Parameter Switching Algorithm Research Articles

Existence of hidden attractors in nonlinear hydro-turbine governing systems and its stability analysis

This work studies the stability and hidden dynamics of the nonlinear hydro-turbine governing system with an output limiting link, and propose a new six-dimensional system, which exhibits some hidden attractors. The parameter switching algorithm is used to numerically study the dynamic behaviors of the system. Moreover, it is investigated that for some parameters the system with a stable equilibrium point can generate strange hidden attractors. A self-excited attractor with the change of its parameters is also recognized. In addition, numerical simulations are carried out to analyze the dynamic behaviors of the proposed system by using the Lyapunov exponent spectra, Lyapunov dimensions, bifurcation diagrams, phase space orbits, and basins of attraction. Consequently, the findings in this work show that the basins of hidden attractors are tiny for which the standard computational procedure for localization is unavailable. These simulation results are conducive to better understanding of hidden chaotic attractors in higher-dimensional dynamical systems, and are also of great significance in revealing chaotic oscillations such as uncontrolled speed adjustment in the operation of hydropower station due to small changes of initial values.

Controlling the dynamics of a COVID‐19 mathematical model using a parameter switching algorithm

In this paper, the dynamics of an autonomous mathematical model of COVID‐19 depending on a real bifurcation parameter is controlled by the parameter switching (PS) algorithm. With this technique, it is proved that every attractor of the considered system can be numerically approximated and, therefore, the system can be determined to evolve along, for example, a stable periodic motion or a chaotic attractor. In this way, the algorithm can be considered as a chaos control or anticontrol (chaoticization) algorithm. Contrarily to existing chaos control techniques that generate modified attractors, the obtained attractors with the PS algorithm belong to the set of the system attractors. It is analytically shown that using the PS algorithm, every system attractor can be expressed as a convex combination of some existing attractors. Interestingly, the PS algorithm can be viewed as a generalization of Parrondo's paradox.

Attractor as a convex combination of a set of attractors

This paper presents an effective approach to constructing numerical attractors of a general class of continuous homogenous dynamical systems: decomposing an attractor as a convex combination of a set of other existing attractors. For this purpose, the convergent Parameter Switching (PS) numerical method is used to integrate the underlying dynamical system. The method is built on a convergent fixed step-size numerical method for ODEs. The paper shows that the PS algorithm, incorporating two binary operations, can be used to approximate any numerical attractor via a convex combination of some existing attractors. Several examples are presented to show the effectiveness of the proposed method.

Hidden hyperchaotic attractors in a new 4D fractional order system and its synchronization.

The research of finding hidden attractors in nonlinear dynamical systems has attracted much consideration because of its practical and theoretical importance. A new fractional order four-dimensional system, which can exhibit some hidden hyperchaotic attractors, is proposed in this paper. The predictor-corrector method of the Adams-Bashforth-Moulton algorithm and the parameter switching algorithm are used to numerically study this system. It is interesting that three different kinds of hidden hyperchaotic attractors with two positive Lyapunov exponents are found, and the fractional order system can have a line of equilibria, no equilibrium point, or only one stable equilibrium point. Moreover, a self-excited attractor is also recognized with the change of its parameters. Finally, the synchronization behavior is studied by using a linear feedback control method.

Hidden Hyperchaotic Attractors in a New 5D System Based on Chaotic System with Two Stable Node-Foci

This paper reports some hidden hyperchaotic attractors and complex dynamics in a new five-dimensional (5D) system with only two nonlinear terms. The system is generated by adding two linear controllers to an unusual 3D autonomous quadratic chaotic system with two stable node-foci. In particular, the hyperchaotic system without equilibrium or with only one stable equilibrium can generate two kinds of hidden hyperchaotic attractors with three positive Lyapunov exponents. Numerical methods not only verify the existence of such attractors and hyperchaotic attractors, but also show the dynamical evolution of this system. The 5D system has self-excited attractors and two types of hidden attractors with the change of its parameter. The parameter switching algorithm is further utilized to numerically approximate the attractor. Specifically, the hidden hyperchaotic attractor can be approximated by switching between two self-excited chaotic attractors. Finally, the circuit realization results are consistent with the numerical results.

