Introduction: Differential Equations and Dynamical Systems

1983 ◽  
pp. 1-65 ◽  
Author(s):  
John Guckenheimer ◽  
Philip Holmes
Keyword(s):  
Dynamical Systems ◽  
Differential Equations

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.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

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