RESUMO
The Koper model is a prototype system with two slow variables and one fast variable that possesses small-amplitude oscillations (SAOs), large-amplitude oscillations (LAOs), and mixed-mode oscillations (MMOs). In this article, we study a pair of identical Koper oscillators that are symmetrically coupled. Strong symmetry breaking rhythms are presented of the types SAO-LAO, SAO-MMO, LAO-MMO, and MMO-MMO, in which the oscillators simultaneously exhibit rhythms of different types. We identify the key folded nodes that serve as the primary mechanisms responsible for the strong nature of the symmetry breaking. The maximal canards of these folded nodes guide the orbits through the neighborhoods of these key points. For all of the strong symmetry breaking rhythms we present, the rhythms exhibited by the two oscillators are separated by maximal canards in the phase space of the oscillator.
RESUMO
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.
RESUMO
Mixed-mode oscillations (MMOs) are a complex dynamical behavior in which each cycle of oscillation consists of one or more large amplitude spikes followed by one or more small amplitude peaks. MMOs typically undergo period-adding bifurcations under parameter variation. We demonstrate here, in a set of three identical, linearly coupled van der Pol oscillators, a scenario in which MMOs exhibit a period-doubling sequence to chaos that preserves the MMO structure, as well as period-adding bifurcations. We characterize the chaotic nature of the MMOs and attribute their existence to a master-slave-like forcing of the inner oscillator by the outer two with a sufficient phase difference between them. Simulations of a single nonautonomous oscillator forced by two sine functions support this interpretation and suggest that the MMO period-doubling scenario may be more general.
RESUMO
We analyze a model of two identical chemical oscillators coupled through diffusion of the slow variable. As a parameter is varied, a single oscillator undergoes a canard explosion-a transition from small amplitude, nearly harmonic oscillations to large-amplitude, relaxation oscillations over a very small parameter interval. In the coupled system, if the two oscillators have the same initial conditions, then the oscillators remain synchronized and exhibit the same canard behavior observed for the single oscillator. If the oscillators are separated initially, then in the region of the canard they display a variety of complex behaviors, including intermittent spiking, mixed-mode oscillation, and quasiperiodicity. Further variation of the parameter leads to a return to synchronized large-amplitude oscillation followed by a post-canard symmetry-breaking, in which one oscillator shows small-amplitude, complex behavior (mixed-mode oscillation, quasiperiodicity, chaos,...) while the other undergoes essentially periodic large amplitude behavior, resembling a master-slave scenario. We analyze the origins of this behavior by looking at several modified coupling schemes.
RESUMO
Symmetrically coupled identical oscillators were once believed to support only totally synchronous or totally asynchronous states. More recently, chimera states, in which a subset of oscillators behaves coherently while the other subset exhibits disorder, have been found in large arrays of oscillators, coupled either locally or globally. We demonstrate for the first time the existence of a chimera state with only two diffusively coupled identical oscillators, one behaving nearly periodically (coherently) and the other chaotically (incoherently). We attribute this behavior to a "master-slave" interaction, which arises via a symmetry-breaking canard explosion.
