Chaos in a fractional order logistic map

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Joakim Munkhammar
Keyword(s):  
Fractional Order ◽  
Discrete Time ◽  
Logistic Map ◽  
Period Doubling ◽  
Fractional Dynamics ◽  
Open Problems ◽  
Time Dynamics ◽  
Route To Chaos ◽  
Special Case ◽  
Discrete Time Dynamical Systems

AbstractIn this paper we investigate a fractional order logistic map and its discrete time dynamics. After a brief introduction to the discrete-time dynamical systems and fractional dynamics we show some basic properties of the fractional logistic map. We then move on to prove that the special case α = 1/2 exhibits a period doubling route to chaos. A bifurcation diagram for the special case of α = 1/2 is also included. Finally a discussion concerning the results and open problems is given.

Dynamic Behaviors of a Fractional Order Viscoelastic Two-Member Truss

2011 ◽  
Vol 90-93 ◽  
pp. 951-957
Author(s):  
Yuan Ping Li ◽  
Wei Zhang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Chaotic Motion ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Fractional Dynamics ◽  
Dynamic Behaviors ◽  
Period Doubling Bifurcations

The fractional dynamics equation of a viscoelastic two-member truss system, in which fractional derivative model introduced to simulate the materials’ characteristics, is proposed. The simplified single DOF differential equation is developed combined with boundary conditions and symmetry. Dynamic behaviors of the fractional single DOF system with harmonic loads are discussed by numerical calculations. The results show that: the system may lead to chaotic motion via period-doubling bifurcations or intermittent routes; the dynamical character is greatly inflected by the varying of excitation amplitude or damping coefficient or fractional order.

Nonlinear Dynamic Analysis of a Simplest Fractional-Order Delayed Memristive Chaotic System

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Wei Hu ◽  
Dawei Ding ◽  
Nian Wang
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Chaotic System ◽  
Chaotic Behavior ◽  
Nonlinear Dynamic Analysis ◽  
Period Doubling ◽  
Transversality Condition ◽  
Memristive System ◽  
Route To Chaos ◽  
The Stability

A simplest fractional-order delayed memristive chaotic system is investigated in order to analyze the nonlinear dynamics of the system. The stability and bifurcation behaviors of this system are initially investigated, where time delay is selected as the bifurcation parameter. Some explicit conditions for describing the stability interval and the transversality condition of the emergence for Hopf bifurcation are derived. The period doubling route to chaos behaviors of such a system is discussed by using a bifurcation diagram, a phase diagram, a time-domain diagram, and the largest Lyapunov exponents (LLEs) diagram. Specifically, we study the influence of time delay on the chaotic behavior, and find that when time delay increases, the transitions from one cycle to two cycles, two cycles to four cycles, and four cycles to chaos are observed in this system model. Corresponding critical values of time delay are determined, showing the lowest orders for chaos in the fractional-order delayed memristive system. Finally, numerical simulations are provided to verify the correctness of theoretical analysis using the modified Adams–Bashforth–Moulton method.

A POSSIBLE NEW ROUTE TO CHAOS VIA SEMICONDUCTOR SUPERLATTICES

2007 ◽  
Vol 21 (23n24) ◽  
pp. 3967-3974
Author(s):  
X. R. WANG ◽  
Z. Z. SUN ◽  
ZHENYU ZHANG
Keyword(s):  
Limit Cycles ◽  
Logistic Map ◽  
Period Doubling ◽  
Current Understanding ◽  
Semiconductor Superlattices ◽  
Frequency Locking ◽  
Routes To Chaos ◽  
Route To Chaos ◽  
Geometric Picture ◽  
Torus Bifurcation

Our current understanding of routes to chaos is mainly based on torus bifurcation where new periods are generated, the period-doubling mechanism revealed in the logistic map, and intermittency where periodic and burst motion appear alternatively. We present a possible new route to chaos based on our geometric picture of the frequency-locking of limit-cycles in semiconductor superlattices. In the period-double route and/or its variations, the period increases exponentially with bifurcation order, whereas the period in the new route increases linearly with the order of bifurcations.

Nonlinear Dynamics of Duffing System With Fractional Order Damping

2009 ◽  
Author(s):  
Junyi Cao ◽  
Chengbin Ma ◽  
Hang Xie ◽  
Zhuangde Jiang
Keyword(s):  
Nonlinear Dynamics ◽  
Fractional Order ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Bifurcation Diagrams ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Route To Chaos ◽  
Euler Methods ◽  
Period Motion

In this paper, nonlinear dynamics of Duffing system with fractional order damping is investigated. The four order Runge-Kutta method and ten order CFE-Euler methods are introduced to simulate the fractional order Duffing equations. The effect of taking fractional order on the system dynamics is investigated using phase diagrams, bifurcation diagrams and Poincare map. The bifurcation diagram is also used to exam the effects of excitation amplitude and frequency on Duffing system with fractional order damping. The analysis results show that the fractional order damped Duffing system exhibits period motion, chaos, period motion, chaos, period motion in turn when the fractional order changes from 0.1 to 2.0. A period doubling route to chaos is clearly observed.

