Basins of attraction in a ring of overdamped bistable systems with delayed coupling

Lyons, Daniel Carl; Mahaffy, Joseph M.; Palacios, Antonio Lozano; In, Visarath; Longhini, Patrick; Kho, Andy

doi:10.1016/j.physleta.2010.04.060

Cited by 6 publications

(3 citation statements)

References 25 publications

(41 reference statements)

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“…This work has been motivated by previous analysis of the basins of attraction of coupled bistable systems subject to delay [Lyons et al, 2010]. Similar behaviors are observed for larger (odd) values of N with the additional appearance of unstable branches of limit cycles through Hopf bifurcations.…”

Section: Discussionsupporting

confidence: 52%

Geometry of Basins of Attraction and Heteroclinic Connections in Coupled Bistable Systems

Lyons

Mahaffy

Wang

et al. 2014

Int. J. Bifurcation Chaos

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The fundamental principle of bistability is widely used across various disciplines, including biology, chemistry, mechanics, physics, electronics and materials science. As the need for more powerful, efficient and sensitive complex-engineered systems grow, networks of coupled bistable systems have gained significant attention in recent years. Modeling and analysis of such higher-dimensional systems is usually focused on finding conditions for the existence and stability of typical invariant sets, i.e. steady states, periodic solutions and chaotic sets. High-dimensionality leads to complex patterns of collective behavior. Which type of behavior is exhibited by a network depends greatly on the initial conditions. Thus, it is also important to study the geometric structure and evolution of the basins of attraction of such patterns. In this manuscript, a complete study of the basins of attraction of a ring of bistable systems, coupled unidirectionally, is presented. 3D visualizations are included to aid the discussion of the changes in the basins of attraction as the coupling parameter varies. The results are broad enough that they can be applied to a wide range of systems with similar coupling topologies.

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Section: Discussionsupporting

confidence: 52%

Geometry of Basins of Attraction and Heteroclinic Connections in Coupled Bistable Systems

Lyons

Mahaffy

Wang

et al. 2014

Int. J. Bifurcation Chaos

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“…In particular, to help us understand the potential effects of delay in the signal processing module that reads out and processes the output of each individual fluxgate before it can be used to drive the dynamics of another fluxgate. In [29] we investigated the behavior of a N-dimensional (N odd for negative feedback) uni-directionally coupled ring of overdamped bistable systems with delayed nearest-neighbor connections. As a test case, we use the model Eq.…”

Section: Effects Of Time Delaymentioning

confidence: 99%

Coupled overdamped bistable systems with applications to sensor devices

Kho

Longhini

et al. 2012

NOLTA

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Theoretical and experimental works have shown that coupling similar overdamped bistable systems can lead, under certain conditions that depend on the topology of connections and the number of units, to self-induced large-amplitude oscillations that emerge through a global bifurcation of heteroclinic connections between saddle-node equilibria. We have exploited this fundamental feature to model, design, and fabricate a new generation of highly-sensitive, low-powered, sensor devices, mainly detectors of magnetic-and electric-fields. In this article we review the fundamental principles and methods behind this new paradigm. These principles are device-independent so they can be readily adapted to a wide range of sensors, such as acoustic and gyroscopic sensors among many types. In this manuscript we describe the design and fabrication of a new class of highly-sensitive fluxgate-and electric-field magnetometers as a case study to review basic ideas and methods.

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“…Orosz [41] developed an effective decomposition method for investigating the dynamics in the vicinity of steady and oscillatory cluster states and showed the coexistence of multiple stable and unstable steady and oscillatory cluster states. For more related studies on the dynamics of coupled systems with time delays, the reader is referred to the work in [14,[42][43][44][45][46][47], and some references cited therein. In previous studies, the authors often limited their research to the models of two or three coupled sub-systems and/or the models with a special structure since such models are fundamental and relatively simple.…”

Section: Introductionmentioning

confidence: 99%

Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

Mao

Wang

2015

Nonlinear Dyn

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This paper reveals the dynamical behaviors of a coupled network consisting of four neural subnetworks and multiple two-way couplings of neurons between the individual sub-networks. Different types of time delays are introduced into the internal connections within the individual sub-networks and the couplings between the sub-networks. Stability switches of the network equilibrium and Hopf bifurcation are analyzed by discussing the associated characteristic equation. The conditions for the existence of different patterns of oscillations are discussed. By using the Lyapunov's second method, the global stability of the network equilibrium is studied. Numerical simulations are performed to validate the theoretical results, and rich dynamical behaviors are observed, such as synchronous oscillations, multiple stability switches between the rest state and oscillations, phase-locked oscillations, asynchronous oscillations, and the coexistence of different patterns of bifurcated oscillations.

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Basins of attraction in a ring of overdamped bistable systems with delayed coupling

Cited by 6 publications

References 25 publications

Geometry of Basins of Attraction and Heteroclinic Connections in Coupled Bistable Systems

Geometry of Basins of Attraction and Heteroclinic Connections in Coupled Bistable Systems

Coupled overdamped bistable systems with applications to sensor devices

Stability switches and bifurcation in a system of four coupled neural networks with multiple time delays

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