A New Chaotic Jerk System with Double-Hump Nonlinearity

2020 ◽  
Vol 29 (14) ◽  
pp. 2050232
Author(s):  
Debabrata Biswas
Keyword(s):  
Chaotic Regime ◽  
Third Order ◽  
Parameter Mismatch ◽  
Chaotic Response ◽  
Variation Of Parameters ◽  
Electronic Hardware ◽  
Jerk System ◽  
Lyapunov Exponent Spectrum ◽  
Two Parameter ◽  
Good Analogy

In this paper, we report a new third-order chaotic jerk system with double-hump (bimodal) nonlinearity. The bimodal nonlinearity is of basic interest in biology, physics, etc. The proposed jerk system is able to exhibit chaotic response with proper choice of parameters. Importantly, the chaotic response is also obtained from the system by tuning the nonlinearity preserving its bimodal form. We analytically obtain the symmetry, dissipativity and stability of the system and find the Hopf bifurcation condition for the emergence of oscillation. Numerical investigations are carried out and different dynamics emerging from the system are identified through the calculation of eigenvalue spectrum, two-parameter and single parameter bifurcation diagrams, Lyapunov exponent spectrum and Kaplan–Yorke dimension. We identify that the form of the nonlinearity may bring the system to chaotic regime. Effect of variation of parameters that controls the form of the nonlinearity is studied. Finally, we design the proposed system in an electronic hardware level experiment and study its behavior in the presence of noise, fluctuations, parameter mismatch, etc. The experimental results are in good analogy with that of the analytical and numerical ones.

scholarly journals Generation of 3-D Grid Multi-Scroll Chaotic Attractors Based on Sign Function and Sine Function

Electronics ◽  
2020 ◽  
Vol 9 (12) ◽  
pp. 2145
Author(s):  
Pengfei Ding ◽  
Xiaoyi Feng ◽  
Lin Fa
Keyword(s):  
Chaotic System ◽  
Circuit Simulation ◽  
Equilibrium Points ◽  
Time Shift ◽  
Chaotic Attractors ◽  
Sine Function ◽  
Sign Function ◽  
Simulation Results ◽  
Jerk System ◽  
Lyapunov Exponent Spectrum

A three directional (3-D) multi-scroll chaotic attractors based on the Jerk system with nonlinearity of the sine function and sign function is introduced in this paper. The scrolls in the X-direction are generated by the sine function, which is a modified sine function (MSF). In addition, the scrolls in Y and Z directions are generated by the sign function series, which are the superposition of some sign functions with different time-shift values. In the X-direction, the scroll number is adjusted by changing the comparative voltages of the MSF, and the ones in Y and Z directions are regulated by the sign function. The basic dynamics of Lyapunov exponent spectrum, phase diagrams, bifurcation diagram and equilibrium points distribution were studied. Furthermore, the circuits of the chaotic system are designed by Multisim10, and the circuit simulation results indicate the feasibility of the proposed chaotic system for generating chaotic attractors. On the basis of the circuit simulations, the hardware circuits of the system are designed for experimental verification. The experimental results match with the circuit simulation results, this powerfully proves the correctness and feasibility of the proposed system for generating 3-D grid multi-scroll chaotic attractors.

Contigious hyperquadrics of coequipped hyperbands sHm

2019 ◽  
pp. 126-132
Author(s):  
Yu. Popov
Keyword(s):  
Projective Space ◽  
Second Order ◽  
Third Order ◽  
Base Surface ◽  
The Third ◽  
Two Parameter ◽  
Order Contact

We consider hyperquadrics that are internally connected to coequipped hyperbands in the projective space. Specifically, a hyperquadric Qn1 tangent to a hyperplane at the point is called a contiguous hyper quadric of a hyperband if it has a second-order contact with the base surface of the hyperband. In a the third order differential neighborhood of the forming element of the hyperband, two two-parameter bundles of fields of adjoining hyperquadrics are internally invariantly joined, their equations are given in a dot frame. The set of hyperquadrics such that the plane and the plane of Cartan are conjugate with respect to hyperquadric Qn1 is considered. The condition is shown under which the normal of the 2nd kind and the Cartan plane are conjugate with respect to the hyperquadric Qn1 . In addition, the following theorem is proved: normalization of a coequipped regular hyperband has a semi-internal equipment if and only if its normals of the first and second kind are polarly conjugate with respect to the hyperquadric.

The geometrical aberrations of general electron optical systems II. The primary (third order) aberrations of orthogonal systems, and the secondary (fifth order) aberrations of round systems

1965 ◽  
Vol 257 (1086) ◽  
pp. 523-552 ◽  
Keyword(s):  
Second Order ◽  
Order Perturbation ◽  
Optical Systems ◽  
Characteristic Functions ◽  
Electron Optics ◽  
Third Order ◽  
Variation Of Parameters ◽  
General Kind ◽  
Fifth Order ◽  
Orthogonal Systems

The theory of characteristic functions, developed by Sturrock for electron optics, is used to calculate the primary aberrations of rectilinear orthogonal systems of the most general kind. In the second part, the secondary aberrations of round systems are calculated with the aid of Sturrock’s second-order perturbation characteristic functions. A proof of the equivalence of the aberration formulae obtained by Melkich, using the variation of parameters method, and those obtained below is offered in an appendix.