scholarly journals Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Paul Yaovi Dousseh ◽  
Cyrille Ainamon ◽  
Clément Hodévèwan Miwadinou ◽  
Adjimon Vincent Monwanou ◽  
Jean Bio Chabi Orou
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Fractional Order Systems ◽  
Uncertain Dynamics ◽  
Dynamical Behaviors ◽  
Order Case ◽  
Route To Chaos

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.13 for the incommensurate order case. Also, period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

Hunting Cooperation in a Discrete-Time Predator–Prey System

2018 ◽  
Vol 28 (07) ◽  
pp. 1850083 ◽  
Author(s):  
Saheb Pal ◽  
Nikhil Pal ◽  
Joydev Chattopadhyay
Keyword(s):  
Numerical Simulations ◽  
Discrete Time ◽  
Period Doubling ◽  
Phase Portraits ◽  
Predator Population ◽  
Predator Prey ◽  
Prey System ◽  
Cooperation Rate ◽  
Route To Chaos ◽  
The Impact

The present paper mainly investigates the impact of hunting cooperation in a discrete-time predator–prey system through numerical simulations. We show that without hunting cooperation, an increase in the growth rate of prey population produces chaotic dynamics. We also show that hunting cooperation has the potential to modify the well-known period-doubling route to chaos by reverse period-halving bifurcations and makes the system stable. However, very high hunting cooperation can be detrimental and populations go to extinction. We observe that hunting cooperation induces strong demographic Allee effect in the system, where predator population persists due to hunting cooperation and would go to extinction without hunting cooperation. We perform extensive numerical simulations of the system and draw phase portraits, bifurcation diagrams, maximum Lyapunov exponents, two-parameter stability regions. We also observe the occurrence of flip and Neimark–Sacker bifurcations by taking the hunting cooperation rate as a bifurcation parameter.

scholarly journals Corrigendum to “Chaos in a Financial System with Fractional Order and Its Control via Sliding Mode”

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Paul Yaovi Dousseh ◽  
Cyrille Ainamon ◽  
Clément Hodévèwan Miwadinou ◽  
Adjimon Vincent Monwanou ◽  
Jean Bio Chabi Orou
Keyword(s):  
Fractional Order ◽  
Chaos Control ◽  
Sliding Mode ◽  
Financial System ◽  
Period Doubling ◽  
Fractional Order Systems ◽  
Uncertain Dynamics ◽  
Dynamical Behaviors ◽  
Order Case ◽  
Route To Chaos

In this paper, the dynamical behaviors and chaos control of a fractional-order financial system are discussed. The lowest fractional order found from which the system generates chaos is 2.49 for the commensurate order case and 2.57 for the incommensurate order case. Also, the period-doubling route to chaos was found in this system. The results of this study were validated by the existence of a positive Lyapunov exponent. Besides, in order to control chaos in this fractional-order financial system with uncertain dynamics, a sliding mode controller is derived. The proposed controller stabilizes the commensurate and incommensurate fractional-order systems. Numerical simulations are carried out to verify the analytical results.

QUADRATIC STOCHASTIC OPERATORS AND PROCESSES: RESULTS AND OPEN PROBLEMS

2011 ◽  
Vol 14 (02) ◽  
pp. 279-335 ◽  
Author(s):  
RASUL GANIKHODZHAEV ◽  
FARRUKH MUKHAMEDOV ◽  
UTKIR ROZIKOV
Keyword(s):  
Dynamical Systems ◽  
Stochastic Processes ◽  
Discrete Time ◽  
Continuous Time ◽  
Full Text ◽  
Open Problems ◽  
Quadratic Stochastic Operators ◽  
History Of ◽  
Discrete Time Dynamical Systems

The history of the quadratic stochastic operators can be traced back to the work of Bernshtein (1924). For more than 80 years, this theory has been developed and many papers were published. In recent years it has again become of interest in connection with its numerous applications in many branches of mathematics, biology and physics. But most results of the theory were published in non-English journals, full text of which are not accessible. In this paper we give all necessary definitions and a brief description of the results for three cases: (i) discrete-time dynamical systems generated by quadratic stochastic operators; (ii) continuous-time stochastic processes generated by quadratic operators; (iii) quantum quadratic stochastic operators and processes. Moreover, we discuss several open problems.

scholarly journals A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Khalid K. Ali ◽  
Mohamed A. Abd El salam ◽  
Emad M. H. Mohamed
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Order ◽  
Linear Functional ◽  
Numerical Technique ◽  
Operational Matrix ◽  
Fractional Order Differential Equations ◽  
The Given ◽  
Functional Argument ◽  
Special Case

AbstractIn this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

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