Computational explorations into the dynamics of rings of coupled oscillators (original) (raw)

Interaction of chaotic rotating waves in coupled rings of chaotic cells

Physica D: Nonlinear Phenomena, 1999

The interaction of two chaotic rotating waves of the type recently reported by Matías et al. [Europhys. Lett. 37 (1997) 379] is studied experimentally with arrays of non-linear electronic circuits arranged in ring geometries. Unidirectional coupling is assumed for the cell-to-cell coupling within the same ring, but between rings, cells are coupled diffusively. Depending on the relative sense of driving, competition between a rotating chaotic wave and a global synchronized state has been observed. The results are rationalized by means of a linear stability analysis around the uniform synchronized behavior, where the circulant symmetry of the system allows to express the problem as a superposition of a series of Fourier modes.

Ring Oscillators Under Nonlinear Coupling: Bifurcation and Chaos

ECTI Transactions on Electrical Engineering, Electronics, and Communications

The dynamics of two non-linearly coupled ring oscillators are examined in this study. Each ring oscillator consists of three-stage inverters, coupled through a resistor and diode. The system is mathematically modeled by non-linear differential equations. A numerical phase plane, bifurcation, and quantitative measures, like the Lyapunov exponent, confirm the transition from periodic to chaotic oscillation in a broad parameter zone. The system is implemented in a prototype hardware electronic circuit with bifurcation and chaos observed experimentally. This circuit can be used in practical applications such as cryptography and random number generation.

Fast transition to chaos in a ring of unidirectionally coupled oscillators

2011

In this paper we study the destabilization mechanism in a ring of unidirectionally coupled oscillators. We derive an amplitude equation of Ginzburg-Landau type that describes the destabilization of the stationary state for systems with a large number of oscillators. Based on this amplitude equation, we are able to provide an explanation for the fast transition to chaos (or hyperchaos) that can be observed in such systems. We show that the parameter interval, where the transition from a stable periodic state to chaos occurs, scales like the inverse square of the number of oscillators in the ring. In particular, for a sufficiently large number of oscillators a practically immediate transition to chaos can be observed. The results are illustrated by a numerical study of a system of unidirectionally coupled Duffing oscillators.

Routes to complex dynamics in a ring of unidirectionally coupled systems

Chaos: An Interdisciplinary Journal of Nonlinear Science, 2010

We study the dynamics of a ring of unidirectionally coupled autonomous Dung oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures and the Eckhaus eect for a large number of oscillators. Our analytical and numerical results are conrmed by a simple experiment based on the electronic implementation of coupled Dung oscillators.

Collision between Chaotic and Periodic Attractors in a Ring of Coupled Chaotic Circuits

In this study, we investigate synchronization phenomena observed in coupled chaotic circuits as a ring topology when two types of the bifurcation parameters to generate chaotic and three-periodic attractors are arranged. By using computer simulations and circuit experiments, we observe a collision between chaotic and periodic attractors depending on the number of chaotic attractors and the location of chaotic attractors in the system.

Desynchronization Transitions in Rings of Coupled Chaotic Oscillators

International Journal of Bifurcation and Chaos, 1998

Rings of chaotic oscillators coupled unidirectionally through driving are studied. While synchronization is observed for small sizes of the ring, beyond a certain critical size a desynchronizing transition occurs. In the two examples studied here the system exhibits a transition to periodic rotating waves for rings of Lorenz systems, while one finds a sort of chaotic rotating waves when Chua's circuit is used.

Rotating Wave Dynamics in Rings of Coupled Oscillators: A Comprehensive Review

Nonlinear Phenomena in Complex Systems, 2023

This comprehensive review paper provides a thorough survey of the extensive research conducted on rotating waves observed in rings of coupled oscillators. These waves manifest as stable periodic, quasiperiodic, or chaotic orbits, arising from the phase difference between neighboring oscillators. The research encompasses a wide range of nonlinear systems, including electrical circuits, neural models, Duffing, Lorenz, and Rössler oscillators, among others. While exploring the behavior of rotating waves, particular emphasis is placed on neural oscillators, as neural rings in the brain play a crucial role in working memory. The intricate dynamics of rotating waves are elucidated through the application of various techniques, including time series analysis, phase-space analysis, bifurcation diagrams, spectral analysis, and basins of attraction. These methods carefully uncover the complex routes from coexisting stable equilibria to hyperchaos. The study highlights a sequence of bifurcations occurring with increasing coupling strength, such as the Andronov-Hopf, torus, and crisis bifurcations. Moreover, the coexistence of multiple rotating waves under the same system parameters is examined. The vast body of research on rotating waves provides insights that are essential for a wide range of scientific disciplines and realworld applications, including lasers, chemical reactions, cardiorespiratory systems, and even beyond, with particular relevance to neural networks and brain functions.

Chaotic Dynamics of Coupled Nonlinear Circuits in Ring Connection

2012

It is generally difficult to synchronize a ring network that features chaotic behaviour, especially if the system's order is too large. In this paper, we consider a ring network of three identical nonlinear and non-autonomous circuits of fourth order, which are bidirectionally coupled through three coupling linear resistances R C . We present simulation and experimental results for synchronization of such a network in low frequency area, and derive a sufficient condition for chaotic synchronization of this type of network.

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.