Mechanical systems with hyperbolic chaotic attractors based on Froude pendulums
Vyacheslav P. Kruglov, Sergey P. Kuznetsov, Yuliya V. Sedova
aa r X i v : . [ n li n . C D ] S e p Mechanical systems with hyperbolic chaotic attractors basedon Froude pendulums
Vyacheslav P. Kruglov
Kotel’nikov Institute of Radioengineeringand Electronics of RASSaratov Branch, RussiaUdmurt State UniversityIzhevsk, Russia [email protected]
Sergey P. Kuznetsov
Kotel’nikov Institute of Radioengineeringand Electronics of RASSaratov Branch, RussiaUdmurt State UniversityIzhevsk, Russia [email protected]
Yuliya V. Sedova
Kotel’nikov Institute of Radioengineeringand Electronics of RASSaratov Branch, Russia [email protected]
September 4, 2019
Abstract
We discuss two mechanical systems with hyperbolic chaotic attractors of Smale– Williams type. Both models are based on Froude pendulums. The first system iscomposed of two coupled Froude pendulums with alternating periodic braking. Thesecond system is Froude pendulum with time-delayed feedback and periodic braking.We demonstrate by means of numerical simulations that proposed models have chaoticattractors of Smale – Williams type. We specify regions of parameter values at whichthe dynamics corresponds to Smale – Williams solenoid. We check numerically hyper-bolicity of the attractors.
Keywords— hyperbolic chaotic attractors, Smale – Williams solenoid, Bernoulli map
Uniformly hyperbolic attractors [1] are genuine chaotic and consist only of saddle trajectories. Atevery point of a uniformly hyperbolic attractor its tangent space can be decomposed into directsum of two subspaces, stable and unstable. It is important part of definition that all possible anglesbetween any vector from stable subspace and any vector from unstable subspace are distanced from ero at every point of the attractor. Uniformly hyperbolic attractors are structurally stable. Thatmeans they occupy an open set in the parameter space [1].Smale – Williams solenoid [2,3] is well-known example of uniformly hyperbolic chaotic attractor.It appears in the phase space of a descrete dynamical system if a torus-form domain undergoes inone discrete time step an M -fold longitudinal stretching ( M ≥ is integer), strong transversecompression and folding in a loop located inside the initial torus. With each repetition of thetransformation, the number of curls increases by factor M and in the limit tends to infinity, resultingin a solenoid with a Cantor-like transverse structure. Chaotic nature of the dynamics is determinedby the fact that the transformation of the angular coordinate in this setup corresponds to anexpanding circle map, or the Bernoulli map φ n +1 = M φ n (mod 2 π ) .Among possible examples of hyperbolic chaos in systems of various nature we outline mechanicalmodels as they are easily perceived and interpreted in a frame of our everyday experience. Wepropose to consider two mechanical systems [4, 5] with the basic element being a self-oscillatingFroude pendulum. The first model is composed of two Froude pendulums on a common shaftinteracting by friction [4]. Pendulums are alternately braked by periodic application of externalfrictional forces. The Smale – Williams solenoid occurs as an attractor of the Poincaré stroboscopicmap as we properly specify the system parameters. The second model is composed of only oneFroude pendulum interacting with mechanical time-delay line [5]. The pendulum undergoes periodicbraking. The Smale – Williams solenoid in this model appears in the phase space of infinitedimension of the system with time-delay.We start with a quick overview of the Froude pendulum. Figure 1: (a) Attracting limit cycles for various parameters corresponding to sustained periodic motions asingle Froude pendulum: oscillatory A, B, C, D respectively, at a = 0 . , . , . , . and rotationalD+, E ± , F ± at a = 0 . , . , . . (b) The dependence of the frequency of the self-oscillating regimeson the value of a . Other parameters are b = 0 . , µ = 0 . . On the panel (a) points are marked thatare equilibrium states: unstable focus O and saddle S . On the panel (b) the arrow indicates the situationwhere the frequency of self-oscillations is half the frequency of small oscillations of the pendulum. Froude pendulum
Froude pendulum is a well-known example of mechanical self-oscillator. It is a weight on a rod ofnegligible mass. The rod is attached to a sleeve placed on a shaft rotating at a constant angularvelocity. In the dimensionless form the governing equation of the Froude pendulum reads ¨ x − (cid:0) a − b ˙ x (cid:1) ˙ x + sin x = µ, (1)where x is an angle of the pendulum displacement from the vertical line. The friction torquebetween the shaft and the sleeve gives rise to self-oscillatory motions of the Froude pendulum atcertain range of parameters.If a > , then self-oscillations arise in the system. On the phase plane Fig. 1 (a) the self-oscillatory modes are represented by limit cycles around the equilibrium state O at ( x, ˙ x ) =(arcsin µ, . For a near zero and small µ , a frequency of the self-oscillations is close to the naturalfrequency of the pendulum f = (2 π ) − . With growth of a , the limit cycles increase