Symmetry-Breaking Bifurcations and Patterns of Oscillations in Rings of Crystal Oscillators (original) (raw)
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Symmetry-breaking rhythms in coupled, identical fast–slow oscillators
Chaos: An Interdisciplinary Journal of Nonlinear Science
Symmetry-breaking in coupled, identical, fast–slow systems produces a rich, dramatic variety of dynamical behavior—such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast–slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel–Epstein model of chemical oscillators.
papangelo2017 Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry.pdf
Snaking bifurcations in a chain of mechanical oscillators are studied. The individual oscillators are weakly nonlinear and subject to self-excitation and subcritical Hopf-bifurcations with some parameter ranges yielding bistability. When the oscillators are coupled to their neighbours, snaking bifurcations result, corresponding to localised vibration states. The snaking patterns do seem to be more complex than in previously studied continuous systems, comprising a plethora of isolated branches and also a large number of similar but not identical states, originating from the weak coupling of the phases of the individual oscillators.
Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry
Snaking bifurcations in a chain of mechanical oscillators are studied. The individual oscil-lators are weakly nonlinear and subject to self-excitation and subcritical Hopf-bifurcations with some parameter ranges yielding bistability. When the oscillators are coupled to their neighbours, snaking bifurcations result, corresponding to localised vibration states. The snaking patterns do seem to be more complex than in previously studied continuous systems , comprising a plethora of isolated branches and also a large number of similar but not identical states, originating from the weak coupling of the phases of the individual oscillators.
Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry.pdf
Snaking bifurcations in a chain of mechanical oscillators are studied. The individual oscillators are weakly nonlinear and subject to self-excitation and subcritical Hopf-bifurcations with some parameter ranges yielding bistability. When the oscillators are coupled to their neighbours, snaking bifurcations result, corresponding to localised vibration states. The snaking patterns do seem to be more complex than in previously studied continuous systems, comprising a plethora of isolated branches and also a large number of similar but not identical states, originating from the weak coupling of the phases of the individual oscillators.
Snaking bifurcations in a chain of mechanical oscillators are studied. The individual oscillators are weakly nonlinear and subject to self-excitation and subcritical Hopf-bifurcations with some parameter ranges yielding bistability. When the oscillators are coupled to their neighbours, snaking bifurcations result, corresponding to localised vibration states. The snaking patterns do seem to be more complex than in previously studied continuous systems, comprising a plethora of isolated branches and also a large number of similar but not identical states, originating from the weak coupling of the phases of the individual oscillators.
Hopf Bifurcation in Symmetric Networks of Coupled Oscillators with Hysteresis
Journal of Dynamics and Differential Equations, 2012
The standard approach to study symmetric Hopf bifurcation phenomenon is based on the usage of the equivariant singularity theory developed by M. Golubitsky et al. In this paper, we present the equivariant degree theory based method which is complementary to the equivariant singularity approach. Our method allows systematic study of symmetric Hopf bifurcation problems in non-smooth/non-generic equivariant settings. The exposition is focused on a network of eight identical van der Pol oscillators with hysteresis memory, which are coupled in a cube-like configuration leading to S 4 -equivariance. The hysteresis memory is the source of non-smoothness and of the presence of an infinite dimensional phase space without local linear structure. Symmetric properties and multiplicity of bifurcating branches of periodic solutions are discussed in the context showing a direct link between the physical properties and the equivariant topology underlying this problem.
Strong Symmetry Breaking in Coupled, Identical Lengyel–Epstein Oscillators via Folded Singularities
Journal of nonlinear science, 2024
We study pairs of symmetrically coupled, identical Lengyel-Epstein oscillators, where the coupling can be through both the fast and slow variables. We find a plethora of strong symmetry breaking rhythms, in which the two oscillators exhibit qualitatively different oscillations, and their amplitudes differ by as much as an order of magnitude. Analysis of the folded singularities in the coupled system shows that a key folded node, located off the symmetry axis, is the primary mechanism responsible for the strong symmetry breaking. Passage through the neighborhood of this folded node can result in splitting between the amplitudes of the oscillators, in which one is constrained to remain of small amplitude, while the other makes a large-amplitude oscillation or a mixed-mode oscillation. The analysis also reveals an organizing center in parameter space, where the system undergoes an asymmetric canard explosion, in which one oscillator exhibits a sequence of limit cycle canards, over an interval of parameter values centered at the explosion point, while the other oscillator executes small amplitude oscillations. Other folded singularities can also impact properties of the strong symmetry breaking rhythms. We contrast these strong symmetry breaking rhythms with asymmetric rhythms that are close to symmetric states, such as in-phase or anti-phase oscillations. In addition to the symmetry breaking rhythms, we also find an explosion of anti-phase limit cycle canards, which mediates the transition from small-amplitude, anti-phase oscillations to large-amplitude, anti-phase oscillations. Keywords Fast-slow systems • Multiple-scale systems • Canards • Small-amplitude oscillations • Multi-mode oscillations • Folded nodes Communicated by Alexander Lohse.
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The local time-average and the symmetry of 2T-periodic forced oscillations
Phys Lett a, 1990
Some 2 T-states of a T-periodically driven nonlinear oscillatory circuit are interpreted as a mutual connection between the nonzero time average over some T-intervals of the response function and the symmetry (evenness or oddness) of the function with respect to some instants inside the intervals. A change in the symmetry of the 2 T-oscillations is also discussed, as being important to the application of the symmetry argument.
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.