Chaotic Dynamics in Duffing System with Two External Forcings

2011 ◽  
Vol 216 ◽  
pp. 777-781
Author(s):  
Shu Guang Zhang ◽  
Zhi Yong Zhu ◽  
Zhi Guo
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Homoclinic Bifurcation ◽  
External Forcing ◽  
Bifurcation Diagrams ◽  
Duffing System ◽  
External Forcings ◽  
Maximum Lyapunov Exponents

Duffing system with two external forcing terms is investigated in detail. The criterion of existence of chaos under the periodic perturbation is given by using Melnikov's method. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, maximum lyapunov exponents and Poincare map are given to illustrate the theoretical analysis.

CHAOS BEHAVIOR IN THE DISCRETE BVP OSCILLATOR

2002 ◽  
Vol 12 (03) ◽  
pp. 619-627 ◽  
Author(s):  
ZHUJUN JING ◽  
ZHIYUAN JIA ◽  
RUIQI WANG
Keyword(s):  
Theoretical Analysis ◽  
Periodic Orbit ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Period Doubling ◽  
Euler Method ◽  
Dynamical Behaviors ◽  
Chaotic Orbits ◽  
Marotto’S Chaos ◽  
Definition Of

The discrete BVP oscillator obtained through the Euler method is investigated, and also first proved that there exist chaotic phenomena in the sense of Marotto's definition of chaos and two-period cycles. And numerical simulations not only show the consistence with the theoretical analysis but also exhibit the complex dynamical behaviors, including the ten-periodic orbit, a cascade of period-doubling bifurcation, quasiperiodic orbits and the chaotic orbits in Marotto's chaos and intermitten's chaos. The computations of Lyapunov exponents confirm the existence of dynamical behaviors.

scholarly journals Harmonic Balance Method for Chaotic Dynamics in Fractional-Order Rössler Toroidal System

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Huijian Zhu
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Harmonic Balance ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Chaotic Motions ◽  
Behavior Based

This paper deals with the problem of determining the conditions under which fractional order Rössler toroidal system can give rise to chaotic behavior. Based on the harmonic balance method, four detailed steps are presented for predicting the existence and the location of chaotic motions. Numerical simulations are performed to verify the theoretical analysis by straightforward computations.

Conservative Chaos in a Class of Nonconservative Systems: Theoretical Analysis and Numerical Demonstrations

2018 ◽  
Vol 28 (07) ◽  
pp. 1850087 ◽  
Author(s):  
Shijian Cang ◽  
Aiguo Wu ◽  
Ruiye Zhang ◽  
Zenghui Wang ◽  
Zengqiang Chen
Keyword(s):  
Theoretical Analysis ◽  
Lyapunov Exponents ◽  
Time Reversal ◽  
Dynamical Evolution ◽  
Phase Portraits ◽  
Time Reversal Symmetry ◽  
Bifurcation Diagrams ◽  
Continuous Curve ◽  
Nonconservative Systems ◽  
Chaotic Flows

This paper proposes a class of nonlinear systems and presents one example system to illustrate its interesting dynamics, including quasiperiodic motion and chaos. It is found that the example system is a subsystem of a non-Hamiltonian system, which has a continuous curve of equilibria with time-reversal symmetry. In this study, the dynamical evolution of the example system with three different kinds of external excitations are fully investigated by using general chaotic analysis methods such as Poincaré sections, phase portraits, Lyapunov exponents and bifurcation diagrams. Both theoretical analysis and numerical simulations show that the example system is nonconservative but has conservative chaotic flows, which are numerically verified by the sum of its Lyapunov exponents. It is also found that the example system has time-reversal symmetry.

scholarly journals A Study of the Dynamics of a New Piecewise Smooth Map

2020 ◽  
Vol 30 (01) ◽  
pp. 2050018 ◽  
Author(s):  
Dhrubajyoti Biswas ◽  
Soumyajit Seth ◽  
Mita Bor
Keyword(s):  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Logistic Map ◽  
Unit Interval ◽  
Piecewise Smooth ◽  
Bifurcation Diagrams ◽  
Tent Map ◽  
Before And After ◽  
Smooth Map ◽  
Analytical Tools

In this article, we have studied a [Formula: see text]D map, which is formed by combining the two well-known maps, i.e. the tent and the logistic maps in the unit interval, i.e. [Formula: see text]. The point of discontinuity of the map (known as border) denotes the transition from tent map to logistic map. The proposed map can behave as the piecewise smooth or nonsmooth map or both (depending on the behavior of the map just before and after the border) and the dynamics of the map has been studied using analytical tools and numerical simulations. Characterization has been done by primarily studying the Lyapunov exponents and the corresponding bifurcation diagrams. Some peculiar dynamics of this map have been shown numerically. Finally, a Simulink implementation of the proposed map has been demonstrated.

