SUDDEN OCCURRENCE OF CHAOS-II IN NONSMOOTH MAPS

2010 ◽  
Vol 20 (12) ◽  
pp. 4159-4174 ◽  
Author(s):  
GAOGAO DONG ◽  
LIXIN TIAN ◽  
JUN XIA ◽  
SONG ZHENG ◽  
RUIJIN DU
Keyword(s):  
Period Doubling ◽  
Band Gaps ◽  
Band Splitting ◽  
Dynamical Behaviors ◽  
Previous Category

Different ways to chaos show different features. The study of Sudden Occurrence of Chaos has generated much interest in exploring new ways to chaos. In this novel way, the Sudden Occurrence of Chaos-II indicates more comprehensive dynamical behaviors than Sudden Occurrence of Chaos. The Sudden Occurrence of Chaos-II not only belongs to the previous category of Sudden Occurrence of Chaos, but also includes inverse period-doubling band splitting, and a series of "band-gaps" by successive splitting of bands, which do not appear in the previous case. The constant of inverse period-doubling splitting band is first calculated, which is completely different from the first Feigenbaum constant. Furthermore, we give some examples of m-period Sudden Occurrence of Chaos-II by regulating periodic parameter m.

THE EFFECT OF IMPULSIVE SPRAYING PESTICIDE ON STAGE-STRUCTURED POPULATION MODELS WITH BIRTH PULSE

2005 ◽  
Vol 13 (01) ◽  
pp. 31-44 ◽  
Author(s):  
BING LIU ◽  
ZHIDONG TENG ◽  
LANSUN CHEN
Keyword(s):  
Discrete Dynamical System ◽  
Period Doubling ◽  
Fixed Time ◽  
Population Models ◽  
Strong Impact ◽  
Structured Population Models ◽  
Birth Pulse ◽  
Structured Population ◽  
Dynamical Behaviors ◽  
Threshold Conditions

In most models of population dynamics, increases in population due to birth are assumed to be time dependent, but many species reproduce only a single period of the year. In this paper, we construct a stage-structured pest model with birth pulse and periodic spraying pesticide at fixed time in each birth period by using impulsive differential equation. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which are with Ricker function or Beverton-Holt function, and obtain the threshold conditions for their stability. Further, we show that the time of spraying pesticide has a strong impact on the number of the mature pest population. Our results imply that the best time of spraying pesticide is at the end of the season, that is before and near the time of birth. Finally, by numerical simulations we find that the dynamical behaviors of the stage-structured population models with birth pulse and impulsive spraying pesticide are very complex, including period-doubling cascade, period-halving cascade, chaotic bands with periodic windows and "period-adding" phenomena.

scholarly journals Bifurcation Analysis in Population Genetics Model with Partial Selfing

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yingying Jiang ◽  
Wendi Wang
Keyword(s):  
Center Manifold ◽  
Evolutionary Game ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Expected Payoff ◽  
Center Manifold Theorem ◽  
Population Evolution ◽  
Dynamical Behaviors ◽  
Period 2 ◽  
Dynamics Of Population

A new model which allows both the effect of partial selfing selection and an exponential function of the expected payoff is considered. This combines ideas from genetics and evolutionary game theory. The aim of this work is to study the effects of partial selfing selection on the discrete dynamics of population evolution. It is shown that the system undergoes period doubling bifurcation, saddle-node bifurcation, and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-3, 6 orbits, cascade of period-doubling bifurcation in period-2, 4, 8, and the chaotic sets. These results reveal richer dynamics of the discrete model compared with the model in Tao et al., 1999. The analysis and results in this paper are interesting in mathematics and biology.

Periodic-Doubling Bifurcation of a Circuit With a Fractional-Order Memristor

2019 ◽  
Author(s):  
Yajuan Yu ◽  
Yangquan Chen
Keyword(s):  
Fractional Order ◽  
Memory Loss ◽  
Period Doubling ◽  
Hysteresis Loops ◽  
State Variables ◽  
Discontinuous Change ◽  
Dynamical Behaviors ◽  
Sinusoidal Current ◽  
Memristive Circuit ◽  
Theoretical Analyses

Abstract A new fractional-order current-controlled memristor is proposed by the fact of the memory loss. Excited by sinusoidal current, the generalized hysteresis loops of the new fractional-order memristor are no longer symmetrical to the origin and the time to reach the steady state is longer than the integer-order memristor’s. The dynamical behaviors of a new fractional-order memristive circuit system whose state variables have different derivation orders are investigated by theoretical analyses and simulated numerically. It is shown that the new fractional-order memristive circuit system goes into chaos by period-doubling bifurcation; the periodic windows are induced by the discontinuous change of derivative order between variables.

