The onset of chaos in the wake of an oscillating cylinder: Experiment and the dynamics of the circle map

This paper deals with a comparison between experimental observations in a low-Reynolds-number wake behind an oscillating cylinder and the universal properties of a sine circle map. When the limit cycle due to the natural vortex shedding in the wake is modulated at a second frequency by oscillating the cylinder transversely, one obtains in phase space a flow on a two torus. The nonlinear interaction between the two oscillators results in Arnol’d tongues due to phase locking, the devil’s staircase along the critical line, and a transition from order to chaosvia the quasiperiodic route. The sine circle map describes these features adequately. A comparison between the experiment and the theory is made in terms of multifractal formalism and trajectory scaling function.     

About Landau–Hopf scenario in a system of coupled self-oscillators

The conditions are discussed for which an ensemble of interacting oscillators may demonstrate the Landau–Hopf scenario of successive birth of multi-frequency quasi-periodic motions. A model is proposed that is a network of five globally coupled oscillators characterized by controlled degree of activation of individual oscillators. Illustrations are given for successive birth of tori of increasing dimension via quasi-periodic Hopf bifurcations.     

The scaling of Arnol'd tongues for differentiable homeomorphisms of the circle

We show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms, that the widths of the Arnol'd tongues in a one parameter family scale asq ^–3 whenq is the denominator of the rotation number.Research supported by NSERC     

Recognition of resonance type in periodically forced oscillators

This paper deals with families of periodically forced oscillators undergoing a Hopf-Ne?marck-Sacker bifurcation. The interest is in the corresponding resonance sets, regions in parameter space for which subharmonics occur. It is a classical result that the local geometry of these sets in the non-degenerate case is given by an Arnol’d resonance tongue. In a mildly degenerate situation a more complicated geometry given by a singular perturbation of a Whitney umbrella is encountered. Our main contribution is providing corresponding recognition conditions, that determine to which of these cases a given family of periodically forced oscillators corresponds. The conditions are constructed from known results for families of diffeomorphisms, which in the current context are given by Poincaré maps. Our approach also provides a skeleton for the local resonant Hopf-Ne?marck-Sacker dynamics in the form of planar Poincaré-Takens vector fields. To illustrate our methods two case studies are included: A periodically forced generalized Duffing-Van der Pol oscillator and a parametrically forced generalized Volterra-Lotka system.     

Dynamics of a pair of coupled nonlinear oscillators

A pair of coupled classical oscillators with a general potential and general form of coupling is investigated. For general potentials, the single-frequency solution is shown to be stable for small excitations. For special potentials, such system remains stable for an arbitrary excitation. In both cases, the stability does not depend on the form of coupling. Transition to the instability regime follows from the way how nonlinear potential entrains the energy transfer between the oscillators. Relation between the existence of multi-frequency quasi-periodic or periodic solutions and the instability of single-frequency ones is discussed.     

Numerical bifurcation analysis of two coupled FitzHugh-Nagumo oscillators

The behavior of neurons can be modeled by the FitzHugh-Nagumo oscillator model, consisting of two nonlinear differential equations, which simulates the behavior of nerve impulse conduction through the neuronal membrane. In this work, we numerically study the dynamical behavior of two coupled FitzHugh-Nagumo oscillators. We consider unidirectional and bidirectional couplings, for which Lyapunov and isoperiodic diagrams were constructed calculating the Lyapunov exponents and the number of the local maxima of a variable in one period interval of the time-series, respectively. By numerical continuation method the bifurcation curves are also obtained for both couplings. The dynamics of the networks here investigated are presented in terms of the variation between the coupling strength of the oscillators and other parameters of the system. For the network of two oscillators unidirectionally coupled, the results show the existence of Arnold tongues, self-organized sequentially in a branch of a Stern-Brocot tree and by the bifurcation curves it became evident the connection between these Arnold tongues with other periodic structures in Lyapunov diagrams. That system also presents multistability shown in the planes of the basin of attractions.     

Desynchronization by phase slip patterns in networks of pulse-coupled oscillators with delays

Coupling delays may cause drastic changes in the dynamics of oscillatory networks. In the present paper we investigate how coupling delays alter synchronization processes in networks of all-to-all coupled pulse oscillators. We derive an analytic criterion for the stability of synchrony and study the synchronization areas in the space of the delay and coupling strength. Specific attention is paid to the scenario of destabilization on the borders of the synchronization area. We show that in bifurcation points the system possesses homoclinic loops, which give rise to complex long- or quasi-periodic solutions. These newly born solutions are characterized by a synchronous group, from which an oscillator periodically escapes, laps one period, and rejoins. We call such a dynamical regime “phase slip patterns”.     