in size, andthe frequency f decreases. This is due to the fact that the pendulum approaches the saddle point S , ( x, ˙ x ) = ( π − arcsin µ, , corresponding to position pointing upwards, where the motion slowsdown. Further growth of the parameter a leads to a change of the oscillatory movements of thependulum to periodic rotational motions, which correspond to the limit cycles going around thephase cylinder. The dependence of the frequency of the self-oscillating regimes on the value of a isshown on Fig. 1 (b).For further consideration, we select parameters to obtain self-oscillations with the frequencyexactly equal to half of the frequency of small self-oscillations at a close to zero. At µ = 0 . and b = 0 . this is the case if we set a = 0 . . Then the oscillatory process has an essential secondharmonic of the fundamental frequency (this is due to the lack of symmetry of the equation (1)with respect to the substitution x ↔ − x ). If the generated signal acts on a linear oscillator ofnatural frequency ω = 1 , one can observe its resonant excitation under the second harmonic of theself-oscillating system. Let us consider two identical Froude pendulums placed on a common shaft and weakly connectedwith each other by viscous friction, so that the torque of the frictional force is proportional to therelative angular velocity. Let the motion of pendulums be decelerated alternately by attaching abrake shoe providing suppression of the self-oscillations due to the incorporated sufficiently strongviscous friction. Denoting the angular coordinate of the first and the second pendulum as x and y ,and the angular velocities as u and v , we write down the equations ˙ x = u, ˙ u = (cid:2) a − d ( t ) − bu (cid:3) u − sin x + µ + ε ( v − u ) , ˙ y = v, ˙ v = (cid:2) a − d ( t + T / − bv (cid:3) v − sin y + µ + ε ( u − v ) ,d ( t ) = , t < T ,D, T < t < T / , , T / < t < T. d ( t + T ) = d ( t ) . (2) arameters are assigned as follows: a = 0 . , b = 0 . , µ = 0 . , ε = 0 . , D = 0 . , T = 250 , T = T / . (3)To explain the operation of the system (2) we start with the situation when one pendulum isself-oscillating, and the second is suppressed to small oscillations by brake. Due to the fact that theparameters are chosen in accordance with the reasoning of the previous section, the basic frequencyof the self-oscillatory mode is half of that of the second pendulum. Therefore, when the brake shoeis removed from the second pendulum, it will begin to swing in a resonant manner due to the actionof the second harmonic from the first pendulum, and the phase of the oscillations that arise willcorrespond to the doubled phase of the oscillations of the first pendulum. As a result, when thesecond pendulum approaches the sustained self-oscillatory state, its phase appears to be doubledin comparison with the initial phase of the first pendulum. Further, the first pendulum undergoesbraking, and at the end of this stage, its oscillations will be stimulated in turn by the action of thesecond harmonic from the second pendulum, and so on.Since the system (2) is non-autonomous, one can go on to discrete time dynamics by constructingthe Poincaré stroboscopic map. In our case, taking into account a symmetry of the system in respectto substitution x ↔ y, u ↔ v, t ↔ t + T / , it is appropriate to use the mapping in half a periodof modulation, determining the state vector at the instants of time as X n = (cid:26) ( x ( t n ) , u ( t n ) , y ( t n ) , v ( t n )) , if n is odd , ( y ( t n ) , v ( t n ) , x ( t n ) , u ( t n )) , if n is even. (4)The Poincaré map for the vector X n = ( x , x , x , x ) n : X n +1 = F T/ ( X n ) . (5)Since each new stage of the excitation transfer to one or another pendulum is accompanied bya doubling of the phase of oscillations, this corresponds to the Bernoulli map for the phase. If avolume contraction takes place along the remaining directions in the state space of the system, thiswill correspond to occurrence of the Smale – Williams solenoid as attractor of the Poincaré map (5).At the stage of the high-amplitude oscillations their waveform differs significantly from thesinusoidal. In this case, evaluation of the phase through the ratio of the variable and its derivativeas arctangent is not so satisfactory. We define the phase using a value of the time shift of thewaveform with respect to a given reference point, normalized to the characteristic period of theself-oscillatory mode. Let t be a fixed time instant at the activity stage of one of the pendulums,and t , t are the preceding moments of the sign change of the angular velocity from plus to minus,and t > t . Then we can define the phase as a variable belonging to the interval [0, 1] by therelation ϕ = ( t − t )( t − t ) − .Fig. 2 (a) shows a diagram for the phases determined at the end parts of successive stages ofexcitation of the first and second pendulums, obtained in numerical calculations for a sufficientlylarge number of the modulation periods. As can be seen, the mapping for the phase in the topologicalsense looks equivalent to the Bernoulli map ϕ n +1 = 2 ϕ n + const (mod 1) . Indeed, one completeround for the pre-image ϕ n (i.e., a unit