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.

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.

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.

Inner-product theory calculations for some rational potentials in a three-dimensional system

1996 ◽  
Vol 74 (1-2) ◽  
pp. 4-9
Author(s):  
M. R. M. Witwit
Keyword(s):  
Energy Levels ◽  
Three Dimensional ◽  
Inner Product ◽  
Dimensional System ◽  
Product Technique ◽  
Special Cases ◽  
Wide Range ◽  
Perturbation Parameters ◽  
Three Dimensional System ◽  
Range Of Values

The energy levels of a three-dimensional system are calculated for the rational potentials,[Formula: see text]using the inner-product technique over a wide range of values of the perturbation parameters (λ, g) and for various eigenstates. The numerical results for some special cases agree with those of previous workers where available.

The influence of mean shear on Alfvén-internal wave propagation

1976 ◽  
Vol 15 (2) ◽  
pp. 197-222
Author(s):  
R. J. Hartman
Keyword(s):  
Three Dimensional ◽  
Dimensional System ◽  
Linear Wave ◽  
Mean Flow ◽  
Wave Processes ◽  
Initial Value ◽  
Initial Wave ◽  
Wave Phenomena ◽  
Three Dimensional System

This paper uses the general solution of the linearized initial-value problem for an unbounded, exponentially-stratified, perfectly-conducting Couette flow in the presence of a uniform magnetic field to study the development of localized wave-type perturbations to the basic flow. The two-dimensional problem is shown to be stable for all hydrodynamic Richardson numbers JH, positive and negative, and wave packets in this flow are shown to approach, asymptotically, a level in the fluid (the ‘isolation level’) which is a smooth, continuous, function of JH that is well defined for JH < 0 as well as JH > 0. This system exhibits a rich complement of wave phenomena and a variety of mechanisms for the transport of mean flow kinetic and potential energy, via linear wave processes, between widely-separated regions of fluid; this in addition to the usual mechanisms for the absorption of the initial wave energy itself. The appropriate three-dimensional system is discussed, and the role of nonlinearities on the development of localized disturbances is considered.

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