Effect of Random Parameter Switching on Commensurate Fractional Order Chaotic Systems

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record.The paper explores the effect of random parameter switching in a fractional order (FO) unified chaotic system which captures the dynamics of three popular sub-classes of chaotic systems i.e. Lorenz, Lu and Chen's family of attractors. The disappearance of chaos in such systems which rapidly switch from one family to the other has been investigated here for the commensurate FO scenario. Our simulation study show that a noise-like random variation in the key parameter of the unified chaotic system along with a gradual decrease in the commensurate FO is capable of suppressing the chaotic fluctuations much earlier than that with the fixed parameter one. The chaotic time series produced by such random parameter switching in nonlinear dynamical systems have been characterized using the largest Lyapunov exponent (LLE) and Shannon entropy. The effect of choosing different simulation techniques for random parameter FO switched chaotic systems have also been explored through two frequency domain and three time domain methods. Such a noise-like random switching mechanism could be useful for stabilization and control of chaotic oscillation in many real-world applications

Fractional order chaotic systems and their electronic design

"Con el desarrollo del cálculo fraccionario y la teoría del caos, los sistemas caóticos de orden fraccionario se han convertido en una forma útil de evaluar las características de los sistemas dinámicos. En esta dirección, esta tesis es principalmente relacionada, es decir, en el estudio de sistemas caóticos de orden fraccionario, basado en sistemas disipativos de inestables, un sistema disipativo de inestable de orden fraccionario es propuesto. Algunas propiedades dinámicas como puntos de equilibrio, exponentes de Lyapunov, diagramas de bifurcación y comportamientos dinámicos caóticos del sistema caótico de orden fraccionario son estudiados. Los resultados obtenidos muestran claramente que el sistema discutido presenta un comportamiento caótico. Por medio de considerar la teoría del cálculo fraccionario y simulaciones numéricas, se muestra que el comportamiento caótico existe en el sistema de tres ecuaciones diferenciales de orden fraccionario acopladas, con un orden menor a tres. Estos resultados son validados por la existencia de un exponente positivo de Lyapunov, además de algunos diagramas de fase. Por otra parte, la presencia de caos es también verificada obteniendo la herradura topológica. Dicha prueba topológica garantiza la generaci´n de caos en el sistema de orden fraccionario propuesto. En orden de verificar la efectividad del sistema propuesto, un circuito electrónico es diseñado con el fin de sintetizar el sistema caótico de orden fraccionario.""With the development of fractional order calculus and chaos theory, the fractional order chaotic systems have become a useful way to evaluate characteristics of dynamical systems and forecast the trend of complex systems. In this direction, this thesis is primarily concerned with the study of fractional order chaotic systems, based on an unstable dissipative system (UDS), a fractional order unstable dissipative system (FOUDS) is proposed. Dynamical properties, such as equilibrium points, Lyapunov exponents, bifurcation diagrams and phase diagrams of the fractional order chaotic system are studied. The obtained results shown that the fractional order unstable dissipative system has a chaotic behavior. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the fractional order three dimensional system with order less than three. The lowest order to yield chaos in this system is 2.4. The results are validated by the existence of one positive Lyapunov exponent, phase diagrams; Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order unstable dissipative system. In order to verify the effectiveness of the proposed system, an electronic circuit is designed with the purpose of synthesize the fractional order chaotic system, the fractional order integral is realized with electronic circuit utilizing the synthesis of a fractance circuit. The realization has been done via synthesis as passive RC circuits connected to an operational amplifier. The continuos fractional expansion have been utilized on fractional integration transfer function which has been approximated to integer order rational transfer function considering the Charef Method. The analogue electronics circuits have been simulated using HSPICE.

Parameter switching in a generalized Duffing system: Finding the stable attractors

This paper presents a simple periodic parameter-switching method which can find any stable limit cycle that can be numerically approximated in a generalized Duffing system. In this method, the initial value problem of the system is numerically integrated and the control parameter is switched periodically within a chosen set of parameter values. The resulted attractor matches with the attractor obtained by using the average of the switched values. The accurate match is verified by phase plots and Hausdorff distance measure in extensive simulations

Opamp-based synthesis of a fractional order switched system

"The analysis, design and circuit synthesis of a fractional order switched system is presented in this paper. That system is capable of showing chaotic oscillations with a fractional order less than three, i.e., 2.4. The dynamical system is called fractional order unstable dissipative system (FOUDS); because it consists of a switching law to display strange attractors. Its dynamical behavior is explored and a circuit synthesis system is realized considering operational amplifiers. SPICE simulations agree with the numerical results.

Hopf bifurcation, antimonotonicity and amplitude controls in the chaotic Toda jerk oscillator: analysis, circuit realization and combination synchronization in its fractional-order form

In this paper, an autonomous Toda jerk oscillator is proposed and analysed. The autonomous Toda jerk oscillator is obtained by converting an autonomous two-dimensional Toda oscillator with an exponential nonlinear term to a jerk oscillator. The existence of Hopf bifurcation is established during the stability analysis of the unique equilibrium point. For a suitable choice of the parameters, the proposed autonomous Toda jerk oscillator can generate antimonotonicity, periodic oscillations, chaotic oscillations and bubbles. By introducing two additional parameters in the proposed autonomous Toda jerk oscillator, it is possible to control partially or totally the amplitude of its signals. In addition, electronic circuit realization of the proposed Toda jerk oscillator is carried out to confirm results found during numerical simulations. The commensurate fractional-order version of the proposed autonomous chaotic Toda jerk oscillator is studied using the stability theorem of fractional-order oscillators and numerical simulations. It is found that periodic oscillations and chaos exist in the fractional-order form of the proposed Toda jerk oscillator with order less than three. Finally, combination synchronization of two fractional-order proposed autonomous chaotic Toda jerk oscillators with another fractional-order proposed autonomous chaotic Toda jerk oscillator is analysed using the nonlinear feedback control method

Strange attractors generated by a fractional order switching system and its topological horseshoe

"Chaos generation in a new fractional order unstable dissipative system with only two equilibrium points is reported. Based on the integer version of an unstable dissipative system (UDS) and using the same systems parameters, chaos behavior is observed with an order less than three, i.e., 2.85. The fractional order can be decreased as low as 2.4 varying the eigenvalues of the fractional UDS in accordance with a switching law that fulfills the asymptotic stability theorem for fractional systems. The largest Lyapunov exponent is computed from the numerical time series in order to prove the chaotic regime. Besides, the presence of chaos is also verified obtaining the topological horseshoe. That topological proof guarantees the chaos generation in the proposed fractional order switching system avoiding the possible numerical bias of Lyapunov exponents. Finally, an electronic circuit is designed to synthesize this fractional order chaotic system.

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...