scholarly journals ESTUDO DO COMPORTAMENTO DINÂMICO DO MODELO NEURONAL DE HINDMARSH-ROSE

2019 ◽  
Vol 11 (4) ◽  
pp. 122-130
Author(s):  
RaildoSantos de Lima ◽  
Fábio Roberto Chavarette ◽  
Luiz Gustavo Pereira Roéfero Roéfero
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Nerve Impulse ◽  
Neuronal Model ◽  
Random Behavior ◽  
Highly Sensitive ◽  
Real Neuron ◽  
Hr Model ◽  
Biological Neuron ◽  
The Stability

Based on the Hindmarsh-Rose (RH) neuronal model for nerve impulse transmission, this paper aims to study the properties and dynamic behavior of the non-linear chaotic system that describes neuronal bursting in a single neuron. On the part of bioengineering, there is great motivation in the study of the HR model because it is well representative of the biological neuron, being able to simulate several behaviors of a real neuron, among them periodic, aperiodic and chaotic behavior. The literature suggests that the chaotic behaviorrepresents in the human being the epileptic or convulsive state. Through computer simulations, considering the system parameters, it was analyzed that the stability is highly sensitive to the initial conditions and producing oscillations, more so, when the oscillation increases the random behavior tends to increase making the system unpredictable.

scholarly journals Stability, Bifurcation, Chaos : Discrete Prey Predator Model with Step Size

2019 ◽  
Vol 9 (1) ◽  
pp. 3382-3387 ◽  
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium States ◽  
Phase Portraits ◽  
Step Size ◽  
Dynamical Nature ◽  
Predator Model ◽  
The Stability ◽  
Parameter Values ◽  
Bifurcation Chaos

In this work titled Stability, Bifurcation, Chaos: Discrete prey predator model with step size, by Forward Euler Scheme method the discrete form is obtained. Equilibrium states are calculated and the stability of the equilibrium states and dynamical nature of the model are examined in the closed first quadrant 2 R with the help of variation matrix. It is observed that the system is sensitive to the initial conditions and also to parameter values. The dynamical nature of the model is investigated with the assistance of Lyapunov Exponent, bifurcation diagrams, phase portraits and chaotic behavior of the system is identified. Numerical simulations validate the theoretical observations.

scholarly journals CONTROLE DE CAOS NO MODELO NEURONAL DE HINDMARSH-ROSE COM PARÂMETROS INCERTOS

2021 ◽  
Vol 12 (4) ◽  
pp. 38-45
Author(s):  
Raildo Santos de Lima ◽  
Fábio Roberto Chavarette
Keyword(s):  
Optimal Control ◽  
Chaotic Dynamics ◽  
Neuron Model ◽  
Chaotic Behavior ◽  
Control Design ◽  
Linear Optimal Control ◽  
Mathematical System ◽  
Real Neuron ◽  
Biological Neuron ◽  
Do So

In bioengineering there is a great motivation in studying the Hindmarsh-Rose (HR) neuron model due to the fact that it represents well the biological neuron, making it possible to simulate several behaviors of a real neuron, including periodic, aperiodic and chaotic behaviors, for example. Based on this model, this article proposes applying a linear optimal control design to the uncertain and chaotic behavior established by changes in the parameters of the system. To do so, the mathematical system of the RH model and its chaotic behavior are presented; afterwards, the fixed parametersare replaced by uncertain ones, and the chaotic dynamics of the system is investigated. At last, the linear optimal control is proposed as a method for controlling the chaotic behavior of the model, and numerical simulations are presented to show the efficiency of this proposal.

