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Limit cycle

About: Limit cycle is a(n) research topic. Over the lifetime, 5635 publication(s) have been published within this topic receiving 115013 citation(s). more


Journal ArticleDOI: 10.1109/9.664150
Michael S. Branicky1Institutions (1)
Abstract: We introduce some analysis tools for switched and hybrid systems. We first present work on stability analysis. We introduce multiple Lyapunov functions as a tool for analyzing Lyapunov stability and use iterated function systems theory as a tool for Lagrange stability. We also discuss the case where the switched systems are indexed by an arbitrary compact set. Finally, we extend Bendixson's theorem to the case of Lipschitz continuous vector fields, allowing limit cycle analysis of a class of "continuous switched" systems. more

Topics: Lyapunov function (64%), Lyapunov equation (62%), Lyapunov stability (60%) more

3,136 Citations

Open accessJournal ArticleDOI: 10.1016/S0006-3495(72)86068-5
Abstract: Coupled nonlinear differential equations are derived for the dynamics of spatially localized populations containing both excitatory and inhibitory model neurons. Phase plane methods and numerical solutions are then used to investigate population responses to various types of stimuli. The results obtained show simple and multiple hysteresis phenomena and limit cycle activity. The latter is particularly interesting since the frequency of the limit cycle oscillation is found to be a monotonic function of stimulus intensity. Finally, it is proved that the existence of limit cycle dynamics in response to one class of stimuli implies the existence of multiple stable states and hysteresis in response to a different class of stimuli. The relation between these findings and a number of experiments is discussed. more

Topics: Wilson–Cowan model (65%), Limit cycle (54%), Population (52%) more

2,980 Citations

Open accessBook
Shijun Liao1, SA Sherif2Institutions (2)
27 Oct 2003-
Abstract: PART I BASIC IDEAS Introduction Illustrative Description Systematic Description Relations to Some Previous Analytic Methods Advantages, Limitations, and Open Questions PART II APPLICATIONS Simple Bifurcation of a Nonlinear Problem Multiple Solutions of a Nonlinear Problem Nonlinear Eigenvalue Problem Thomas-Fermi Atom Model Volterra's Population Model Free Oscillation Systems with Odd Nonlinearity Free Oscillation Systems with Quadratic Nonlinearity Limit Cycle in a Multidimensional System Blasius' viscous Flow Boundary-layer Flow with Exponential Property Boundary-layer Flow with Algebraic Property Von Karman Swirling Flow Nonlinear Progressive Waves in Deep Water BIBLIOGRAPHY INDEX more

Topics: Homotopy analysis method (58%), Nonlinear system (55%), Limit cycle (54%) more

2,737 Citations

Journal ArticleDOI: 10.1098/RSPB.1984.0024
J. L. Hindmarsh1, R. M. Rose1Institutions (1)
Abstract: We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron into one that fires repetitively. Addition of a third equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge. more

Topics: Theta model (59%), Equilibrium point (56%), Phase plane (53%) more

1,459 Citations

Open access
01 Jan 2004-
Abstract: Complex dynamics in economics arise from nonlinear systems that do not converge to a fixed point, a limit cycle, or explode or implode exponentially due to endogenous factors. They arise from cybernetics, catastrophe theory, chaos theory, or the varieties of modern complexity theory, including models with heterogeneous, interacting agents. This major three-volume collection presents the most important papers in the area of complexity in economics. more

Topics: Catastrophe theory (57%), Chaos theory (53%), Cybernetics (53%) more

1,408 Citations

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Topic's top 5 most impactful authors

Jaume Llibre

73 papers, 1.4K citations

Maoan Han

72 papers, 1.6K citations

Pei Yu

33 papers, 528 citations

Somanath Majhi

25 papers, 472 citations

Hiroya Nakao

21 papers, 509 citations

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