Asymptotic stability of heteroclinic cycles in systems with symmetry

Krupa, Martin; Melbourne, [No Value]

doi:10.1017/s0143385700008270

Cited by 172 publications

(324 citation statements)

References 21 publications

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“…The individual saddles appear to attract for a certain time, but any small components in the unstable directions grow, leading to eventual "switching" between saddles. In terms of nonlinear dynamics, such attractors have been studied for some time as heteroclinic networks and there is an extensive literature on their robustness, their stability (attractiveness) [12,13,24] and structure [1,21].…”

Section: Winnerless Competition Models For Cognitive Processesmentioning

confidence: 99%

A low-dimensional model of binocular rivalry using winnerless competition

Ashwin

Lavric

2010

Physica D: Nonlinear Phenomena

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We discuss a novel minimal model for binocular rivalry (and more generally perceptual dominance) effects. The model has only three state variables, but nonetheless exhibits a wide range of input and noise-dependent switching. The model has two reciprocally inhibiting input variables that represent perceptual processes active during the recognition of one of two possible states and a third variable that represents the perceived output. Sensory inputs only affect the input variables.We observe, for rivalry-inducing inputs, the appearance of winnerless competition in the perceptual system. This gives rise to behaviour that conforms to well-known principles describing binocular rivalry (Levelt's propositions, in particular proposition IV: monotonic response of residence time as a function of image contrast) down to very low levels of stimulus intensity.

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Section: Winnerless Competition Models For Cognitive Processesmentioning

confidence: 99%

A low-dimensional model of binocular rivalry using winnerless competition

Ashwin

Lavric

2010

Physica D: Nonlinear Phenomena

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“…For a more precise description of heteroclinic cycles and their stability, see Melbourne et al [7], Krupa and Melbourne [8], the monograph by Field [9] and the survey articles by Krupa [10,11]. Such behavior is unusual in a general dynamical system.…”

Section: Topology Of Heteroclinic Cyclesmentioning

confidence: 99%

Symmetry-Breaking as a Paradigm to Design Highly-Sensitive Sensor Systems

Palacios

Longhini

2015

Symmetry

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A large class of dynamic sensors have nonlinear input-output characteristics, often corresponding to a bistable potential energy function that controls the evolution of the sensor dynamics. These sensors include magnetic field sensors, e.g., the simple fluxgate magnetometer and the superconducting quantum interference device (SQUID), ferroelectric sensors and mechanical sensors, e.g., acoustic transducers, made with piezoelectric materials. Recently, the possibilities offered by new technologies and materials in realizing miniaturized devices with improved performance have led to renewed interest in a new generation of inexpensive, compact and low-power fluxgate magnetometers and electric-field sensors. In this article, we review the analysis of an alternative approach: a symmetry-based design for highly-sensitive sensor systems. The design incorporates a network architecture that produces collective oscillations induced by the coupling topology, i.e., which sensors are coupled to each other. Under certain symmetry groups, the oscillations in the network emerge via an infinite-period bifurcation, so that at birth, they exhibit a very large period of oscillation. This characteristic renders the oscillatory wave highly sensitive to symmetry-breaking effects, thus leading to a new detection mechanism. Model equations and bifurcation analysis are discussed in great detail. Results from experimental works on networks of fluxgate magnetometers are also included.

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“…Along the heteroclinic cycle, there is the expanding eigenvalue e i (i.e, the largest eigenvalue of Df (p i ) corresponding to an eigenvector v / ∈ R {σ(i)} ) and the contracting eigenvalue c i (i.e., the largest eigenvalue of −Df (p i ) corresponding to an eigenvector v / ∈ R {σ(i)} ). With these definitions in place, we get the following lemma which is based upon results of [2,4,10,16].…”

Section: Includes a Heteroclinic Orbit γ That Passes Through The Ormentioning

confidence: 99%

“…Let f ∂ denote f restricted to ∂R 3 + . When γ is an attractor for f ∂ , the assertions of this lemma are well-known and have been proven using either average Lyapunov functions [10] or Poincaré sections [2,16]. In fact these authors prove that if…”

Section: Then γ Is An Isolated Invariant Set Andmentioning

confidence: 99%

Successional stability of vector fields in dimension three

Schreiber

1999

Proc. Amer. Math. Soc.

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Abstract. A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the C 1 Whitney topology by a successionally stable vector field.

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Asymptotic stability of heteroclinic cycles in systems with symmetry

Cited by 172 publications

References 21 publications

A low-dimensional model of binocular rivalry using winnerless competition

A low-dimensional model of binocular rivalry using winnerless competition

Symmetry-Breaking as a Paradigm to Design Highly-Sensitive Sensor Systems

Successional stability of vector fields in dimension three

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