ON THE NUMERICAL METHOD OF CONSTRUCTION OF UNSTABLE SOLUTIONS OF DYNAMICAL SYSTEMS WITH QUADRATIC NONLINEARITIES

2018 ◽  
pp. 555-565
Author(s):  
Alexander Nikolaevich Pchelintsev
Keyword(s):  
Dynamical Systems ◽  
Power Series ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Approximate Solutions ◽  
Convergence Of Series ◽  
Quadratic Nonlinearities ◽  
Unstable Solutions ◽  
Region Of Convergence

In this paper, the author considers the modification of the method of power series for the numerical construction of unstable solutions of systems of ordinary differential equations of chaotic type with quadratic nonlinearities in general form. A region of convergence of series is found and an algorithm for constructing approximate solutions is proposed.

scholarly journals A New Reliable Numerical Method for Computing Chaotic Solutions of Dynamical Systems: The Chen Attractor Case

2015 ◽  
Vol 25 (13) ◽  
pp. 1550187 ◽  
Author(s):  
René Lozi ◽  
Alexander N. Pchelintsev
Keyword(s):  
Dynamical Systems ◽  
Power Series ◽  
Numerical Method ◽  
Approximate Solutions ◽  
Chen System ◽  
Convergence Of Series ◽  
Special Cases ◽  
Backward Control ◽  
Numerical Precision ◽  
Region Of Convergence

This paper describes a new reliable numerical method for computing chaotic solutions of dynamical systems and, in special cases, is applied to Chen strange attractor. The numerical precision of the computation is finely mastered. We introduce a modification of the method of power series for the construction of approximate solutions of the Chen system together with forward/backward control of the precision. As a test for the method, we obtained the region of convergence of series and researched the behavior of the trajectories on this attractor. The results of a numerical experiment are presented.

THE THEORY OF CONFINORS IN CHUA’S CIRCUIT: ACCURATE ANALYSIS OF BIFURCATIONS AND ATTRACTORS

1993 ◽  
Vol 03 (02) ◽  
pp. 333-361 ◽  
Author(s):  
RENÉ LOZI ◽  
SHIGEHIRO USHIKI
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Phase Portrait ◽  
Chaotic Attractors ◽  
Chua’S Circuit ◽  
Bifurcation Diagrams ◽  
Accurate Analysis ◽  
Chua's Circuit

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincaré maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

scholarly journals Numerical method for parameter inference of systems of nonlinear ordinary differential equations with partial observations

2021 ◽  
Vol 8 (7) ◽  
pp. 210171
Author(s):  
Yu Chen ◽  
Jin Cheng ◽  
Arvind Gupta ◽  
Huaxiong Huang ◽  
Shixin Xu
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Gaussian Process ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Optimization Algorithm ◽  
Nonlinear Ordinary Differential Equations ◽  
Parameter Inference ◽  
Partial Observations ◽  
Deterministic Optimization

Parameter inference of dynamical systems is a challenging task faced by many researchers and practitioners across various fields. In many applications, it is common that only limited variables are observable. In this paper, we propose a method for parameter inference of a system of nonlinear coupled ordinary differential equations with partial observations. Our method combines fast Gaussian process-based gradient matching and deterministic optimization algorithms. By using initial values obtained by Bayesian steps with low sampling numbers, our deterministic optimization algorithm is both accurate, robust and efficient with partial observations and large noise.

scholarly journals On attracting sets in artificial networks: cross activation

2018 ◽  
Vol 16 ◽  
pp. 01005
Author(s):  
Felix Sadyrbaev
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Mathematical Models ◽  
Ordinary Differential Equations ◽  
Critical Points ◽  
Regulatory Networks ◽  
Main Diagonal ◽  
Sigmoidal Function ◽  
Genomic Regulatory Networks ◽  
Over Time

Mathematical models of artificial networks can be formulated in terms of dynamical systems describing the behaviour of a network over time. The interrelation between nodes (elements) of a network is encoded in the regulatory matrix. We consider a system of ordinary differential equations that describes in particular also genomic regulatory networks (GRN) and contains a sigmoidal function. The results are presented on attractors of such systems for a particular case of cross activation. The regulatory matrix is then of particular form consisting of unit entries everywhere except the main diagonal. We show that such a system can have not more than three critical points. At least n–1 eigenvalues corresponding to any of the critical points are negative. An example for a particular choice of sigmoidal function is considered.

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