Control of Chaos in Networks with Delay: A Model for Synchronization of Cortical Tissue

1994 ◽  
Vol 6 (6) ◽  
pp. 1141-1154 ◽  
Author(s):  
C. Lourenço ◽  
A. Babloyantz
Keyword(s):  
Differential Equations ◽  
Periodic Orbits ◽  
Delay Differential Equations ◽  
Chaotic Dynamics ◽  
Cortical Activity ◽  
Inhibitory Neurons ◽  
Unstable Periodic Orbits ◽  
Cortical Tissue ◽  
Control Of Chaos ◽  
Delay Differential

The unstable periodic orbits of chaotic dynamics in systems described by delay differential equations are considered. An orbit is stabilized successfully, using a method proposed by Pyragas. The system under investigation is a network of excitatory and inhibitory neurons of moderate size, describing cortical activity. The relevance of the results for synchronized cortical activity is discussed.

Periodic Orbits and Chaos in Nonsmooth Delay Differential Equations

2019 ◽  
Vol 29 (10) ◽  
pp. 1950137
Author(s):  
Andrea Bel ◽  
Romina Cobiaga ◽  
Walter Reartes
Keyword(s):  
Differential Equations ◽  
Periodic Orbits ◽  
Delay Differential Equations ◽  
Order Differential Equation ◽  
Stable Periodic Solution ◽  
Unstable Periodic Orbits ◽  
First Order ◽  
Stable Periodic ◽  
Delay Differential ◽  
Discrete Map

In this paper, we present a method to find periodic solutions for certain types of nonsmooth differential equations or nonsmooth delay differential equations. We apply the method to three examples, the first is a second-order differential equation with a nonsmooth term, in this case the method allows us to find periodic orbits in a nonlinear center. The two remaining examples are first-order nonsmooth delay differential equations. In the first one, there is a stable periodic solution and in the second, the presence of a chaotic attractor was detected. In the latter, the method allows us to obtain unstable periodic orbits within the attractor. For large values of the delay, both examples can be seen as singularly perturbed delay differential equations. For them, an analysis is performed with an associated discrete map which is obtained in the limit of large delays.

Search for Periodic Orbits in Delay Differential Equations

2014 ◽  
Vol 24 (06) ◽  
pp. 1450084 ◽  
Author(s):  
Romina Cobiaga ◽  
Walter Reartes
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Heuristic Search ◽  
Delay Differential Equations ◽  
Anharmonic Oscillator ◽  
Analysis Method ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Delay Differential

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback.

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