consideration. As λ increases, the arising out-of-phaserotations can also undergo the following period-doubling bifurcations, leading to the appear-ance of π , π , π , etc. regimes. For example, for λ = 0 . the mode undergoes severalperiod doubling bifurcations on the interval . < K < . . . . : 1) (Fig. 3).A further increase in the dissipation parameter λ leads to the in-phase mode instabilityintervals intersections appearance. For λ = 0 . four regions corresponding to the smallestcoupling parameter K values intersect (Fig. 4), while the π -periodic motions correspondingto the (1 : 1 : . . . : 1) regime are unstable and rotations with a larger number of turns by π ,which arise during subsequent π -periodic trajectories bifurcations, are realized. In the “left”region corresponding to (1 : 1 : . . . : 1) regime with chaotic motions appears as a result of aperiod doubling bifurcations cascade, what is confirmed by the positive values of the largestLyapunov exponent. In the case λ = 0 . , when five instability regions intersect, chaotic12egimes are observed already at several intervals. The transition from periodic to chaoticdynamics can occur through the Neimark-Sacker bifurcation leading to the emergence ofan invariant torus, and its subsequent destruction through the torus destruction bifurcationwhich ultimately leads to the chaotic attractor birth. Here, the curves corresponding to π -periodic rotational regimes have a rather nontrivial structure with different closures witheach other (Fig. 5). With a further increase in the dissipation parameter λ the regions ofchaotic dynamics gradually increase and merge with one another (Fig. 6).Fixed dissipation parameter. Let us consider the regimes evolution in the case of thefixed dissipation parameter value, for example, λ = 0 . , with a change in the chain elementsnumber N and the coupling parameter K . First of all, we will be interested in the transitionsfrom regular to chaotic regimes and the role played by the appearance and evolution of theIPR instability regions during these transitions.Using the general expression (5), we can determine the fraction u N ( K ) of modes ψ n thatlose their stability for some fixed value of the coupling parameter K : u N ( K ) = θ ( K − K ∗ / N − (cid:36) Nπ arccos (cid:32)(cid:114) K ∗ K (cid:33)(cid:37) − θ ( K − K ∗ / N − (cid:36) Nπ arccos (cid:32)(cid:114) K ∗ K (cid:33)(cid:37) , (17)where θ is the Heaviside step function. Since K ∗ /K ∗ > , then for any N all instabilityregions of ψ n modes overlap, which leads to the formation of a single, global in-phase rotationinstability region. Find out what happens in the limit N → ∞ . For this case, we introducethe fraction u ∞ ( K ) determined by the expression u ∞ ( K ) = 2 π (cid:34) θ ( K − K ∗ /
4) arccos (cid:32)(cid:114) K ∗ K (cid:33) − θ ( K − K ∗ /
4) arccos (cid:32)(cid:114) K ∗ K (cid:33)(cid:35) . (18)The Fig. 7 (c) demonstrates the dependence of the unstable modes fraction on the couplingparameter K for fixed N = 7 and in the limit N → ∞ . The figure shows that the largestunstable modes number is observed at small and intermediate coupling parameter values( . < K < ). As a result of the IPR stability loss new OPR appear, which in turn canalso become unstable, which leads to the new and new OPRs emergence (see Figs. 3-6).The general property of the obtained dependence (18) is its maximum at the point K = K ∗ / , which is demonstrated in Fig. 7c, obtained for the considered case λ = 0 . . Thus,the maximum ψ N mode instability regions overlapping density is observed at small andintermediate K values located near the left boundary of the in-phase rotation instabilityinterval, i.e. where the “left” instability regions appear.13ur computational experiments show that chaotic regime precisely observed in the areaof greatest accumulation and intersection of various instability regions. Fig. 7 (a) shows thearea where chaotic dynamics is observed on the ( K, N ) plane at λ = 0 . . It can be seen thatalthough the right boundary K ( N − of the instability region moves rather quickly to theright as N increases, chaotic regimes are realized only in a rather narrow range of couplingstrength values . < K < . At small K , due to weak interaction, chaotization of initiallyregular rotations does not occur. In the case of strong couplings, the pendulums interactionleads to the rotations regularization. In this case, the resulting modes are not necessarilyin-phase. The in-phase rotational mode is established only with a large coupling strength(see the K ( N − curve in Fig. 7 (a)).Analyzing Fig. 7, the following conclusions can be drawn. For a fixed value of K witha change in the chain length N , two main scenarios of the chaotic behavior emergence andexistence are realized:(i) for small K ∼ . , the chaos appearing at N = 2 does not disappear with an increase in N ;(ii) for . (cid:46) K (cid:46) . : a) the addition of one (sometimes two or more) new elements canlead to both chaos and regularization of modes; b) when two (sometimes more) elementsare added, the chaotic regime leaves it chaotic, and the regular regime remains regular. Theparity of the elements number in the chain plays an important role here. In this case, thecloser K is to 1, the less dense of the overall chaos area becomes, i.e. the regular behaviorislands