Stochastic phase dynamics and noise-induced mixed-mode oscillations in coupled oscillators

Synaptically coupled neurons show in-phase or antiphase synchrony depending on the chemical and dynamical nature of the synapse. Deterministic theory helps predict the phase differences between two phase-locked oscillators when the coupling is weak. In the presence of noise, however, deterministic theory faces difficulty when the coexistence of multiple stable oscillatory solutions occurs. We analyze the solution structure of two coupled neuronal oscillators for parameter values between a subcritical Hopf bifurcation point and a saddle node point of the periodic branch that bifurcates from the Hopf point, where a rich variety of coexisting solutions including asymmetric localized oscillations occurs. We construct these solutions via a multiscale analysis and explore the general bifurcation scenario using the lambda-omega model. We show for both excitatory and inhibitory synapses that noise causes important changes in the phase and amplitude dynamics of such coupled neuronal oscillators when multiple oscillatory solutions coexist. Mixed-mode oscillations occur when distinct bistable solutions are randomly visited. The phase difference between the coupled oscillators in the localized solution, coexisting with in-phase or antiphase solutions, is clearly represented in the stochastic phase dynamics.     

Synchronization regions of two pulse-coupled electronic piecewise linear oscillators

Stable synchronous states of different order were analytically, numerically and experimentally characterized in pulse-coupled light-controlled oscillators (LCOs). The Master-Slave (MS) configuration was studied in conditions where different time-scale parameters were tuned under varying coupling strength. Arnold tongues calculated analytically – based on the piecewise two-time-scale model for LCOs – and obtained numerically were consistent with experimental results. The analysis of the stability pattern and tongue shape for (1 : n) synchronization was based on the construction of return maps representing the Slave LCO evolution induced by the action of the Master LCO. The analysis of these maps showed that both tongue shape and stability pattern remained invariant. Considering the wide variation range of LCO parameters, the obtained results could have further applications on ethological models.     

Quasi-periodic bifurcations and “amplitude death” in low-dimensional ensemble of van der Pol oscillators

The dynamics of the four dissipatively coupled van der Pol oscillators is considered. Lyapunov chart is presented in the parameter plane. Its arrangement is discussed. We discuss the bifurcations of tori in the system at large frequency detuning of the oscillators. Here are quasi-periodic saddle-node, Hopf and Neimark–Sacker bifurcations. The effect of increase of the threshold for the “amplitude death” regime and the possibilities of complete and partial broadband synchronization are revealed.     

Study of the Nonlinear Dynamics of a Traveling-Wave-Tube Oscillator with Delayed Feedback

We present the results of numerical simulations of the nonlinear dynamics of a traveling-wave-tube (TWT) oscillator with delayed feedback. Basic properties of stationary single-frequency oscillation regimes are considered, and the onset of self-modulation is studied in detail. Various route-to-chaos scenarios corresponding to successively increasing values of the beam current are simulated numerically. It is shown that the basic scenario is a quasi-periodic route to chaos, while the beam deceleration in strongly nonlinear regimes causes transitions via intermittency to regimes based on modes with higher frequencies. Competition between these two scenarios leads to a complex picture of regular and chaotic self-modulation regimes in the parameter space. Such a behavior is typical of distributed electron–wave self-excited oscillators with delayed feedback.     

Multi-scale energy exchanges between a nonlinear oscillator of Bouc-Wen type and another coupled nonlinear system

The concept of energy exchange between coupled oscillators can be endowed for wide variety of applications such as control and energy harvesting. It has been proved that by coupling an essential nonlinear oscillator (cubic nonlinearity) to a main system (mostly linear), the latter system can be controlled in a one way and almost irreversible manner. The phenomenon is called energy pumping and the coupled nonlinear system is named as nonlinear energy sink (NES). The process of energy transfer from the main system to the nonlinear smooth or non-smooth attachment at different scales of time can present several scenarios: It can be attracted to periodic behaviors which present low or high energy levels for the main system and/or to quasi-periodic responses of two oscillators by persistent bifurcations between their stable zones. In this paper we analyze multi-scale dynamics of two attached oscillators: a Bouc-Wen type in general (in particular: a Dahl type and a modified hysteresis system) and a NES (nonsmooth and cubic). The system behavior at fast and first slow times scales by detecting its invariant manifold, its fixed points and singularities will be analyzed. Analytical developments will be accompanied by some numerical examples for systems that present quasi-periodic responses. The endowed Bouc-Wen models correspond to the hysteretic behavior of materials or structures. This paper is clearly connected with the dynamics of systems with hysteresis and nonlinear dynamics based energy harvesting.     

Complex dynamics analysis of impulsively coupled Duffing oscillators with ring structure

Impulsively coupled systems are high-dimensional non-smooth systems that can exhibit rich and complex dynamics.This paper studies the complex dynamics of a non-smooth system which is unidirectionally impulsively coupled by three Duffing oscillators in a ring structure.By constructing a proper Poincare map of the non-smooth system,an analytical expression of the Jacobian matrix of Poincare map is given.Two-parameter Hopf bifurcation sets are obtained by combining the shooting method and the Runge-Kutta method.When the period is fixed and the coupling strength changes,the system undergoes stable,periodic,quasi-periodic,and hyper-chaotic solutions,etc.Floquet theory is used to study the stability of the periodic solutions of the system and their bifurcations.     