shift) corresponds to a double round for the image ϕ n +1 .Fig. 2 (b) shows attractor of the Poincaré stroboscopic map. Although visually the objectlooks like a closed curve, in fact it has a fine transverse structure, visualization of which requireshigh-accuracy calculations, and evolution in discrete time corresponds to jumps of the representingpoint around the loop accordingly to iterations of the Bernoulli map. igure 2: (a) A diagram illustrating transformation of the phases of pendulums in successive stages ofactivity every half a period of modulation. (b) Attractor of the Poincaré stroboscopic map, which is aSmale – Williams solenoid in projection onto the plane of two of variables. At the parameters assigned according to (3), the Lyapunov exponents for the Poincaré mapattractor are Λ = 0 . ± . , Λ = − . ± . , Λ = − . ± . , Λ = − . ± . . The presence of a positive exponent Λ indicates chaotic nature of the dynamics. Its value isclose ln 2 = 0 . . . . , which agrees with the approximate description of the evolution of the phasevariable ϕ by the Bernoulli map. The action of the Poincaré map F in the four-dimensional spaceis accompanied by stretching in the direction corresponding to the phase ϕ and contracting alongthe remaining three directions. This corresponds to the Smale – Williams construction, namely, inthe four-dimensional space.Fig. 3 (a) shows a chart of regimes on the parameter plane ( ε, D ) , i.e. of the coupling parameterversus the dissipation parameter introduced by the brake pad during the braking stages. Theremaining parameters correspond to the situation when the period of relaxation self-oscillations atthe activity stage is exactly twice the period of small oscillations. The structure of the regionsis determined by the excitation transfer from one stage of activity to the next one in the courseof the system operation. In the central part of the chart one can see a broad area SW, wherehyperbolic chaos takes place. To identify Smale – Williams solenoid the topological equivalence ofthe map for the phase to the Bernoulli map was tested visually or automatically, using a speciallydeveloped algorithm. Exit this area down corresponds to the fact that dissipation at the stagesof breaking decreases and becomes too small to provide a sufficient degree of damping of thenatural oscillations of the pendulum after the previous activity stage, which makes a competingcontribution to the stimulation of the oscillatory process at the new activity stage, so that thephase doubling transfer mechanism is violated. The exit through the upper boundary of the SW igure 3: (a) Chart of regimes of the system (2) on the plane ( ε, D ) , where areas of chaos and of periodicmotions are shown. The region of hyperbolic chaos associated with the Smale – Williams attractor isshown in gray and marked as SW, other chaotic regimes are shown in white. The legend for periodicregimes is shown on the right. The periods indicated in colors are measured in units of modulation period.Fixed parameters are a = 0 . , b = 0 . , µ = 0 . , T = 250 , T = T / . (b) Histogram of the anglesof intersection of stable and unstable subspaces for the hyperbolic attractor of the Poincaré map of thesystem (2). a = 0 . , b = 0 . , µ = 0 . , ε = 0 . , D = 0 . , T = 250 , T = T / . region corresponds to the fact that at the curve representing the graph for the phases, a bend isformed on one of the branches, and then a maximum and a minimum appear, so that the monotonyproperty is lost. This means that when a stretched double loop is inserted into the original toroidalregion in the solenoid construction procedure, there appears a local fold on the loop, which leadsto disruption of the proper solenoid structure. When the parameters at the upper edge of theregion SW are varied, periodic motions become possible; on the chart one can see there a set ofperiodicity tongues. Visually, they look similar to the classical synchronization Arnold tongues, butthe principal difference from the classical picture is that between them we have chaotic dynamicsrather than the quasiperiodicity.Fig. 3 (b) shows the histogram of the angles of intersection of stable and unstable subspaces fora trajectory on the attractor of the Poincaré map of the system with parameters assigned accordingto (3). The fact that the distribution is separated from zero, confirms the hyperbolic nature ofthe attractor. Similar results are obtained for parameters of the system in a certain range, whichcorresponds to the structural stability inherent to the hyperbolic attractor. The technique of theperformed numerical test was developed in [6]. Let us consider a Froude pendulum placed on a rotating shaft and contacted with mechanical time-delay transmission line (a spring with a free end). Let the motion of pendulum be deceleratedperiodically by attaching a brake shoe. Denoting the angular coordinate of the pendulum as x , and he angular velocity as u , we write down the equations ˙ x = u, ˙ u = [ a − d ( t ) − bu ] u − sin x + µ + ε ( u ( t − τ ) − u ( t )) ,d ( t ) = , t < T ,D, T < t < T / , , T / < t < T. d ( t + T ) = d ( t ) . (6)The system (6) is non-autonomous, with periodic parameter