A SYMMETRIC PIECEWISE-LINEAR CHAOTIC SYSTEM WITH A SINGLE EQUILIBRIUM POINT

2005 ◽  
Vol 15 (04) ◽  
pp. 1411-1415 ◽  
Author(s):  
RAFAEL GONZÁLEZ LÓPEZ ◽  
MANUEL PRIAN RODRÍGUEZ ◽  
MIGUEL A. FERNÁNDEZ GRANERO ◽  
JUAN L. ROJAS OJEDA ◽  
EDUARDO ROMERO BRUZÓN
Keyword(s):  
Mathematical Model ◽  
Phase Space ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Operational Amplifier ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
The Mathematical Model ◽  
Electronic Oscillator

In this paper, we propose a new autonomous electronic oscillator designed with some modifications of the well-known Wien bridge oscillator. In the mathematical model planned for such a circuit, the nonlinearity in the operational amplifier saturation is considered and reference is made to the only equilibrium point at the origin of phase-space. We show how the relation between the bifurcation parameters starts stable oscillations, providing an example for chaotic behavior and bifurcations diagrams. Finally, we conclude with a brief summary of the oscillators operation using a parameters plane.

Dynamics, Synchronization and Electronic Implementations of a New Lorenz-like Chaotic System with Nonhyperbolic Equilibria

2019 ◽  
Vol 29 (14) ◽  
pp. 1950197 ◽  
Author(s):  
P. D. Kamdem Kuate ◽  
Qiang Lai ◽  
Hilaire Fotsin
Keyword(s):  
Chaotic System ◽  
Chaotic Systems ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Adaptive Synchronization ◽  
The Novel ◽  
Algebraic Equations ◽  
The Lorenz System

The Lorenz system has attracted increasing attention on the issue of its simplification in order to produce the simplest three-dimensional chaotic systems suitable for secure information processing. Meanwhile, Sprott’s work on elegant chaos has revealed a set of 19 chaotic systems all described by simple algebraic equations. This paper presents a new piecewise-linear chaotic system emerging from the simplification of the Lorenz system combined with the elegance of Sprott systems. Unlike the majority, the new system is a non-Shilnikov chaotic system with two nonhyperbolic equilibria. It is multiplier-free, variable-boostable and exclusively based on absolute value and signum nonlinearities. The use of familiar tools such as Lyapunov exponents spectra, bifurcation diagrams, frequency power spectra as well as Poincaré map help to demonstrate its chaotic behavior. The novel system exhibits inverse period doubling bifurcations and multistability. It has only five terms, one bifurcation parameter and a total amplitude controller. These features allow a simple and low cost electronic implementation. The adaptive synchronization of the novel system is investigated and the corresponding electronic circuit is presented to confirm its feasibility.

MATHEMATICAL ANALYSIS OF THE ENDEMIC EQUILIBRIUM OF MALARIAHYGIENE MODEL

2021 ◽  
Vol 5 (10) ◽  
Author(s):  
Oluwafemi Temidayo J. ◽  
Azuaba E. ◽  
Lasisi N. O.
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Equilibrium Point ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
The Mathematical Model ◽  
Globally Stable ◽  
Well Posed ◽  
The Basic Reproduction Number ◽  
Invariant Principle

In this study, we analyzed the endemic equilibrium point of a malaria-hygiene mathematical model. We prove that the mathematical model is biological and meaningfully well-posed. We also compute the basic reproduction number using the next generation method. Stability analysis of the endemic equilibrium point show that the point is locally stable if reproduction number is greater that unity and globally stable by the Lasalle’s invariant principle. Numerical simulation to show the dynamics of the compartment at various hygiene rate was carried out.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

scholarly journals Colpitts Chaotic Oscillator Coupling with a Generalized Memristor

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Ling Lu ◽  
Changdi Li ◽  
Zicheng Zhao ◽  
Bocheng Bao ◽  
Quan Xu
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Fourth Order ◽  
Chaotic Attractors ◽  
Chaotic Oscillator ◽  
Nonlinear Phenomena ◽  
Unique Equilibrium ◽  
First Order ◽  
The Mathematical Model ◽  
Order Parallel

By introducing a generalized memristor into a fourth-order Colpitts chaotic oscillator, a new memristive Colpitts chaotic oscillator is proposed in this paper. The generalized memristor is equivalent to a diode bridge cascaded with a first-order parallel RC filter. Chaotic attractors of the oscillator are numerically revealed from the mathematical model and experimentally captured from the physical circuit. The dynamics of the memristive Colpitts chaotic oscillator is investigated both theoretically and numerically, from which it can be found that the oscillator has a unique equilibrium point and displays complex nonlinear phenomena.

scholarly journals The Verhulst-Like Equations: Integrable OΔE and ODE with Chaotic Behavior

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1446 ◽  
Author(s):  
Igor Andrianov ◽  
Galina Starushenko ◽  
Sergey Kvitka ◽  
Lelya Khajiyeva
Keyword(s):  
Mathematical Model ◽  
Dynamical Systems ◽  
Difference Equations ◽  
Chaotic Behavior ◽  
Logistic Equation ◽  
Deterministic System ◽  
Characteristic Parameters ◽  
Chaotic Dynamical Systems ◽  
The Mathematical Model ◽  
Continualization Procedure

In this paper, we study various variants of Verhulst-like ordinary differential equations (ODE) and ordinary difference equations (O Δ E). Usually Verhulst ODE serves as an example of a deterministic system and discrete logistic equation is a classic example of a simple system with very complicated (chaotic) behavior. In our paper we present examples of deterministic discretization and chaotic continualization. Continualization procedure is based on Padé approximants. To correctly characterize the dynamics of obtained ODE we measured such characteristic parameters of chaotic dynamical systems as the Lyapunov exponents and the Lyapunov dimensions. Discretization and continualization lead to a change in the symmetry of the mathematical model (i.e., group properties of the original ODE and O Δ E). This aspect of the problem is the aim of further research.