number and size increase. For K > , no chaotic behavior was found for any N .A series of computational experiments was carried out in which the chain length N variedwith time. In one series, the number of elements was changed by adding (or excluding) oneelement. In the second series, several elements were added to the chain (excluded from thechain). In all cases, both of the above scenarios were observed.For fixed N , the regular and chaotic regimes evolution with increasing K is qualitativelythe same for different chain lengths N . Namely, chaos appears practically at the same valueof K ≈ . and is realized continuously without regular behavior windows up to K ≈ . .Further, the chaotic regime can alternate with the regular one.In studied chain of pendulums there is the oppotunity of existence of hyperchaos. Wecomputed the spectrum of Lyapunov exponents in the chain of length N = 7 and γ = 0 . , λ = 0 . in dependence on coupling strength K (Fig. 7(b)). There are several intervals of K ψ n . FIG. 3. Local bifurcation diagram of periodic rotational regimes (a). Ξ – synchronicity parameter.Round markers show π -periodic rotational regimes. Filled markers correspond to stable rotationalregimes, unfilled markers – to unstable ones. The line without markers corresponds to the in-phase π -periodic rotational regime, the solid line, to the stable one, and the dotted line, to theunstable one. Diagram is obtained in two ways: with increasing and decreasing parameter K . Localfrequency maxima max ˙ ϕ m (b), Parameters: N = 7 , γ = 0 . , λ = 0 . , m = 2 . IG. 4. The same as in Fig. 3. Parameters: N = 7 , γ = 0 . , λ = 0 . , m = 4 .FIG. 5. The same as in Fig. 3. Parameters: N = 7 , γ = 0 . , λ = 0 . , m = 3 .FIG. 6. Local frequency maxima max ˙ ϕ m . Parameters: N = 7 , γ = 0 . , λ = 0 . , m = 3 . IG. 7. (a) Chaotic regimes map in the system (1) depending on the coupling strength K and theelements number N at γ = 0 . and λ = 0 . . Black lines denote the left K (1)1 and right K ( N − boundaries of the IPR φ ( t ) instability range. White color denotes regular rotational modes (largestLyapunov exponent Λ = 0 ). In colored area the largest Lyapunov exponent is positive. (b) Thespectrum (first six maximal) Lyapunov exponents. There are intervals with four, three, two andone positive exponents. (c) Dependence of the unstable modes ψ n for N = 7 (blue lines) and for N → ∞ (blue dashed curve). The red dashed line indicates the in-phase regime instability regionfor N = 7 . I. CONCLUSION
The paper investigates the appearance and disappearance of a chaotic rotational regimein a chain of locally coupled identical pendulums. The discovered scenarios of the space-timechaos emergence in the ensemble under consideration are typical both for low-dimensionallumped dynamic systems and for multidimensional systems distributed over space. Thisis the birth of a chaotic attractor: a) through a period doubling bifurcations sequence ofperiodic motions; b) through the invariant tori destruction. The chaos appearance studywas carried out with a change in: a) the dissipation parameter of an individual element; b)the coupling parameter and c) the interacting elements number. It is shown that an increasein dissipation in an ensemble with a fixed value of the coupling and the elements number canlead to the chaos appearance. This is due to the fact that with an increase in the dissipationparameter λ (this leads to an approach to the Tricomi curve), the individual pendulumrotations become significantly inhomogeneous in time: intervals of fast and slow changes inthe phase φ can be distinguished. A chaotic rotational regime can arise when even two suchpendulums interact. Chaos in an ensemble can arise when the interacting elements numberchanges. Since in this case the control parameter is discrete, the chaos occurrence is rigid.With a fixed dissipation and coupling parameters values and with a change in the chainlength N two main scenarios of the chaotic behavior emergence and existence are realized:a) the occuring chaos at N = 2 does not disappear with increasing N ; b) the addition ofone (sometimes two or more) new elements can lead to both chaotization and regularizationof modes, and when two (sometimes more) elements are added, the chaotic regime leaveschaotic and the regular regime remains regular. In this case, the parity of the chain elementsnumber plays an important role.The coupling strength influence on the chaos occurrence is specific. For any chain lengthsconsidered in this paper, the region of chaos existence is bounded and for N > does notdepend on N . This is the range of small and intermediate coupling . (cid:46) K (cid:46) . . Thechaos existence in this range is due to the various out-of-phase rotational regimes existenceregions overlap. There are no chaotic regimes were found in the chain with a strong coupling.It is important to note, that with increase of number of coupled elements the transition tohyperchaotic behavior is possible. 18 CKNOWLEDGMENTS
The numerical calculations in this work were supported by the Russian Science Founda-tion (grant No. 19-12-00367) and the analytical studies were supported by the Ministry ofScience and Higher Education of Russian Federation (project No. 0729-2020-0036).19
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