CHAOTIC DYNAMICS OF A FIVE-DIMENSIONAL NONLINEAR NETWORK

2007 ◽  
Vol 18 (03) ◽  
pp. 335-342
Author(s):  
XUEWEI JIANG ◽  
DI YUAN ◽  
YI XIAO
Keyword(s):  
Lyapunov Exponent ◽  
Power Spectrum ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Chinese Traditional Medicine ◽  
Bifurcation Diagrams ◽  
Nonlinear Network ◽  
Dynamical Behaviors ◽  
Lyapunov Exponent Spectrum ◽  
Period Doubling Bifurcations

The dynamics of a five-dimensional nonlinear network based on the theory of Chinese traditional medicine is studied by the Lyapunov exponent spectrum, Poincaré, power spectrum and bifurcation diagrams. The result shows that this system has complex dynamical behaviors, such as chaotic ones. It also shows that the system evolves into chaos through a series of period-doubling bifurcations.

scholarly journals Chaotic dynamics of a pulse modulated control system for a heating unit

2018 ◽  
Vol 224 ◽  
pp. 02055
Author(s):  
Yuriy A. Gol’tsov ◽  
Alexander S. Kizhuk ◽  
Vasiliy G. Rubanov
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classical Period ◽  
Nonlinear Phenomena ◽  
Pulse Control ◽  
Heating Unit ◽  
Border Collision Bifurcation ◽  
Period Doubling Bifurcations

The dynamic modes and bifurcations in a pulse control system of a heating unit, the condition of which is described through differential equations with discontinuous right–hand sides, have been studied. It has been shown that the system under research can demonstrate a great variety of nonlinear phenomena and bifurcation transitions, such as quasiperiodicity, multistable behaviour, chaotization of oscillations through a classical period–doubling bifurcations cascade and border–collision bifurcation.

Generating Multiple Chaotic Attractors from Sprott B System

2016 ◽  
Vol 26 (11) ◽  
pp. 1650177 ◽  
Author(s):  
Qiang Lai ◽  
Shiming Chen
Keyword(s):  
Lyapunov Exponent ◽  
Function Method ◽  
Polynomial Function ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Nonlinear Phenomenon ◽  
Bifurcation Diagrams ◽  
Initial Values ◽  
Index 2 ◽  
Lyapunov Exponent Spectrum

Multiple chaotic attractors, implying several independent chaotic attractors generated simultaneously in a system from different initial values, are a very interesting and important nonlinear phenomenon, but there are few studies that have previously addressed it to our best knowledge. In this paper, we propose a polynomial function method for generating multiple chaotic attractors from the Sprott B system. The polynomial function extends the number of index-2 saddle foci, which determines the emergence of multiple chaotic attractors in the system. The analysis of the equilibria is presented. Two coexisting chaotic attractors, three coexisting chaotic attractors and four coexisting chaotic attractors are investigated for verifying the effectiveness of the method. The chaotic characteristics of the attractors are shown by bifurcation diagrams, Lyapunov exponent spectrum and phase portraits.

Periodic and chaotic behaviour in a reduction of the perturbed Korteweg-de Vries equation

1994 ◽  
Vol 445 (1923) ◽  
pp. 1-21 ◽  
Keyword(s):  
Lyapunov Exponent ◽  
Vries Equation ◽  
Period Doubling ◽  
Homoclinic Orbits ◽  
Dynamical Behaviour ◽  
Chaotic Behaviour ◽  
Subharmonic Bifurcations ◽  
Melnikov Functions ◽  
De Vries ◽  
Period Doubling Bifurcations

The dynamical behaviour of a reduction of the forced (and damped) Korteweg-de Vries equation is studied numerically. Chaos arising from subharmonic instability and homoclinic crossings are observed. Both period-doubling bifurcations and the Melnikov sequence of subharmonic bifurcations are found and lead to chaotic behaviour, here characterised by a positive Lyapunov exponent. Supporting theoretical analysis includes the construction of periodic solutions and homoclinic orbits, and their behaviour under perturbation using Melnikov functions.

scholarly journals Predicting Tipping Points in Chaotic Maps with Period-Doubling Bifurcations

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Changzhi Li ◽  
Dhanagopal Ramachandran ◽  
Karthikeyan Rajagopal ◽  
Sajad Jafari ◽  
Yongjian Liu
Keyword(s):  
Chaotic Maps ◽  
Period Doubling ◽  
Tipping Points ◽  
Bifurcation Parameter ◽  
Bifurcation Diagrams ◽  
Bifurcation Points ◽  
Slowing Down ◽  
Sine Map ◽  
Autocorrelation Method ◽  
Period Doubling Bifurcations

In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

scholarly journals Chaotic Dynamics by Some Quadratic Jerk Systems

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 227
Author(s):  
Mei Liu ◽  
Bo Sang ◽  
Ning Wang ◽  
Irfan Ahmad
Keyword(s):  
Hopf Bifurcation ◽  
Cross Sections ◽  
Chaotic Dynamics ◽  
Stable Equilibrium ◽  
Period Doubling ◽  
Dynamical Evolution ◽  
Chaotic Attractors ◽  
Bifurcation Diagrams ◽  
Hidden Attractors ◽  
Subcritical Hopf Bifurcation

This paper is about the dynamical evolution of a family of chaotic jerk systems, which have different attractors for varying values of parameter a. By using Hopf bifurcation analysis, bifurcation diagrams, Lyapunov exponents, and cross sections, both self-excited and hidden attractors are explored. The self-exited chaotic attractors are found via a supercritical Hopf bifurcation and period-doubling cascades to chaos. The hidden chaotic attractors (related to a subcritical Hopf bifurcation, and with a unique stable equilibrium) are also found via period-doubling cascades to chaos. A circuit implementation is presented for the hidden chaotic attractor. The methods used in this paper will help understand and predict the chaotic dynamics of quadratic jerk systems.

