Nonlinear Dynamics of the BZ Reaction: A Simple Experiment that Illustrates Limit Cycles, Chaos, Bifurcations, and Noise

1996 ◽  
Vol 73 (9) ◽  
pp. 868 ◽  
Author(s):  
Peter Strizhak ◽  
Michael Menzinger
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Simple Experiment ◽  
Bz Reaction

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.

Nonlinear Phenomena in Thermoacoustic Systems With Premixed Flames

2012 ◽  
Author(s):  
Karthik Kashinath ◽  
Santosh Hemchandra ◽  
Matthew P. Juniper
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycle ◽  
Release Rate ◽  
Time Domain ◽  
Limit Cycles ◽  
Integral Relation ◽  
Single Mode ◽  
Rate Of Change ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

Nonlinear analysis of thermoacoustic instability is essential for prediction of frequencies, amplitudes and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, but can be used to calculate the amplitudes of limit cycles, as well as their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modelled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearised momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state, for single-mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitude-dependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, as well as twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact resulting in a larger limit cycle amplitude. Multi-mode simulations show that in some situations the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems.

scholarly journals A KICKED OSCILLATOR AS A MODEL OF A PULSED MEMS SYSTEM

2009 ◽  
Vol 19 (01) ◽  
pp. 187-202 ◽  
Author(s):  
ELENA BLOKHINA ◽  
ORLA FEELY
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Initial Conditions ◽  
Control Parameters ◽  
Rotation Numbers ◽  
Mems Oscillator ◽  
Iterative Maps ◽  
Damped Oscillator

In this paper, we study the behavior of a MEMS oscillator by applying methods of nonlinear dynamics. We model the system as a kicked damped oscillator and obtain the iterative maps that describe the MEMS system. The dynamics of the maps are studied numerically: we present a study of limit cycles and their rotation numbers and show how the control parameters and initial conditions may affect the output frequencies.

Nonlinear Dynamics and Symbolic Dynamics of Neural Networks

1992 ◽  
Vol 4 (5) ◽  
pp. 621-642 ◽  
Author(s):  
John E. Lewis ◽  
Leon Glass
Keyword(s):  
Nonlinear Dynamics ◽  
Neural Networks ◽  
Efficient Method ◽  
Limit Cycles ◽  
Symbolic Dynamics ◽  
Chaotic Dynamics ◽  
Piecewise Linear ◽  
Stable Limit ◽  
Model Networks ◽  
Stable Limit Cycles

A piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks. A symbolic representation of the dynamics in this equation is given as a directed graph on an N-dimensional hypercube. This provides a formal link with discrete neural networks such as the original Hopfield models. Analytic criteria are given to establish steady states and limit cycle oscillations independent of network dimension. Model networks that display multiple stable limit cycles and chaotic dynamics are discussed. The results show that such equations are a useful and efficient method of investigating the behavior of neural networks.

scholarly journals SINDy-PI: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200279
Author(s):  
Kadierdan Kaheman ◽  
J. Nathan Kutz ◽  
Steven L. Brunton
Keyword(s):  
Nonlinear Dynamics ◽  
Measurement Data ◽  
Noisy Data ◽  
Rational Functions ◽  
Double Pendulum ◽  
Bz Reaction ◽  
Systems Models ◽  
Noise Robust ◽  
Pendulum Dynamics ◽  
Robust Variant

Accurately modelling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and simplified model for the Belousov–Zhabotinsky (BZ) reaction.

FREQUENCY DOMAIN APPROACH TO COMPUTATION AND ANALYSIS OF BIFURCATIONS AND LIMIT CYCLES: A TUTORIAL

1993 ◽  
Vol 03 (04) ◽  
pp. 843-867 ◽  
Author(s):  
JORGE L. MOIOLA ◽  
GUANRONG CHEN
Keyword(s):  
Nonlinear Dynamics ◽  
Frequency Domain ◽  
Control Systems ◽  
Time Domain ◽  
Limit Cycles ◽  
Nonlinear Dynamical Systems ◽  
Graphical Analysis ◽  
Nonlinear Dynamical ◽  
Feedback Control Systems ◽  
Frequency Domain Approach

This paper introduces a frequency domain approach together with some techniques and methodologies for the computation and analysis of bifurcations and limit cycles arising in nonlinear dynamical systems. The frequency domain approach discussed in this paper originates from the classical feedback control systems theory, which has been proven to be successful and efficient for the computation and analysis of regular as well as singular bifurcations and stable as well as unstable limit cycles. While describing these techniques and methods, two representative yet distinct applications of the approach are studied in detail: The graphical analysis of multiple parametric bifurcation curves and the numerical computation of multiple limit cycles. Compared to the classical time domain methods, both the advantages and the limitations of the frequency domain approach are analyzed and discussed. It is believed that this frequency domain approach to the study of nonlinear dynamics has great potential and promising future in both theory and applications.

Sign in / Sign up

Export Citation Format

Share Document