modulation, so we can use thedescription of the dynamics in terms of discrete time by means of the stroboscopic Poincaré map X n = F T ( X n − ) . (7)Here vectors X n denote the sets of quantities x ( t n ) , ˙ x ( t n ) together with functions ˙ x ( t − τ ) , t ∈ [ t n − τ, t n ) determining the state of the system at the time instants t n = nT , and should beinterpreted as elements of infinite-dimensional state space.Parameters are assigned as follows: a = 0 . , b = 0 . , µ = 0 . , ε = 0 . ,D = 0 . , T = 250 , T = T / , τ = T / . (8)Let us start with a situation when the braking is not applied, and the pendulum performsrelaxation self-oscillations, in which, due to the selection of the parameters, the frequency is half ofthat for the small oscillations of the pendulum. The signal generated at this stage is being sendingto the time-delay feedback transmission line. Further, the pendulum oscillations are suppressed byapplication of the brake pad. When braking stops, a new stage of the buildup of the oscillationsbegins, starting from the practically unexcited state of the pendulum. As we properly select thedelay time, the oscillatory process will be stimulated by resonant action of the second harmonic ofthe signal received through the feedback circuit and emitted just on the previous stage of intenseoscillations. Therefore, the phase of the developing oscillations corresponds to the doubled phaseof the main component of oscillations at the prior activity stage. As a result, when the newlyarisen oscillations of the pendulum approach the sustained relaxation self-oscillations, their phasewill be doubled compared to the phase at the preceding activity stage. Further, the process isrepeated again and again. A full cycle corresponding to the modulation period T is accompaniedby multiplying the initial phase of the oscillatory process by factor of , i.e. for it, a doublyexpanding circle map takes place. Due to compression in remaining directions in the state space ofthe map (7), the Smale – Williams attractor arises.Numerical simulations were undertaken. It was demonstrated that dynamics of phases indeedis described by a map topologically equivalent to the Bernoulli mapping. The calculation of thelargest three Lyapunov exponents of the Poincaré map for the attractor at the given parameters (8)yields Λ = 0 . , Λ = − . , Λ = − . . The positive value of the first exponent indicates chaotic nature of the dynamics, and, as we cansee, it is quite close to the value ln 2 = 0 . . . . (Lyapunov exponent of the Bernoulli map). OtherLyapunov exponents are negative. This corresponds to the Smale – Williams solenoid embedded inthe infinite-dimensional space. igure 4: (a) Chart of regimes of the system (6) on the plane ( ε, D ) , where areas of chaos and of periodicmotions are shown. The region of hyperbolic chaos associated with the Smale – Williams attractor is shownin gray and marked as SW, other chaotic regimes are shown in white. The legend for periodic regimes isshown on the right. The periods indicated in colors are measured in units of the modulation period. Fixedparameters are a = 0 . , b = 0 . , µ = 0 . , T = 250 , T = T / , τ = T / . (b) Histogram of the anglesof intersection of stable and unstable subspaces for the hyperbolic attractor of the Poincaré map of thesystem (6). a = 0 . , b = 0 . , µ = 0 . , ε = 0 . , D = 0 . , T = 250 , T = T / , τ = T / . Fig. 4 (a) shows a chart of regimes on the parameter plane ( ε, D ) , that is the time-delayfeedback intensity parameter versus the dissipation parameter introduced by the brake pad duringthe braking stages. One can see a broad domain SW of hyperbolic chaos. Observe remarkablesimilarity of the chart to that in Fig. 3; the difference is distinct coloring of the periodicity areas,as in the system of coupled pendulums the phase transfer happens twice on the modulation period,while in the time-delay system it takes place one time on a modulation period. Fig. 4 (b) showsthe histogram of the angles of intersection of stable and unstable subspaces for a trajectory on theattractor of the Poincaré map of the system (6) with parameters assigned according to (8). Thedistribution is distanced from zero, that confirms the hyperbolicity of the attractor. The specialmethod of numerical test of hyperbolicity for time-delayed systems was developed in [7]. We discussed two mechanical models based on Froude pendulum with Smale – Williams hyperbolicattractors. Both models were simulated numerically. Areas of hyperbolic dynamics were identifiedin the parameter space checking the topological nature of the map for the angular variable. Thehyperbolicity of the chaotic attractors was also confirmed using a criterion based on the analysisof the intersection angles of stable and unstable invariant subspaces of small perturbation vectorsand checking the absence of tangencies between these subspaces. unding S. P. Kuznetsov and V. P. Kruglov acknowledge support of the grant of Russian Science FoundationNo. 15-12-20035 (Sections 1-3). Yu.V. Sedova acknowledges support of the grant of Russian ScienceFoundation No. 17-12-01008 (Section 4).
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