Emulating “Chaos + Chaos = Order” in Chen’s Circuit of Fractional Order by Parameter Switching

In this paper, the effect of the parameter switching (PS) algorithm in a fractional order chaotic circuit is investigated both in simulation and experiment. The Chen system of fractional order is focused and realized in an electronic circuit. By designing a switching circuit, the PS algorithm is implemented and it is the first time, the paradoxical “Chaos [Formula: see text] Chaos [Formula: see text] Order” is presented in an electronic circuit. Both the simulation and experimental results confirm that the obtained attractor under switching approximates the attractor of the time-averaged model. Some important design issues for the circuitry realization of the PS scheme are pointed out. Finally, our work confirms the practical usage of PS algorithm in potential applications such as attractor synthesis and chaos control.

Chaos control of Hastings-Powell model by combining chaotic motions.

In this paper, we propose a Parameter Switching (PS) algorithm as a new chaos control method for the Hastings-Powell (HP) system. The PS algorithm is a convergent scheme that switches the control parameter within a set of values while the controlled system is numerically integrated. The attractor obtained with the PS algorithm matches the attractor obtained by integrating the system with the parameter replaced by the averaged value of the switched parameter values. The switching rule can be applied periodically or randomly over a set of given values. In this way, every stable cycle of the HP system can be approximated if its underlying parameter value equalizes the average value of the switching values. Moreover, the PS algorithm can be viewed as a generalization of Parrondo's game, which is applied for the first time to the HP system, by showing that losing strategy can win: "losing + losing = winning." If "loosing" is replaced with "chaos" and, "winning" with "order" (as the opposite to "chaos"), then by switching the parameter value in the HP system within two values, which generate chaotic motions, the PS algorithm can approximate a stable cycle so that symbolically one can write "chaos + chaos = regular." Also, by considering a different parameter control, new complex dynamics of the HP model are revealed.

Note on a parameter switching method for nonlinear ODEs

Abstract In this paper we study analytically a parameter switching (PS) algorithm applied to a class of systems of ODE, depending on a single real parameter. The algorithm allows the numerical approximation of any solution of the underlying system by simple periodical switches of the control parameter. Near a general approach of the convergence of the PS algorithm, some dissipative properties are investigated and the dynamical behavior of solutions is investigated with the Lyapunov function method. A numerical example is presented.

Parrondo’s paradox for chaos control and anticontrol of fractional-order systems

We present the generalized forms of Parrondo’s paradox existing in fractional-order nonlinear systems. The generalization is implemented by applying a parameter switching (PS) algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems. The PS algorithm switches a system parameter within a specific set of N ≥ 2 values when solving the system with some numerical integration method. It is proven that any attractor of the concerned system can be approximated numerically. By replacing the words “winning” and “loosing” in the classical Parrondo’s paradox with “order” and “chaos", respectively, the PS algorithm leads to the generalized Parrondo’s paradox: chaos1 + chaos2 + ··· + chaosN = order and order1 + order2 + ··· + orderN = chaos. Finally, the concept is well demonstrated with the results based on the fractional-order Chen system.

Approximations of chaotic attractors and its circuit design based on the parameter switching algorithm

