Chapter 3. Chaos in Deterministic Systems and the Melnikov Function

2002 ◽  
pp. 51-75
Keyword(s):  
Melnikov Function ◽  
Deterministic Systems

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

Self-organization

2018 ◽  
Author(s):  
Stuart P. Wilson
Keyword(s):  
Large Deviation ◽  
Natural World ◽  
Self Organization ◽  
Global Order ◽  
Deterministic Systems ◽  
Working Definition ◽  
Complex Forms ◽  
Robust To Noise ◽  
Organizing Principles ◽  
Self Organizing

Self-organization describes a dynamic in a system whereby local interactions between individuals collectively yield global order, i.e. spatial patterns unobservable in their entirety to the individuals. By this working definition, self-organization is intimately related to chaos, i.e. global order in the dynamics of deterministic systems that are locally unpredictable. A useful distinction is that a small perturbation to a chaotic system causes a large deviation in its trajectory, i.e. the butterfly effect, whereas self-organizing patterns are robust to noise and perturbation. For many, self-organization is as important to the understanding of biological processes as natural selection. For some, self-organization explains where the complex forms that compete for survival in the natural world originate from. This chapter outlines some fundamental ideas from the study of simulated self-organizing systems, before suggesting how self-organizing principles could be applied through biohybrid societies to establish new theories of living systems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

scholarly journals Modeling Noisy Time Series: Physiological Tremor

1998 ◽  
Vol 08 (07) ◽  
pp. 1505-1516 ◽  
Author(s):  
J. Timmer
Keyword(s):  
Time Series ◽  
Stochastic Systems ◽  
State Space Model ◽  
Physiological Tremor ◽  
Space Model ◽  
Deterministic Systems ◽  
Estimated Parameters ◽  
Noisy Time Series ◽  
Observational Noise ◽  
Small Amplitude Oscillation

Empirical time series often contain observational noise. We investigate the effect of this noise on the estimated parameters of models fitted to the data. For data of physiological tremor, i.e. a small amplitude oscillation of the outstretched hand of healthy subjects, we compare the results for a linear model that explicitly includes additional observational noise to one that ignores this noise. We discuss problems and possible solutions for nonlinear deterministic as well as nonlinear stochastic processes. Especially we discuss the state space model applicable for modeling noisy stochastic systems and Bock's algorithm capable for modeling noisy deterministic systems.

Global Asymptotic Stabilization of Nonlinear Deterministic Systems Using Wiener Processes

2016 ◽  
Vol 61 (8) ◽  
pp. 2318-2323 ◽  
Author(s):  
Kenta Hoshino ◽  
Yuki Nishimura ◽  
Yuh Yamashita ◽  
Daisuke Tsubakino
Keyword(s):  
Asymptotic Stabilization ◽  
Global Asymptotic Stabilization ◽  
Deterministic Systems ◽  
Wiener Processes
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