Switching control of limit cycles in planar systems: a nonlinear dynamics approach

2004 ◽  
Author(s):  
F. Angulo ◽  
M. di Bernardo ◽  
G. Olivar
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Switching Control ◽  
Planar Systems

PERIODIC INHIBITION OF LIVING PACEMAKER NEURONS (I): LOCKED, INTERMITTENT, MESSY, AND HOPPING BEHAVIORS

1991 ◽  
Vol 01 (03) ◽  
pp. 549-581 ◽  
Author(s):  
J. P. SEGUNDO ◽  
E. ALTSHULER ◽  
M. STIBER ◽  
A. GARFINKEL
Keyword(s):  
Nonlinear Dynamics ◽  
Point Process ◽  
Limit Cycles ◽  
Nonlinear Oscillator ◽  
Strange Attractors ◽  
Synaptic Inhibition ◽  
Pacemaker Neurons ◽  
Pacemaker Neuron ◽  
Exhaustive List

This communication is concerned with an embodiment of periodic nonlinear oscillator driving, the synaptic inhibition of one spike-producing pacemaker neuron by another. Data came from a prototypical living synapse. Analyses centered on a prolonged condition between the transients following the onset and cessation of inhibition. Evaluations were guided by point process mathematics and nonlinear dynamics. A rich and exhaustive list of discharge forms, described precisely and canonically, was observed across different inhibitory rates. Previously unrecognized at synapses, most forms were identified with several well known types from nonlinear dynamics. Ordered by decreasing regularities, they were locked, intermittent (including walk-throughs), messy (including erratic and stammerings) and hopping. Each is discussed within physiological and formal contexts. It is conjectured that (i) locked, intermittent and messy forms reflect limit cycles on 2-tori, quasiperiodic orbits and strange attractors, (ii) noise in neurons hovering around threshold contributes to certain intermittent and stammering behaviors, and (iii) hopping either reflects an attractor with several portions or is nonstationary and noise-induced.

scholarly journals DYNAMICS OF DELAY-COUPLED EXCITABLE NEURAL SYSTEMS

2009 ◽  
Vol 19 (02) ◽  
pp. 745-753 ◽  
Author(s):  
M. A. DAHLEM ◽  
G. HILLER ◽  
A. PANCHUK ◽  
E. SCHÖLL
Keyword(s):  
Nonlinear Dynamics ◽  
Fixed Point ◽  
Limit Cycles ◽  
Coupling Strength ◽  
Neural Systems ◽  
Stable Fixed Point ◽  
Bifurcation Of Limit Cycles ◽  
Large Delay ◽  
Saddle Node Bifurcation ◽  
Delay Times

We study the nonlinear dynamics of two delay-coupled neural systems each modeled by excitable dynamics of FitzHugh–Nagumo type and demonstrate that bistability between the stable fixed point and limit cycle oscillations occurs for sufficiently large delay times τ and coupling strength C. As the mechanism for these delay-induced oscillations, we identify a saddle-node bifurcation of limit cycles.

Revealing Nonlinear Dynamics Phenomena in Mooring Due to Slowly-Varying Drift

2005 ◽  
Author(s):  
Michael M. Bernitsas ◽  
Joa˜o Paulo J. Matsuura
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Resonance Phenomenon ◽  
Nonlinear Phenomena ◽  
Mooring System ◽  
Stable Limit ◽  
Dynamic Interactions ◽  
Wave Drift

The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drift and mooring systems. We were able to reveal new phenomena based on the design methodology developed at the University of Michigan for autonomous mooring systems and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or non-periodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes 2–3 orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.

NEURONAL OSCILLATIONS AND STOCHASTIC LIMIT CYCLES

1996 ◽  
Vol 07 (04) ◽  
pp. 399-402 ◽  
Author(s):  
CHRISTIAN KURRER ◽  
KLAUS SCHULTEN
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Neural Activity ◽  
Oscillatory Behavior ◽  
Neuronal Oscillations ◽  
Coupled Oscillator ◽  
Stochastic Limit ◽  
Excitable Systems ◽  
The Individual ◽  
Coupled Oscillator Systems

We investigate a model for synchronous neural activity in networks of coupled neurons. The individual systems are governed by nonlinear dynamics and can continuously vary between excitable and oscillatory behavior. Analytical calculations and computer simulations show that coupled excitable systems can undergo two different phase transitions from synchronous to asynchronous firing behavior. One of the transitions is akin to the synchronization transitions in coupled oscillator systems, while the second transition can only be found in coupled excitable systems. Using the concept of Stochastic Limit Cycles, we present an analytical derivation of the two transitions and discuss implications for synchronization transitions in biological neural networks.

MONODROMY AND STABILITY FOR NILPOTENT CRITICAL POINTS

2005 ◽  
Vol 15 (04) ◽  
pp. 1253-1265 ◽  
Author(s):  
M. J. ÁLVAREZ ◽  
A. GASULL
Keyword(s):  
Critical Points ◽  
Limit Cycles ◽  
Stability Problem ◽  
Short Proof ◽  
Sufficient Conditions ◽  
Planar Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Lyapunov Constants

We give a new and short proof of the characterization of monodromic nilpotent critical points. We also calculate the first generalized Lyapunov constants in order to solve the stability problem. We apply the results to several families of planar systems obtaining necessary and sufficient conditions for having a center. Our method also allows us to generate limit cycles from the origin.

On the limit cycles for a class of planar systems

1995 ◽  
Vol 24 (4) ◽  
pp. 605-614 ◽  
Author(s):  
Zheng Zuo-Huan
Keyword(s):  
Limit Cycles ◽  
Planar Systems
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