The dynamics of two coupled Van der Pol oscillators with attractive and repulsive coupling

We consider a variant of two coupled Van der Pol oscillators with both attractive and repulsive mean-field interactions. In the presence of attractive coupling, the system is in the complete synchrony, while repulsive coupling shows anti-synchronization state leading to suppression of oscillations with increasing interaction strength. The coupled system with both attractive and repulsive interactions shows competitive tendencies of being complete synchronization and anti-synchronization resulting in the stabilization of the fixed point. We have also studied the effect of the damping coefficient of the VdP oscillator on the nature of the transition from oscillatory to a steady-state. These oscillators stabilize to unstable equilibrium point or coupling dependent inhomogeneous steady state via second or first-order transitions respectively depending upon the damping coefficient and coupling strength. These transitions are analyzed in the parameter plane by analytical and numerical studies of the two coupled Van der Pol oscillators.     

Extreme sensitivity to detuning for globally coupled phase oscillators

We discuss the sensitivity of a population of coupled oscillators to differences in their natural frequencies, i.e., to detuning. We argue that for three or more oscillators, one can get great sensitivity even if the coupling is strong. For N globally coupled phase oscillators we find there can be bifurcation to extreme sensitivity, where frequency locking can be destroyed by arbitrarily small detuning. This extreme sensitivity is absent for N = 2, appears at isolated parameter values for N = 3 and N = 4, and can appear robustly for open sets of parameter values for N > or = 5 oscillators.     

Differentiability at the Tip of Arnold Tongues for Diophantine Rotations: Numerical Studies and Renormalization Group Explanations

We study numerically the regularity of Arnold tongues corresponding to Diophantine rotation numbers of circle maps at the edge of validity of KAM theorem. This serves as a good test for the numerical stability of two different algorithms. We find empirically that Arnold tongues are only finitely differentiable at the tip. We also find several scaling properties of the Sobolev norms of the conjugacy near the breakdown. We also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the Sobolev regularity. We also uncover empirically some other patterns which require explanation.     

Arnol’d diffusion in a system with 2.5 degrees of freedom: Classical and quantum mechanical approaches

Arnol’d diffusion, a universal phenomenon in nonlinear dynamics, is analyzed for a model system with 2.5 degrees of freedom. Only the three primary order resonances are taken into account, and the results obtained by using classical and quantum mechanical approaches are compared. It is shown that the parameter dependence of the rate of quantum Arnol’d diffusion is similar to the classical one, but the quantum diffusion coefficient is smaller by approximately an order of magnitude. It is found that the existence of a threshold with respect to perturbation parameters, pointed out earlier, is not an indispensable feature of quantum Arnol’d diffusion. It is shown that a quantum system with weakly overlapping resonances can exhibit mixed dynamics that has no classical counterpart (diffusion along a resonance superimposed by oscillations across the overlapped resonances).     

Coupled van der Pol-Duffing oscillators: Phase dynamics and structure of synchronization tongues

Synchronization in the system of coupled non-identical non-isochronous van der Pol-Duffing oscillators with inertial and dissipative coupling is discussed. Generalized Adler’s equation is obtained and investigated in the presence of all relevant factors affecting the synchronization (non-isochronism of the oscillators, their non-identity, coupling of the dissipative and inertial types). Characteristic symmetries are revealed for the Adler’s equation responsible for equivalence of some of the factors. Numerical study of the parameters space of the initial differential equations is carried out using the method of charts of dynamic regimes in the parameter planes. Results obtained by both these approaches are compared and discussed.     

Nonlinear dynamics of the heartbeat: I. The AV junction: Passive conduit or active oscillator?

Under physiologic conditions, the AV junction is traditionally regarded as a passive conduit for the conduction of impulses from the atria to the ventricles. An alternative view, namely that subsidiary pacemakers play an active role in normal electrophysiologic dynamics during sinus rhythm, has been suggested based on nonlinear models of cardiac oscillators. A central problem has been the development of a simple but explicit mathematical model for coupled nonlinear oscillators relevant both to stable and perturbed cardiac dynamics. We use equations describing an analog electrical circuit with an external d.c. voltage source (V₀) and two nonlinear oscillators with intrinsic frequencies in the ratio of 3:2, comparable to the SA node and AV junction rates. The oscillators are coupled by means of a resistor. 1:1 (SA:AV) phase-locking of the oscillators occurs over a critical range of V₀. Externally driving the SA oscillator at increasing rates results in 3:2 AV Wenckebach periodicity and a 2:1 AV block. These findings appear with no assumptions about conduction time or refractoriness. This dynamical model is consistent with the new interpretation that normal sinus rhythm may represent 1:1 coupling of two or more active nonlinear oscillators and also accounts for the appearance of an AV block with critical changes in a single parameter such as the pacing rate.     

Lag and anticipating synchronization without time-delay coupling

We describe a new method for achieving approximate lag and anticipating synchronization in unidirectionally coupled chaotic oscillators. The method uses a specific parameter mismatch between the drive and response that is a first-order approximation to true time-delay coupling. As a result, an adjustable lag or anticipation effect can be achieved without the need for a variable delay line, making the method simpler and more economical to implement in many physical systems. We present a stability analysis, demonstrate the method numerically, and report experimental observation of the effect in radio-frequency electronic oscillators. In the circuit experiments, both lag and anticipation are controlled by tuning a single capacitor in the response oscillator.