scholarly journals Transient Periodicity in a Morris-Lecar Neural System

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Sreenivasan Rajamoni Nadar ◽  
Vikas Rai
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Dynamical Behavior ◽  
Chaotic Oscillation ◽  
Three Dimensional ◽  
Neural System ◽  
Inhibitory Synapses ◽  
Dimensional System ◽  
Dynamical Complexity ◽  
Three Dimensional System

The dynamical complexity of a system of ordinary differential equations (ODEs) modeling the dynamics of a neuron that interacts with other neurons through on-off excitatory and inhibitory synapses in a neural system was investigated in detail. The model used Morris-Lecar (ML) equations with an additional autonomous variable representing the input from interaction of excitatory neuronal cells with local interneurons. Numerical simulations yielded a rich repertoire of dynamical behavior associated with this three-dimensional system, which included periodic, chaotic oscillation and rare bursts of episodic periodicity called the transient periodicity.

The theorem of the carrying simplex for competitive system defined on the n-rectangle and its application to a three-dimensional system

2014 ◽  
Vol 07 (06) ◽  
pp. 1450063 ◽  
Author(s):  
Jifa Jiang ◽  
Lei Niu
Keyword(s):  
Dynamical Behavior ◽  
Three Dimensional ◽  
Linear Structure ◽  
Positive Equilibrium ◽  
Hopf Bifurcations ◽  
Dimensional System ◽  
Asymptotically Stable ◽  
Competitive System ◽  
Globally Asymptotically Stable ◽  
Three Dimensional System

First, we show that the theorem by Hirsch which guarantees the existence of carrying simplex for competitive system on any n-rectangle: {x ∈ Rn : 0 ≤ xi ≤ ki, i = 1, …, n} still holds. Next, based on the theorem a competitive system with the linear structure saturation defined on the n-rectangle is investigated, which admits a unique (n - 1)-dimensional carrying simplex as a global attractor. Further, we focus on the whole dynamical behavior of the three-dimensional case, which has a unique locally asymptotically stable positive equilibrium. Hopf bifurcations do not occur. We prove that any limit set is either this positive equilibrium or a limit cycle. If limit cycles exist, the number of them is finite. We also give a criterion for the positive equilibrium to be globally asymptotically stable.

scholarly journals Generalized Attracting Horseshoe in the Rössler Attractor

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 30
Author(s):  
Karthik Murthy ◽  
Ian Jordan ◽  
Parth Sojitra ◽  
Aminur Rahman ◽  
Denis Blackmore
Keyword(s):  
Differential Equations ◽  
Strange Attractor ◽  
Electronic Circuit ◽  
Three Dimensional ◽  
Dimensional System ◽  
Numerical Techniques ◽  
Circuit Realization ◽  
Trapping Region ◽  
Potential Applications ◽  
Three Dimensional System

We show that there is a mildly nonlinear three-dimensional system of ordinary differential equations—realizable by a rather simple electronic circuit—capable of producing a generalized attracting horseshoe map. A system specifically designed to have a Poincaré section yielding the desired map is described, but not pursued due to its complexity, which makes the construction of a circuit realization exceedingly difficult. Instead, the generalized attracting horseshoe and its trapping region is obtained by using a carefully chosen Poincaré map of the Rössler attractor. Novel numerical techniques are employed to iterate the map of the trapping region to approximate the chaotic strange attractor contained in the generalized attracting horseshoe, and an electronic circuit is constructed to produce the map. Several potential applications of the idea of a generalized attracting horseshoe and a physical electronic circuit realization are proposed.

Symmetry and conservation laws for tuberculosis model

2017 ◽  
Vol 10 (03) ◽  
pp. 1750042 ◽  
Author(s):  
Maba Boniface Matadi
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Three Dimensional ◽  
Symmetry And Conservation Laws ◽  
Second Order ◽  
Dimensional System ◽  
First Order ◽  
Tuberculosis Model ◽  
Nonlinear Lagrangian ◽  
Three Dimensional System

In this paper, three-dimensional system of the tuberculosis (TB) model is reduced into a two-dimensional first-order and one-dimensional second-order differential equations. We use the method of Jacobi last multiplier to construct linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation. The Noether's theorem is used for determining conservation laws. We apply the techniques of symmetry analysis to a model to identify the combinations of parameters which lead to the possibility of the linearization of the system and provide the corresponding solutions.

TRAVELLING WAVES IN A CIRCULAR ARRAY OF CHUA’S CIRCUITS

1996 ◽  
Vol 06 (03) ◽  
pp. 473-484 ◽  
Author(s):  
VLADIMIR I. NEKORKIN ◽  
VICTOR B. KAZANTSEV ◽  
MANUEL G. VELARDE
Keyword(s):  
Differential Equations ◽  
Periodic Orbits ◽  
Travelling Waves ◽  
Three Dimensional ◽  
Dimensional System ◽  
Circular Array ◽  
One Dimensional ◽  
Numerical Studies ◽  
The Individual ◽  
Three Dimensional System

The possibility of travelling waves in a one-dimensional circular array of Chua's circuits is investigated. It is shown that the problem can be reduced to the analysis of the periodic orbits of a three-dimensional system of ordinary differential equations (ODEs) describing the individual dynamics of Chua's circuit. The results of analytical and numerical studies of the bifurcation associated with the appearance of the periodic orbits are presented. A criterion for stability of the travelling waves is also provided.

Integrable Deformations of Three-Dimensional Chaotic Systems

2018 ◽  
Vol 28 (05) ◽  
pp. 1850066 ◽  
Author(s):  
Cristian Lăzureanu
Keyword(s):  
Differential Equations ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Dimensional System ◽  
Poisson System ◽  
System Of Differential Equations ◽  
Deformation Method ◽  
Integrable Deformations ◽  
Three Dimensional System

The integrable deformation method for a three-dimensional Hamilton–Poisson system consists in alteration of its constants of motion in order to obtain a new Hamilton–Poisson system. We assume that a three-dimensional system of differential equations has a Hamilton–Poisson part and a nonconservative part. We give integrable deformations of the Hamilton–Poisson part and, adding the nonconservative part, we obtain integrable deformations of the considered three-dimensional system of differential equations. In particular, applying this method to chaotic systems may lead to new systems with chaotic behavior. We use this method to obtain integrable deformations of Lorenz, Chen, and Rössler systems. Using particular deformation functions, we have pointed out some deformations of the above-mentioned attractors.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

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