A Probabilistic Theory of Nonlinear Dynamical Systems Based on the Cell State Space Concept

1982 ◽  
Vol 49 (4) ◽  
pp. 895-902 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical Systems ◽  
State Space ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Cell Structure ◽  
The State ◽  
Probabilistic Theory ◽  
Nonlinear Dynamical ◽  
Linear Ordinary Differential Equations ◽  
Highly Nonlinear

Developed in the paper is a probabilistic theory for nonlinear dynamical systems. The theory is based on discretizing the state space into a cell structure and using the cell probability functions to describe the state of a system. Although the dynamical system may be highly nonlinear the probabilistic formulation always leads to a set of linear ordinary differential equations. The evolution of the probability distribution among the cells can then be studied by applying the theory of Markov processes to this set of equations. It is believed that this development possibly offers a new approach to the global analysis of nonlinear systems.

scholarly journals Phase Synchronization Is the Amplified Result by the Hilbert Transform

2015 ◽  
Vol 2015 ◽  
pp. 1-3 ◽  
Author(s):  
Ming-Chi Lu ◽  
Hsing-Chung Ho ◽  
Chen-An Chan ◽  
Chia-Ju Liu ◽  
Jiann-Shing Lih ◽  
...  
Keyword(s):  
Time Series ◽  
Dynamical Systems ◽  
Hilbert Transform ◽  
Phase Synchronization ◽  
Nonlinear Dynamical Systems ◽  
The State ◽  
Nonlinear Dynamical ◽  
Large Correlation ◽  
Current Usage ◽  
The Hilbert Transform

We investigate the interplay between phase synchronization and amplitude synchronization in nonlinear dynamical systems. It is numerically found that phase synchronization intends to be established earlier than amplitude synchronization. Nevertheless, amplitude synchronization (or the state with large correlation between the amplitudes) is crucial for the maintenance of a high correlation between two time series. A breakdown of high correlation in amplitudes will lead to a desynchronization of two time series. It is shown that these unique features are caused essentially by the Hilbert transform. This leads to a deep concern and criticism on the current usage of phase synchronization.

GLOBAL ANALYSIS OF DYNAMICAL SYSTEMS USING POSETS AND DIGRAPHS

1995 ◽  
Vol 05 (04) ◽  
pp. 1085-1118 ◽  
Author(s):  
C. S. HSU
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Ordered Sets ◽  
Cell Mapping ◽  
Basic Notion ◽  
Nonlinear Dynamical ◽  
Global Evolution ◽  
Computation Algorithms ◽  
Primitive Notion

In this paper the resources of the theory of partially ordered sets (posets) and the theory of digraphs are used to aid the task of global analysis of nonlinear dynamical systems. The basic idea underpinning this approach is the primitive notion that a dynamical systems is simply an ordering machine which assigns fore-and-after relations for pairs of states. In order to make the linkage between the theory of posets and digraphs and dynamical systems, cell mapping is used to put dynamical systems in their discretized form and an essential concept of self-cycling sets is used. After a discussion of the basic notion of ordering, appropriate results from the theory of posets and digraphs are adapted for the purpose of determining the global evolution properties of dynamical systems. In terms of posets, evolution processes and strange attractors can be studied in a new light. It is believed that this approach offers us a new way to examine the multifaceted complex behavior of nonlinear systems. Computation algorithms are also discussed and an example of application is included.

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