Dynamical Effects of Neuron Activation Gradient on Hopfield Neural Network: Numerical Analyses and Hardware Experiments

2019 ◽  
Vol 29 (04) ◽  
pp. 1930010 ◽  
Author(s):  
Bocheng Bao ◽  
Chengjie Chen ◽  
Han Bao ◽  
Xi Zhang ◽  
Quan Xu ◽  
...  
Keyword(s):  
Neural Network ◽  
Control Parameter ◽  
Period Doubling ◽  
Response Speed ◽  
Hopfield Neural Network ◽  
Activation Function ◽  
Numerical Analyses ◽  
Dynamical Behaviors ◽  
Neuron Activation ◽  
Crisis Scenarios

Hyperbolic tangent function, a bounded monotone differentiable function, is usually taken as a neuron activation function, whose activation gradient, i.e. gain scaling parameter, can reflect the response speed in the neuronal electrical activities. However, the previously published literatures have not yet paid attention to the dynamical effects of the neuron activation gradient on Hopfield neural network (HNN). Taking the neuron activation gradient as an adjustable control parameter, dynamical behaviors with the variation of the control parameter are investigated through stability analyses of the equilibrium states, numerical analyses of the mathematical model, and experimental measurements on a hardware level. The results demonstrate that complex dynamical behaviors associated with the neuron activation gradient emerge in the HNN model, including coexisting limit cycle oscillations, coexisting chaotic spiral attractors, chaotic double scrolls, forward and reverse period-doubling cascades, and crisis scenarios, which are effectively confirmed by neuron activation gradient-dependent local attraction basins and parameter-space plots as well. Additionally, the experimentally measured results have nice consistency to numerical simulations.

Complex dynamic behaviors of a discrete predator–prey model with stage structure and harvesting

2016 ◽  
Vol 10 (01) ◽  
pp. 1750013 ◽  
Author(s):  
Boshan Chen ◽  
Jiejie Chen
Keyword(s):  
Numerical Simulations ◽  
Period Doubling ◽  
Stage Structure ◽  
Model Parameters ◽  
Flip Bifurcation ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Dynamical Behaviors ◽  
Prey Model

First, a discrete stage-structured and harvested predator–prey model is established, which is based on a predator–prey model with Type III functional response. Then theoretical methods are used to investigate existence of equilibria and their local properties. Third, it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of [Formula: see text], by using the normal form of discrete systems, the center manifold theorem and the bifurcation theory, as varying the model parameters in some range. In particular, the direction and the stability of the flip bifurcation and the Neimark–Sacker bifurcation are showed. Finally, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as cascades of period-doubling bifurcation and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors. In addition, we show also the stabilizing effect of the harvesting by using numerical simulations.

Complex Dynamics in a Memristive Diode Bridge-Based MLC Circuit: Coexisting Attractors and Double-Transient Chaos

2021 ◽  
Vol 31 (03) ◽  
pp. 2150049
Author(s):  
A. Chithra ◽  
T. Fonzin Fozin ◽  
K. Srinivasan ◽  
E. R. Mache Kengne ◽  
A. Tchagna Kouanou ◽  
...  
Keyword(s):  
Complex Dynamics ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Transient Chaos ◽  
Multiple Attractors ◽  
Simulation Tools ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Crisis Scenarios ◽  
Complex Phenomena

This paper uncovers some striking and new complex phenomena in a memristive diode bridge-based Murali–Lakshmanan–Chua (MLC) circuit. These striking dynamical behaviors include the coexistence of multiple attractors and double-transient chaos. Also, period-doubling, chaos, crisis scenarios are observed in the system when varying the amplitude of the external excitation. Numerical simulation tools like phase portrait, cross-section basin of attraction, Lyapunov spectrum, bifurcation diagrams and time series are used to highlight the complex dynamical behaviors in the memristive system. Further, practical realizations of the circuit both in PSpice and real-laboratory measurements match well with the observed numerical simulations.

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.

Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing

2017 ◽  
Vol 27 (08) ◽  
pp. 1750125 ◽  
Author(s):  
Tao Jiang ◽  
Zhiyan Yang ◽  
Zhujun Jing
Keyword(s):  
Parametric Excitation ◽  
Chaotic Motion ◽  
Period Doubling ◽  
Melnikov Method ◽  
Original System ◽  
Duffing Equation ◽  
Periodic Perturbation ◽  
Phase Portraits ◽  
External Forcing ◽  
Dynamical Behaviors

We study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation [Formula: see text] (where [Formula: see text] is not rational relative to [Formula: see text]) and [Formula: see text] is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when [Formula: see text],[Formula: see text]7–15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincaré maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.

DYNAMICAL BEHAVIORS AND NUMERICAL SIMULATION OF A LORENZ-TYPE SYSTEM FOR THE INCOMPRESSIBLE FLOW BETWEEN TWO CONCENTRIC ROTATING CYLINDERS

2012 ◽  
Vol 22 (05) ◽  
pp. 1250124 ◽  
Author(s):  
HEYUAN WANG
Keyword(s):  
Numerical Simulation ◽  
Incompressible Flow ◽  
Stokes Equations ◽  
Type System ◽  
Period Doubling ◽  
Navier Stokes ◽  
Rotating Cylinders ◽  
Taylor Flow ◽  
Dynamical Behaviors ◽  
Low Dimensional

In this paper, we investigate the problem of dynamical behaviors and numerical simulation of Lorenz systems for the incompressible flow between two concentric rotating cylinders. A spectral Galerkin method is used to derive a model system of axisymmetric Couette–Taylor flow, a three-mode system, which is structurally similar to the Lorenz system, is obtained by a suitable three-mode truncation of the Navier–Stokes equations for the incompressible flow between two concentric rotating cylinders. The stability of the three-mode system is discussed, the existence of its attractor is given. Moreover, numerical simulation results indicate that this low-dimensional model exhibits a route to chaos via a period doubling cascade. Using these numerical results we explain successive transitions of Couette–Taylor flow from Laminar flow to turbulence in the experiment.

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.

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