Instability and Chaos in Quadruped Gallop

1994 ◽  
Vol 116 (4) ◽  
pp. 1096-1101 ◽  
Author(s):  
P. Nanua ◽  
K. J. Waldron
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
The Body ◽  
Main Body ◽  
Viscous Damper ◽  
System Parameters ◽  
Constant Stiffness ◽  
Speed Controller ◽  
Symmetry Principles ◽  
The Stability

A dynamic model for the two-dimensional quadruped has been developed. The main body is modelled as a rigid bar and each leg consists of a constant stiffness spring, a viscous damper and a force actuator. Based on symmetry principles, a controller has been devised that will enable the quadruped to gallop at constant speed. The controller consists of two parts: an energy controller which will apply the required amount of force through the legs, and the speed controller that will control the forward speed by appropriately placing the legs. It will be shown that the body pitch need not be explicitly controlled. The stability of this controller will be examined using Poincare maps. Stable systems show either periodic or quasi-periodic response. This system also exhibits chaotic behavior and chaotic response results in instability. The stability of the system with changes in the initial conditions, as well as variations in the system parameters, will also be examined. It will be shown that the system is stable for a range of leg stiffnesses. Outside this range, the system shows chaotic behavior.

Discrete Time Chaotic Maps With Application to Random Bit Generation

2021 ◽  
pp. 542-582
Author(s):  
Lazaros Moysis ◽  
Ahmad Taher Azar ◽  
Aleksandra Tutueva ◽  
Denis N. Butusov ◽  
Christos Volos
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Statistical Tests ◽  
Solution Space ◽  
Small Perturbations ◽  
Highly Sensitive ◽  
Different Shapes ◽  
Parameter Values

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, circuits, engineering, computer science, and more. This chapter reports on some new chaotic discrete time two-dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-Sine, and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions, and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes. The application to random bit generation is then considered using a collection of the proposed maps by applying a simple rule. The resulting bit generator successfully passes all statistical tests performed.

New Discrete Time 2D Chaotic Maps

2017 ◽  
Vol 6 (1) ◽  
pp. 77-104 ◽  
Author(s):  
Lazaros Moysis ◽  
Ahmad Taher Azar
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Single Point ◽  
Solution Space ◽  
Small Perturbations ◽  
Highly Sensitive ◽  
Different Shapes ◽  
Parameter Values

Chaotic behavior is a term that is attributed to dynamical systems whose solutions are highly sensitive to initial conditions. This means that small perturbations in the initial conditions can lead to completely different trajectories in the solution space. These types of chaotic dynamical systems arise in various natural or artificial systems in biology, meteorology, economics, electrical circuits, engineering, computer science and more. Of these innumerable chaotic systems, perhaps the most interesting are those that exhibit attracting behavior. By that, the authors refer to systems whose trajectories converge with time to a set of values, called an attractor. This can be a single point, a curve or a manifold. The attractor is called strange if it is a set with fractal structure. Such systems can be both continuous and discrete. This paper reports on some new chaotic discrete time two dimensional maps that are derived from simple modifications to the well-known Hénon, Lozi, Sine-sine and Tinkerbell maps. Numerical simulations are carried out for different parameter values and initial conditions and it is shown that the mappings either diverge to infinity or converge to attractors of many different shapes.

Instability and Chaos in Quadruped Gallop

1992 ◽  
Author(s):  
Prabjot Nanua ◽  
Kenneth J. Waldron
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
The Body ◽  
Main Body ◽  
Viscous Damper ◽  
System Parameters ◽  
Constant Stiffness ◽  
Speed Controller ◽  
Symmetry Principles ◽  
The Stability

Abstract A dynamic model for the two dimensional quadruped has been developed. The main body is modelled as a rigid bar and each leg consists of a constant stiffness spring, a viscous damper and a force actuator. Based on symmetry principles, a controller has been devised that will enable the quadruped to gallop at constant speed. The controller consists of two parts: an energy controller which will apply the required amount of force through the legs, and the speed controller that will control the forward speed by appropriately placing the legs. It will be shown that the body pitch need not be explicitly controlled. The stability of this controller will be examined using Poincare maps. Stable systems show either periodic or quasi-periodic response. This system also exhibits chaotic behavior and chaotic response leads to instability. The stability of the system with changes in the initial conditions, as well as variations in the system parameters, will also be examined. It will be shown that the system is stable for a range of leg stiffnesses. Outside this range, the system shows chaotic behavior.