Dynamics of a Rolling Wheelset

1993 ◽  
Vol 46 (7) ◽  
pp. 438-444 ◽  
Author(s):  
Hans True
Keyword(s):  
Mathematical Model ◽  
Equilibrium Point ◽  
Chaotic Motion ◽  
Dynamical Behavior ◽  
Railway Track ◽  
Kinematics And Dynamics ◽  
Bifurcation Diagrams ◽  
Speed Range ◽  
Set Up ◽  
The Mathematical Model

We discuss the kinematics and dynamics of a wheelset rolling on a railway track. The mathematical model of a suspended wheelset rolling with constant speed on a straight track is set up and its dynamics is investigated numerically. The results are presented mainly on bifurcation diagrams. Several kinds of dynamical behavior is identified within the investigated speed range. We find a stationary equilibrium point at low speeds and at higher speeds symmetric and asymmetric oscillations are found and ranges with chaotic motion are identified. The bifurcations are described.

scholarly journals Analysis, Stabilization, and DSP-Based Implementation of a Chaotic System with Nonhyperbolic Equilibrium

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Xuan-Bing Yang ◽  
Yi-Gang He ◽  
Chun-Lai Li ◽  
Chang-Qing Liu
Keyword(s):  
Dynamical System ◽  
Phase Space ◽  
Equilibrium Point ◽  
Chaotic System ◽  
Poincaré Map ◽  
Topological Horseshoe ◽  
Control Scheme ◽  
Theoretical Analyses ◽  
Zero Equilibrium ◽  
Autonomous Dynamical System

This paper reports an autonomous dynamical system, and it finds that one nonhyperbolic zero equilibrium and two hyperbolic nonzero equilibria coexist in this system. Thus, it is difficult to demonstrate the existence of chaos by Šil’nikov theorem. Consequently, the topological horseshoe theory is adopted to rigorously prove the chaotic behaviors of the system in the phase space of Poincaré map. Then, a single control scheme is designed to stabilize the dynamical system to its zero-equilibrium point. Besides, to verify the theoretical analyses physically, the attractor and stabilization scheme are further realized via DSP-based technique.

BIPOLAR CONTROL (OR PARADOXICALLY UNILATERAL) OF IMBALANCED LIMIT-CYCLES AND STRANGE ATTRACTORS.: CHAOS AND HOMEOSTASIS

1993 ◽  
Vol 01 (03) ◽  
pp. 311-333 ◽  
Author(s):  
E. BERNARD WEIL
Keyword(s):  
Mathematical Model ◽  
Phase Space ◽  
Limit Cycles ◽  
Biological System ◽  
Chaotic Dynamics ◽  
Control Method ◽  
Strange Attractors ◽  
Periodic Attractors ◽  
The Mathematical Model ◽  
Biological Modelling

The functioning of the mathematical model for the regulation of agonistic antagonistic couples (MRCAA) is first recalled: it intends to simulate normal and pathological states concerning some biological (im)balances, just as a control method allowing to reestablish the balances, if necessary. Using the MRCAA in the frame of AA networks, it was permitted to obtain strange attractors (SA). By approaching in that manner the problem of chaotic dynamics, we may understand that SA have important characteristics other than their aperiodicity. Their topology in the phase-space has equally to be considered. The position of a SA in the phase-space allows us to simulate the functioning of a biological system; if this does not correspond to the physiological position, we consider it as an imbalanced SA. Therefore, SA could take the place in biological modelling of limit-cycles, which were perhaps an approximation of these SA. Then, the problem of correction of these imbalances should be considered. Using the mathematical model of AA networks, it was possible to propose an outline as yet theoretical of this problem: how can we model imbalanced SA, then how can the balance be restored from a method already used in the correction of imbalanced limit-cycles? By this technique, SA have also became quasi-periodic attractors, but this was not the result which was particularly wished. This paper mainly concerns chaotic dynamics (CD) and the problems elicited by the control of corresponding systems. However, given that we will use a general model (or a model of function in the sense proposed by Rosen), i.e., the “model for the regulation of agonistic antagonistic couples” (MRAAC), it has seemed firstly necessary to recall the structure of this model and also to establish a comparison between the results of computer simulations with this model and the results of the control in concrete systems (bipolar or paradoxically unilateral therapies in bio-medicine).

MEMRISTOR MODEL AND ITS APPLICATION FOR CHAOS GENERATION

2012 ◽  
Vol 22 (08) ◽  
pp. 1250205 ◽  
Author(s):  
LIDAN WANG ◽  
EMMANUEL DRAKAKIS ◽  
SHUKAI DUAN ◽  
PENGFEI HE ◽  
XIAOFENG LIAO
Keyword(s):  
Mathematical Model ◽  
Chaotic System ◽  
Initial Conditions ◽  
Third Order ◽  
Nonlinear Part ◽  
The Third ◽  
Simulation Results ◽  
Application Fields ◽  
Model C ◽  
The Mathematical Model

This paper contributes to the understanding of memristor operation and its possible application fields through: (a) derivation of a complete mathematical model for the HP memristor which takes into consideration the inter-dependence between memristance, charge and flux along with the boundary and initial conditions of operation; (b) an introduction of detailed charge- and flux-controlled SPICE memristor models realizing the proposed mathematical memristor model; (c) The incorporation of the memristor model in the SPICE realization of a third-order chaotic system where a single HP memristor acts as the nonlinear part of the system. Simulation results are provided to validate the mathematical model and the synthesis and operation of the third-order chaotic system.

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