Stability of the synchronization manifold in nearest neighbors non identical van der Pol-like oscillators (original) (raw)

Synchronization of two coupled self-excited systems with multi-limit cycles

Chaos, 2007

We analyze the stability and optimization of the synchronization process between two coupled self-excited systems modeled by the multi-limit cycles van der Pol oscillators through the case of an enzymatic substrate reaction with ferroelectric behavior in brain waves model. The one-way and two-way couplings synchronization are considered. The stability boundaries and expressions of the synchronization time are obtained using the properties of the Hill equation. Numerical simulations validate and complement the results of analytical investigations.

A Bifurcation Approach to the Synchronization of Coupled Van der Pol Oscillators

We undertake a bifurcation analysis of a velocity coupled system of two classical nonidentical Van der Pol oscillators to understand the appearance and structure of 1:k parameter regions with synchronized states as we vary the coupling and the frequency mismatch. These regions include multistability of solutions and are formed by classical tongues bordered by curves of limit point bifurcation of periodic orbits with an isola structure and an additional subregion surrounded by curves of torus bifurcations and a curve characterized by a geometrical tangency condition. Symmetry arguments explain the difference between the even and odd k cases.

Synchronization dynamics in a ring of four mutually coupled biological systems

Communications in Nonlinear Science and Numerical Simulation, 2008

This paper considers the synchronization dynamics in a ring of four mutually coupled biological systems described by coupled Van der Pol oscillators. The coupling parameter are non-identical between oscillators. The stability boundaries of the process are first evaluated without the influence of the local injection using the eigenvalues properties and the fourthorder Runge-Kutta algorithm. The effects of a locally injected trajectory on the stability boundaries of the synchronized states are performed using numerical simulations. In both cases, the stability boundaries and the main dynamical states are reported on the stability maps in the (K 1 , K 2 ) plane.

Rhythm synchronization and chaotic modulation of coupled Van der Pol oscillators in a model for the heartbeat

Physica A: Statistical Mechanics and its Applications, 2004

We investigate a phenomenological model for the heartbeat consisting of two coupled Van der Pol oscillators. The coupling between these nodes can be both unidirectional or bidirectional, and an external driving produced by a pacemaker is also included in this model. In order to warrant a robust operation, it is desirable that both units oscillate in a synchronized way, even though in the presence of external in uences or parameter mismatches which are unavoidable in a physiological setting. We study the synchronization properties of such an association with respect to the nature and intensity of coupling. We analyze in particular the (generalized) synchronization of rhythms characterized by a chaotic modulation of the oscillator frequencies. We also investigate the shadowing breakdown of numerically generated chaotic trajectories of the coupled oscillator system via unstable dimension variability in its chaotic invariant set.

Synchronization of diffusively coupled oscillators near the homoclinic bifurcation

Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1999

It has been known that a diffusive coupling between two limit cycle oscillations typically leads to the in-phase synchronization and also that it is the only stable state in the weak-coupling limit. Recently, however, it has been shown that the coupling of the same nature can result in the distinctive dephased synchronization when the limit cycles are close to the homoclinic bifurcation, which often occurs especially for the neuronal oscillators. In this paper we propose a simple physical model using the modified van der Pol equation, which unfolds the generic synchronization behaviors of the latter kind and in which one may readily observe changes in the sychronization behaviors between the distinctive regimes as well. The dephasing mechanism is analyzed both qualitatively and quantitatively in the weak-coupling limit. A general form of coupling is introduced and the synchronization behaviors over a wide range of the coupling parameters are explored to construct the phase diagram u...