scholarly journals Stability Analysis of the Chaotic Lorenz System with a State-Feedback Controller

2021 ◽  
Author(s):  
Dan Jones
Keyword(s):  
Strange Attractor ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Geometric Approach ◽  
Stable Equilibrium Point ◽  
The Lorenz System ◽  
The Stability ◽  
Parameter Values

The Lorenz model is considered a benchmark system in chaotic dynamics in that it displays extraordinary sensitivity to initial conditions and the strange attractor phenomenon. Even though the system tends to amplify perturbations, it is indeed possible to convert a strange attractor to a non-chaotic one using various control schemes. In this work it is shown that the chaotic behavior of the Lorenz system can be suppressed through the use of a feedback loop driven by a quotient controller. The stability of the controlled Lorenz system is evaluated near its equilibrium points using Routh-Hurwitz testing, and the global stability of the controlled system is established using a geometric approach. It is shown that the controlled Lorenz system has only one globally stable equilibrium point for the set of parameter values under consideration.

The Tangled Tale of Phase Space

2018 ◽  
Author(s):  
David D. Nolte
Keyword(s):  
Phase Space ◽  
Chaotic Behavior ◽  
Three Body Problem ◽  
Henri Poincaré ◽  
History Of ◽  
The Stability ◽  
Three Body ◽  
Space Phase ◽  
Central Paradigm ◽  
System Phase

This chapter presents the history of the development of the concept of phase space. Phase space is the central visualization tool used today to study complex systems. The chapter describes the origins of phase space with the work of Joseph Liouville and Carl Jacobi that was later refined by Ludwig Boltzmann and Rudolf Clausius in their attempts to define and explain the subtle concept of entropy. The turning point in the history of phase space was when Henri Poincaré used phase space to solve the three-body problem, uncovering chaotic behavior in his quest to answer questions on the stability of the solar system. Phase space was established as the central paradigm of statistical mechanics by JW Gibbs and Paul Ehrenfest.

scholarly journals Formation of supermassive black hole seeds in nuclear star clusters via gas accretion and runaway collisions

2021 ◽  
Author(s):  
Arpan Das ◽  
Dominik R G Schleicher ◽  
Nathan W C Leigh ◽  
Tjarda C N Boekholt
Keyword(s):  
Black Holes ◽  
Initial Conditions ◽  
High Redshift ◽  
Star Clusters ◽  
Gas Reservoir ◽  
Highly Sensitive ◽  
Stellar Collisions ◽  
Key Variables ◽  
Chaotic Nature ◽  
Gas Accretion

Abstract More than two hundred supermassive black holes (SMBHs) of masses ≳ 109 M⊙ have been discovered at z ≳ 6. One promising pathway for the formation of SMBHs is through the collapse of supermassive stars (SMSs) with masses ∼103 − 5 M⊙ into seed black holes which could grow upto few times 109 M⊙ SMBHs observed at z ∼ 7. In this paper, we explore how SMSs with masses ∼103 − 5 M⊙ could be formed via gas accretion and runaway stellar collisions in high-redshift, metal-poor nuclear star clusters (NSCs) using idealised N-body simulations. We explore physically motivated accretion scenarios, e.g. Bondi-Hoyle-Lyttleton accretion and Eddington accretion, as well as simplified scenarios such as constant accretions. While gas is present, the accretion timescale remains considerably shorter than the timescale for collisions with the most massive object (MMO). However, overall the timescale for collisions between any two stars in the cluster can become comparable or shorter than the accretion timescale, hence collisions still play a crucial role in determining the final mass of the SMSs. We find that the problem is highly sensitive to the initial conditions and our assumed recipe for the accretion, due to the highly chaotic nature of the problem. The key variables that determine the mass growth mechanism are the mass of the MMO and the gas reservoir that is available for the accretion. Depending on different conditions, SMSs of masses ∼103 − 5 M⊙ can form for all three accretion scenarios considered in this work.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

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