Fractional Birkhoffian method for equilibrium stability of dynamical systems

In this paper, we present a new method, i.e. fractional Birkhoffian method, for stability of equilibrium positions of dynamical systems, in terms of Riesz derivatives, and study its applications. For an actual dynamical system, the fractional Birkhoffian method of constructing a fractional dynamical model is given, and then the seven criterions for fractional Birkhoffian method of equilibrium stability are established. As applications, by using the fractional Birkhoffian method, we construct four kinds of actual fractional dynamical models, which include a fractional Duffing oscillator model, a fractional Whittaker model, a fractional Emden model and a fractional Hojman–Urrutia model, and we explore the equilibrium stability of these models respectively. This work provides a general method for studying the equilibrium stability of an actual fractional dynamical system that is related to science and engineering.     

New class of chaotic systems with equilibrium points like a three-leaved clover

This paper presents a new class of chaotic systems with infinite number of equilibrium points like a three-leaved clover. They signify an exciting class of dynamical systems which represent many major characteristics of regular and chaotic motions. These chaotic systems belong to the general class of chaotic systems with hidden attractors. By using a systematic computer search, three chaotic systems with three-leaved-clover-shaped equilibria were found which are classified into dissipative systems. Dynamics of the chaotic system with the three-leaved-clover-equilibria has been investigated by using phase portraits, bifurcation diagram, Lyapunov exponents, Kaplan–Yorke dimension and Poincaré map. Moreover, an electronic circuit implementation of the theoretical system is designed to check its effectiveness. Random number generator design has been realized with newly developed chaotic systems. The obtained random bit sequences are used for image encryption. Security analysis of image encryption processes has been performed.     

OGY method for a class of discontinuous dynamical systems

In this paper, we prove that the OGY method to control unstable periodic orbits (UPOs) of continuous-time systems can be applied to a class of systems discontinuous with respect the state variable, by using a generalized derivative. Because the discontinuous problem may have not classical solutions, the initial value problem is transformed into a set-valued problem via Filippov regularization. The existence of the ingredients necessary to apply OGY method (UPO, Poincaré map and stable and unstable directions) is proved and the numerically implementation is explained. Another possible way analyzed in this paper is the continuous approximation of the underlying initial value problem, via Cellina??s theorem for differential inclusions. Thus, the problem is approximated by a continuous initial value problem, and the OGY method can be applied as usual.     

New results on stability and stabilization of a class of nonlinear fractional-order systems

The asymptotic stability and stabilization problem of a class of fractional-order nonlinear systems with Caputo derivative are discussed in this paper. By using of Mittag–Leffler function, Laplace transform, and the generalized Gronwall inequality, a new sufficient condition ensuring local asymptotic stability and stabilization of a class of fractional-order nonlinear systems with fractional-order α:1<α<2 is proposed. Then a sufficient condition for the global asymptotic stability and stabilization of such system is presented firstly. Finally, two numerical examples are provided to show the validity and feasibility of the proposed method.     

Adaptive explicit Magnus numerical method for nonlinear dynamical systems

Based on the new explicit Magnus expansion developed for nonlinear equations defined on a matrix Lie group, an efficient numerical method is proposed for nonlinear dynamical systems. To improve computational efficiency, the integration step size can be adaptively controlled. Validity and effectiveness of the method are shown by application to several nonlinear dynamical systems including the Duffing system, the van der Pol system with strong stiffness, and the nonlinear Hamiltonian pendulum system.     

A refined asymptotic perturbation method for nonlinear dynamical systems

In this paper, a refined asymptotic perturbation method for general nonlinear dynamical systems is proposed for the first time. This method can be considered as an alternative means for the traditional multiple scales method. Moreover, it is easier to be understood and used to carry out higher-order perturbation analysis. In addition, three examples including the Duffing equation, a system with quadratic and cubic nonlinearities to a subharmonic excitation, as well as the coupled van der Pol oscillator with parametrical excitations are investigated to illustrate the validity and usefulness of the proposed technique. The analytical and numerical results show good agreement.     

A new model reduction method for nonlinear dynamical systems

The present research work proposes a new systematic approach to the problem of model-reduction for nonlinear dynamical systems. The formulation of the problem is conveniently realized through a system of singular first-order quasi-linear invariance partial differential equations (PDEs), and a rather general explicit set of conditions for solvability is derived. In particular, within the class of analytic solutions, the aforementioned set of conditions guarantees the existence and uniqueness of a locally analytic solution. The solution to the above system of singular PDEs is then proven to represent the slow invariant manifold of the nonlinear dynamical system under consideration exponentially attracting all dynamic trajectories. As a result, an exact reduced-order model for the nonlinear system dynamics is obtained through the restriction of the original system dynamics on the aforementioned slow manifold. The local analyticity property of the solution’s graph that corresponds to the system’s slow manifold enables the development of a series solution method, which allows the polynomial approximation of the system dynamics on the slow manifold up to the desired degree of accuracy and can be easily implemented with the aid of a symbolic software package such as MAPLE. Finally, the proposed approach and method is evaluated through an illustrative biological reactor example.     

A novel embedding method for characterization of low-dimensional nonlinear dynamical systems

Nonlinear Dynamics - Characterization of dynamical systems remains a central challenge in real-world applications because, in most cases, governing equations of the systems cannot be obtained...     

Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

A method for seeking main bifurcation parameters of a class of nonlinear dynamical systems is proposed. The method is based on the effects of parametric variation of dynamical systems on eigenvalues of the Frechet matrix. The singularity theory is used to study the engineering unfolding(EU) and the universal unfolding(UU) of an arch structure model, respectively. Unfolding parameters of EU are combination of concerned physical parameters in actual engineering, and equivalence of unfolding parameters and physical parameters is verified. Transient sets and bifurcation behaviors of EU and UU are compared to illustrate that EU can reflect main bifurcation characteristics of nonlinear systems in engineering. The results improve the understanding and the scope of applicability of EU in actual engineering systems when UU is difficult to be obtained.     

The direct method of Lyapunov for nonlinear dynamical systems with fractional damping

Nonlinear Dynamics - In this paper, we introduce a generalization of Lyapunov’s direct method for dynamical systems with fractional damping. Hereto, we embed such systems within the...     

On attraction of equilibrium points of fractional-order systems and corresponding asymptotic stability criteria

Nonlinear Dynamics - This paper investigates the problem of the stability analysis for fractional-order nonlinear systems. First, we investigate the asymptotic behavior of the system solution by...     

A model-reduction method for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty

The present research work proposes a new systematic approach to the problem of model reduction for nonlinear discrete-time skew-product dynamical systems in the presence of model uncertainty. The problem of interest is addressed within the context of functional equation theory, and in particular, through a system of invariance functional equations for which a general set of conditions for solvability is provided. Within the class of analytic solutions, this set of conditions guarantees the existence and uniqueness of a locally analytic solution which represents the system’s slow invariant manifold attracting all dynamic trajectories in the absence of model uncertainty. An exact reduced-order model is then obtained through the restriction of the original discrete-time system dynamics on the slow manifold. The analyticity property of the solution to the invariance functional equations enables the development of a series solution method that can be easily implemented using MAPLE leading to polynomial approximations up to the desired degree of accuracy. Furthermore, the aforementioned attractivity property and the system’s transition towards the above manifold is analyzed and characterized in the presence of model uncertainty. Finally, the proposed method is evaluated through an illustrative biological reactor example.