Based on the parameter switching algorithm and the discrete chaotic system, a new chaotic system based parameter switching algorithm is proposed. The principles of parameter switching algorithm and chaotic system based parameter switching algorithm are presented in detail by means of flow chart and step description. By applying phase diagram observation method, chaotic attractor approximation of the unified chaotic system is investigated based on parameter switching algorithm and chaotic system based parameter switching algorithm. It shows that chaos can be obtained by switching two periodic parameters and periodic state can be observed by switching two chaotic parameters. Thus the formulas chaos+ chaos = periodic and period+ period = chaos are proved to be workable in this paper. Chaotic attractor approximation of Rössler chaotic system is also studied by employing the two switching methods. Two cases are investigated. Firstly, a chaotic switching system is obtained by switching a chaotic parameter and a periodic parameter. Then a more complex switching scheme is carried out. Periodic system is switched by two periodic parameters and a chaotic parameter. So, the formulas chaos+ periodic = chaos and periodic+ period+ chaos = periodic are proved to be workable. It shows that the switching system is the approximation of the original system under specified parameter, and the attractor is in accordance with the attractor of the targeting system. The outputs of the Logistic map based parameter switching algorithm are more complex than those of existing parameter switching algorithm. As the distribution of logistic map is not uniform, the approximate attractor does not consist of the targeting system and shows more complicated structure. But approximate attractors can be obtained when the distribution of discrete sequence is uniform. In addition, the chaotic map based parameter switching algorithm has larger secret key space since it has the initial values and parameter of the chaotic map. Finally, the parameter switching circuit of Rössler system is designed by introducing a square wave generator. Compared with the traditional switching chaotic circuit (switching between different systems), the design of parameter switch circuit is simpler as it does not need to change the original structure of the system. The output is affected by the frequency of the square wave. By adding an appropriate frequency square wave generator, the circuit simulation agrees with the numerical simulation. It presents a theoretical and experimental base for the practical application of the parameter switching chaotic systems.

Finding attractors of continuous-time systems by parameter switching

The review presents a parameter switching algorithm and his applications which allows numerical approximation of any attractor of a class of continuous-time dynamical systems depending linearly on a real parameter. The considered classes of systems are modeled by a general initial value problem which embeds dynamical systems continuous and discontinuous with respect to the state variable, of integer, and fractional order. The numerous results, presented in several papers, are systematized here on four representative known examples representing the four classes. The analytical proof of the algorithm convergence for the systems belonging to the continuous class is presented briefly, while for the other categories of systems, the convergence is numerically verified via computational tools. The utilized numerical tools necessary to apply the algorithm are contained in Appendices A, B, C, D and E.

Finding stable attractors of a class of dissipative dynamical systems by numerical parameter switching

In this article we prove numerically, via computer graphic simulations and specific examples, that switching the control parameter of a dynamical system belonging to a class of dissipative continuous dynamical systems, one can obtain a stable attractor. In this purpose, while a fixed step-size numerical method approximates the solution of the mathematical model, the parameter control is switched every few integration steps, the switching scheme being time periodic. The switch occurs within a considered set of admissible parameter values. Moreover, we show via numerical experiments that the obtained synthesized attractor belongs to the class of all admissible attractors for the considered system and matches to the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This switched strategy may force the system to evolve along on a stable attractor whatever the parameter values and introduces a convex structure inside of the attractor set via a bijection between the set of parameter control values and the attractors set. The algorithm besides its utility in systems stabilization, when some desired parameter control cannot be directly accessed, may serve as a model for the dynamics encountered in reality or in experiments, e.g. three species food chain models, electronic circuits, etc. This method, compared, for example, to the OGY algorithm where only small perturbations of parameter control can be issued, allows relatively large parameter perturbations. Also, it does not allow to stabilize an unstable orbit but, using an appropriate parameter switching algorithm, it allows to reach an already existing attractor. The present work extends the results we obtained previously and is applied to Lorenz, Rössler and Chen systems.

Multiple-scale analysis of optical bistability in semiconductor periodic structures

Multiple-scale analysis is employed for the study of nonlinear wave propagation in periodic layered media. In a first step, wave propagation in each individual layer is modeled by a corresponding equivalent nonlinear transmission line. The multiple-scale analysis is then employed to establish a system of nonlinear equations for the amplitudes of the forward and backward waves in the transmission lines of the model mentioned. This system of nonlinear equations is solved with the aid of a continuation technique for derivation of transmission characteristics of the periodic structure. To study optical bistability, a parameter-switching algorithm is utilized for obtaining the solutions of the system including both stable and unstable ones. For the sake of verification, we have also utilized a nonlinear finite-difference time-domain (NFDTD) method to analyze the wave propagation in the aforementioned structure. While there is an acceptable agreement between the bistability results obtained using our method and the NFDTD, there is a considerable difference between our results and the ones derived using the conventional coupled-mode theory.