A Hyperchaotic System and its Synchronization via Scalar Controller

2011 ◽  
Vol 50-51 ◽  
pp. 254-257
Author(s):  
Wu Chun Dai ◽  
Zheng Fu Cheng
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Lyapunov Exponents ◽  
Mathematical Proof ◽  
Hyperchaotic System ◽  
Scalar Control ◽  
Response System ◽  
Dynamical Behaviors ◽  
Synchronization Scheme ◽  
New Hyperchaotic System

In this paper, a 4D hyperchaotic system is proposed. Some basic dynamical behaviors are explored by calculating its Lyapunov exponents, Poincar´e mapping, etc.. Finally, synchronization for this new hyperchaotic system is achieved via scalar control. The nonlinear terms in the response system are not dropped. The proposed synchronization scheme is simple and theoretically rigorous. The mathematical proof of this method is provided. Some numerical simulations are obtained. The numerical simulations coincide with the theoretical analysis.

Suppressing Chaos in a Nonideal Double-Well Oscillator Using an Based Electromechanical Damped Device

2014 ◽  
Vol 706 ◽  
pp. 25-34 ◽  
Author(s):  
G. Füsun Alişverişçi ◽  
Hüseyin Bayiroğlu ◽  
José Manoel Balthazar ◽  
Jorge Luiz Palacios Felix
Keyword(s):  
Numerical Simulations ◽  
Chaotic Dynamics ◽  
Duffing Oscillator ◽  
Vibration Energy ◽  
Vibration Absorber ◽  
Electrical System ◽  
Chaotic Vibration ◽  
System A ◽  
Ideal System ◽  
Double Well Potential

In this paper, we analyzed chaotic dynamics of an electromechanical damped Duffing oscillator coupled to a rotor. The electromechanical damped device or electromechanical vibration absorber consists of an electrical system coupled magnetically to a mechanical structure (represented by the Duffing oscillator), and that works by transferring the vibration energy of the mechanical system to the electrical system. A Duffing oscillator with double-well potential is considered. Numerical simulations results are presented to demonstrate the effectiveness of the electromechanical vibration absorber. Lyapunov exponents are numerically calculated to prove the occurrence of a chaotic vibration in the non-ideal system and the suppressing of chaotic vibration in the system using the electromechanical damped device.

scholarly journals Theoretical Analysis and Design of an Innovative Coil Structure for Transcranial Magnetic Stimulation

2021 ◽  
Vol 11 (4) ◽  
pp. 1960
Author(s):  
Naming Zhang ◽  
Ziang Wang ◽  
Jinhua Shi ◽  
Shuya Ning ◽  
Yukuo Zhang ◽  
...  
Keyword(s):  
Electromagnetic Field ◽  
Transcranial Magnetic Stimulation ◽  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Magnetic Stimulation ◽  
Influential Factors ◽  
Intersection Angle ◽  
Electromagnetic Field Distribution ◽  
Analysis And Design ◽  
Coil Structure

Previous research showed that pulsed functional magnetic stimulation can activate brain tissue with optimum intensity and frequency. Conventional stimulation coils are always set as a figure-8 type or Helmholtz. However, the magnetic fields generated by these coils are uniform around the target, and their magnetic stimulation performance still needs improvement. In this paper, a novel type of stimulation coil is proposed to shrink the irritative zone and strengthen the stimulation intensity. Furthermore, the electromagnetic field distribution is calculated and measured. Based on numerical simulations, the proposed coil is compared to traditional coil types. Moreover, the influential factors, such as the diameter and the intersection angle, are also analyzed. It was demonstrated that the proposed coil has a better performance in comparison with the figure-8 coil. Thus, this work suggests a new way to design stimulation coils for transcranial magnetic stimulation.

Mathematical Model Establishment and Analysis on the Pipe Deformation under External Pressure with the Ovality

2013 ◽  
Vol 760-762 ◽  
pp. 2263-2266
Author(s):  
Kang Yong ◽  
Wei Chen
Keyword(s):  
Mathematical Model ◽  
Theoretical Analysis ◽  
Yield Strength ◽  
Residual Stresses ◽  
Numerical Simulations ◽  
Significant Influence ◽  
External Pressure ◽  
Pipe Diameter ◽  
Axial Loads ◽  
The Stability

Beside the residual stresses and axial loads, other factors of pipe like ovality, moment could also bring a significant influence on pipe deformation under external pressure. The Standard of API-5C3 has discussed the influences of deformation caused by yield strength of pipe, pipe diameter and pipe thickness, but the factor of ovality degree is not included. Experiments and numerical simulations show that with the increasing of pipe ovality degree, the anti-deformation capability under external pressure will become lower, and ovality affecting the stability of pipe shape under external pressure is significant. So it could be a path to find out the mechanics relationship between ovality and pipe deformation under external pressure by the methods of numerical simulations and theoretical analysis.

scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

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