Simple Chaotic Hyperjerk System

2016 ◽  
Vol 26 (11) ◽  
pp. 1650189 ◽  
Author(s):  
Fatma Yildirim Dalkiran ◽  
J. C. Sprott
Keyword(s):  
Dynamical System ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Nonlinear Function ◽  
Nonlinear Functions ◽  
Third Order ◽  
Experimental Realization ◽  
Numerical Studies ◽  
Hyperjerk System ◽  
Jerk System

In literature many chaotic systems, based on third-order jerk equations with different nonlinear functions, are available. A jerk system is taken to be a part of dynamical systems that can exhibit regular and chaotic behavior. By extension, a hyperjerk system can be described as a dynamical system with [Formula: see text]th-order ordinary differential equations where [Formula: see text] is 4 or up to. Hyperjerk systems have been investigated in literature in the last decade. This paper consists of numerical studies and experimental realization on FPAA for fourth-order hyperjerk system with exponential nonlinear function.

Generation and Circuit Implementation of Multi-Scroll Chaotic Attractors with Controllable Direction

2014 ◽  
Vol 513-517 ◽  
pp. 4559-4562
Author(s):  
Xiao Wen Luo ◽  
Chun Hua Wang
Keyword(s):  
Lyapunov Exponent ◽  
Control Function ◽  
Equilibrium Points ◽  
Nonlinear Function ◽  
Chaotic Attractors ◽  
Circuit Implementation ◽  
Relative Coefficient ◽  
Jerk System ◽  
Lyapunov Exponent Spectrum

An approach for generating multi-scroll chaotic attractors with controllable direction in one plane is proposed. Firstly, an appropriate nonlinear function is selected to control the number and direction of multi-scroll chaotic attractors in the three-order Jerk system. Then, we add new control function to Jerk system and observe Lyapunov exponent spectrum of relative coefficient and the change of equilibrium points. Different multi-scroll chaotic attractors with controllable direction are generated by adjusting the coefficient of the control function in a plane. The implementation of circuit verifies the feasibility of this method.

IMPULSIVE SYNCHRONIZATION FOR A NEW CHAOTIC OSCILLATOR

2007 ◽  
Vol 17 (02) ◽  
pp. 617-623 ◽  
Author(s):  
SINUHÉ BENÍTEZ ◽  
LEONARDO ACHO
Keyword(s):  
Impulsive Control ◽  
Chaotic Behavior ◽  
Polynomial System ◽  
Van Der Pol Oscillator ◽  
Chaotic Oscillator ◽  
Third Order ◽  
Van Der Pol ◽  
Impulsive Synchronization ◽  
Jerk System ◽  
Fourier Spectrum Analysis

Synchronization for a new proposed chaotic system based on impulsive control theory is presented. This new chaotic oscillator is a third order polynomial system (Jerk system), which was developed after the addition of a third state and innovation terms to the well known second order Van der Pol oscillator. The chaotic behavior of this new system is confirmed by using Lyapunov exponents, Poincaré maps, Fourier spectrum analysis and numerical experiments. Impulsive synchronization is achieved using just one channel of communication.

scholarly journals An Efficient Method for Solving System of Third-Order Nonlinear Boundary Value Problems

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Muhammad Noor ◽  
Khalida Noor ◽  
Asif Waheed ◽  
Eisa A. Al-Said
Keyword(s):  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Contact Problems ◽  
Convergent Series ◽  
Nonlinear Boundary Value Problems ◽  
Third Order ◽  
Variation Of Parameters ◽  
Suggested Technique ◽  
Nonlinear Boundary Value

we use the modified variation of parameters method for finding the analytical solution of a system of third-order nonlinear boundary value problems associated with obstacle, unilateral, and contact problems. The results are calculated in terms of convergent series with easily computable components. The suggested technique is applied without any discretization, perturbation, transformation, and restrictive assumptions. Moreover, it is free from round off errors. Some examples are given to illustrate the implementation and efficiency of the modified variation of parameters method.

Dynamics and Synchronization of a Novel Hyperchaotic System Without Equilibrium

2014 ◽  
Vol 24 (06) ◽  
pp. 1450087 ◽  
Author(s):  
Viet-Thanh Pham ◽  
Fadhil Rahma ◽  
Mattia Frasca ◽  
Luigi Fortuna
Keyword(s):  
Continuous Time ◽  
State Feedback ◽  
Dimensional System ◽  
Hyperchaotic System ◽  
Third Order ◽  
Circuit Realization ◽  
Diffusive Coupling ◽  
Dynamical Properties ◽  
Time Autonomous ◽  
Lyapunov Exponent Spectrum

A novel four-dimensional continuous-time autonomous hyperchaotic system which has no equilibrium is proposed in this paper. By starting from a third-order chaotic system and introducing a further variable performing state feedback, a four-dimensional system exhibiting hyperchaos is obtained. The basic dynamical properties of this system are investigated, such as equilibria and stability, Lyapunov exponent spectrum, and bifurcation diagrams. Furthermore, synchronization via diffusive coupling or control has been addressed. In the latter, parameter identification and synchronization are performed simultaneously. The circuit realization and experimental results are also presented.

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