Emergent rhythms in coupled nonlinear oscillators due to dynamic interactions

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2021

The role of a new form of dynamic interaction is explored in a network of generic identical oscillators. The proposed design of dynamic coupling facilitates the onset of a plethora of asymptotic states including synchronous states, amplitude death states, oscillation death states, a mixed state (complete synchronized cluster and small amplitude desynchronized domain), and bistable states (coexistence of two attractors). The dynamical transitions from the oscillatory to death state are characterized using an average temporal interaction approximation, which agrees with the numerical results in temporal interaction. A first order phase transition behavior may change into a second order transition in spatial dynamic interaction solely depending on the choice of initial conditions in the bistable regime. However, this possible abrupt first order like transition is completely non-existent in the case of temporal dynamic interaction. Besides the study on periodic Stuart-Landau systems, we present results for paradigmatic chaotic model of Rössler oscillators and Mac-arthur ecological model. Population biology of ecological networks, person to person communication networks, brain functional networks, possibility of outbreaks and spreading of disease through human contact networks, to name but a few examples which attest to the importance of researches based on temporal interaction approach. Studies based on representating several complex systems as time-varying networks of dynamical units have been shown to be extremely beneficial in understanding real life processes. Surprisingly, in all the previous studies on time-varying interaction, death state receives little attention in a network of coupled oscillators. In addition, only a few studies on dynamic interaction have considered the proximity of the individual systems' trajectories in the context of their interaction. In this paper, we propose a simple yet effective dynamic interaction scheme among nonlinear oscillators, which is capable of relaxing the collective oscillatory dynamics towards the dynamical equilibrium under appropriate choices of parameters. The dynamics of coupled oscillators can show fascinating complex behaviors including various dynamical phenomena. A qualitative explanation of the numerical observation is validated through linear stability analysis and interestingly, a linear stability analysis is persued even when the system is time-dependent. An elaborate study is contemplated to reveal the influences of our proposed dynamic interaction in terms of all the network parameters.

Bistable solutions in the onset of the synchronization of a array of oscillators

The existence of multi-stable solutions in the onset of synchronization of the lckm has been investigated in recent years by **, in what the topology of the system proved to be tantamount of existence/non-existence of multistable solutions.In our paper we investigated a array of oscillators synchronized with external forces, we show that this systems presents bi-stability, we deduce analitical expressions for symmetrical cases.

Regular oscillations, chaos, and multistability in a system of two coupled van der Pol oscillators: numerical and experimental studies

Nonlinear Dynamics, 2014

In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated. The coupling between the two oscillators consists of adding to each one's amplitude a perturbation proportional to the other one. The coupling between two laser oscillators and the coupling between two vacuum tube oscillators are examples of physical/experimental systems related to the model considered in this paper. The stability of fixed points and the symmetries of the model equations are discussed. The bifurcations structures of the system are analyzed with particular attention on the effects of frequency detuning between the two oscillators. It is found that the system exhibits a variety of bifurcations including symmetry breaking, period doubling, and crises when monitoring the frequency detuning parameter in tiny steps. The multistability property of the system for special sets of its parameters is also analyzed. An experimental study of the coupled system is carried out in this work. An appropriate elec

Synchronization of coupled oscillators

In this work we begin by introducing the Kuramoto model, constructing its solutions in the thermodynamic limit and showing the close connection between statistical physics and dynamical systems that lead to the main theoretical insights. The systematic study of a finite population of self sustained oscillators began in the first decade of this century. Unlike most of the papers we have found, we are not interested in the synchronization transition in itself but rather in phase locked patterns and their relation with frequency distribution among oscillators. The problem of stability, as we have already mentioned, experienced great advances in recent years. In a brief discussion we only address the problem of stability of the simplest solution allowed by the Kuramoto model: the incoherent solution. After that we introduce Chimera states, First noticed by Kuramoto and his colleagues in which the introduction of a non local coupling gives origin to a split in a region with synchronised oscillators and other with asynchronous one. Then we proceed by exploring the literature and the results with a fnite number of oscillators, model explored with persistence only since mainly 2004. But here we are yet in Kuramoto framework which is abandoned, in a rigorous terminology, when we pursuit structured and not all-to-all coupling. Although we could introduce the same mean models quantities if well defined in each situation, this did not help us in making sense of the results and is not an help in any analytical work. In our analysis of a ring of coupled oscillators we construct a space that allows us to relate the stable solutions with the eigenvectors of the laplacian of the graph in which we work. work.