Chaos in a Meminductor-Based Circuit

2016 ◽  
Vol 26 (08) ◽  
pp. 1650130 ◽  
Author(s):  
Fang Yuan ◽  
Guangyi Wang ◽  
Peipei Jin ◽  
Xiaoyuan Wang ◽  
Guojin Ma
Keyword(s):  
Lyapunov Exponent ◽  
Equivalent Circuit ◽  
Smooth Curve ◽  
Experimental Results ◽  
Chaotic Oscillator ◽  
Coexisting Attractors ◽  
Dynamical Behaviors ◽  
Equilibrium Set ◽  
Curve Model ◽  
Lyapunov Exponent Spectrum

A smooth curve model of meminductor and its equivalent circuit have been designed, on the condition that the meminductor is commonly unavailable. The equivalent circuit can be used for breadboard experiments for various application circuit designs of meminductor. Based on the meminductor, a new chaotic oscillator is proposed. The dynamical behaviors of the oscillator are investigated, including equilibrium set, Lyapunov exponent spectrum, bifurcations and dynamical map of the system. Particularly, the amplitude controlling is analyzed and coexisting attractors are found for conditions of different parameters. Furthermore, the experimental results are given to confirm the correction of the proposed meminductor model and the meminductor-based oscillator.

scholarly journals Stability Analysis of Nonlinear Glue Flow Control System With Delay

2021 ◽  
Author(s):  
Yuting Ding ◽  
Liyuan Zheng ◽  
Jining Guo
Keyword(s):  
Control System ◽  
Flow Control ◽  
Normal Forms ◽  
Stable Equilibrium ◽  
Period Doubling ◽  
Dynamical Behaviors ◽  
Existence And Stability ◽  
Stable Periodic ◽  
Flow Control System ◽  
Period Doubling Bifurcations

Abstract In this paper, we improve a new mathematical model associated with glue flow control system for glue applying of particleboard. Firstly, we study the existence and stability of the equilibria and the existence of fold, Hopf and Bogdanov-Takens bifurcations in above system. Next, the normal forms of Hopf bifurcation and Bogdanov-Takens bifurcation are derived, and the classifications of local dynamics near above bifurcation critical values are analyzed. Then, numerical simulation results show that the flow control system associated with glue applying of particleborad exists stable equilibrium, stable periodic-1, periodic-2, and periodic-4 solutions, and chaotic attractor phenomenon from a sequence of period-doubling bifurcations. Finally, we compare the dynamical phenomena of flow control system with and without cubic terms, showing that cubic terms can effect the dynamical behaviors of flow control system.

A fractional-order ship power system: chaos and its dynamical properties

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Prakash Duraisamy ◽  
Goitom Tadesse ◽  
Christos Volos ◽  
Fahimeh Nazarimehr ◽  
...  
Keyword(s):  
Power System ◽  
Fractional Order ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Order Form ◽  
Dynamical Behaviors ◽  
Fractional Order Controller ◽  
Parameter Values ◽  
Ship Power System ◽  
D Model

Abstract In this research, the ship power system is studied with a fractional-order approach. A 2-D model of a two-generator parallel-connected is considered. A chaotic attractor is observed for particular parameter values. The fractional-order form is calculated with the Adam–Bashforth–Moulton method. The chaotic response is identified even for the order 0.99. Phase portrait is generated using the Caputo derivative approach. Wolf’s algorithm is used to calculate Lyapunov exponents. For the considered values of parameters, one positive Lyapunov exponent confirms the existence of chaos. Bifurcation diagrams are presented to analyze the various dynamical behaviors and bifurcation points. Interestingly, the considered system is multistable. Also, antimonotonicity, period-doubling, and period halving are observed in the bifurcation diagram. As the last step, a fractional-order controller is designed to remove chaotic dynamics. Time plots are simulated to show the effectiveness of the controller.

BIFURCATION AND CHAOS IN THE TINKERBELL MAP

2011 ◽  
Vol 21 (11) ◽  
pp. 3137-3156 ◽  
Author(s):  
SHAOLIANG YUAN ◽  
TAO JIANG ◽  
ZHUJUN JING
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Phase Portraits ◽  
Transient Chaos ◽  
Flip Bifurcation ◽  
Maximum Lyapunov Exponent ◽  
Invariant Circle ◽  
Dynamical Behaviors ◽  
Fold Bifurcation ◽  
Period Doubling Bifurcations

In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.

EXPERIMENTAL DETERMINATION OF POINT LOCALIZED TIME CORRELATION AND POWER SPECTRUM IN A FARADAY EXPERIMENT

2000 ◽  
Vol 10 (09) ◽  
pp. 2233-2244 ◽  
Author(s):  
C. CABEZA ◽  
C. NEGREIRA ◽  
A. C. SICARDI SCHIFINO ◽  
V. GIBIAT
Keyword(s):  
Power Spectrum ◽  
Experimental Determination ◽  
Time Domain ◽  
Cross Correlation ◽  
Period Doubling ◽  
Time Correlation ◽  
Experimental Results ◽  
Capacitive Transducer ◽  
Period Doubling Bifurcations

Experimental results from a classical Faraday experiment have been obtained using a point-localized measurement of the displacement of the fluid surface. We used a capacitive transducer as probe, and have been able to study the period-doubling bifurcations in time domain through the determination of cross-correlation and power spectrum.

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