Entropy analysis of integer and fractional dynamical systems

This paper investigates the adoption of entropy for analyzing the dynamics of a multiple independent particles system. Several entropy definitions and types of particle dynamics with integer and fractional behavior are studied. The results reveal the adequacy of the entropy concept in the analysis of complex dynamical systems.     

On stability of discontinuous systems via vector Lyapunov functions

This paper deals with the stability of systems with discontinuous right- hand side(with solutions in Filippov's sense)via locally Lipschitz continuous and regular vector Lyapunov functions.A new type of"set-valued derivative"of vector Lyapunov functions is introduced,some generalized comparison principles on discontinuous systems are shown.Furthermore,Lyapunov stability theory is developed for a class of discontinu- ous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.     

A new Lyapunov approach for global synchronization of non-autonomous chaotic systems

This paper proposes a new approach for finding the Lyapunov function to study the sufficient global synchronization criterion of master-slave non-autonomous chaotic systems via linear state error feedback control. The approach is first demonstrated in a synchronization scheme for the second-order non-autonomous chaotic systems and then generalized to the schemes for the nth-order non-autonomous chaotic systems. Some algebraic synchronization criteria for the second-order chaotic systems are obtained. The sharpness of the new criteria is compared with that of the existing criteria of the same type by numerical examples.     

Coexisting solutions and their chaotic neighborhood in the dynamical systems of nonlinear optics

Coexisting periodic solutions of a dynamical system describing nonlinear optical processes of the second-order are studied. The analytical results concern both the simplified autonomous model and the extended nonautonomous model, including the pump and damping mechanism. The neighborhood of periodic solutions is studied numerically, mainly in phase portraits. As a result of disturbance, for example detuning, the periodic solutions are shown to escape to other states, periodic, quasiperiodic, or chaotic. The chaotic behavior is indicated by the Lyapunov exponents. We also investigate selected aspects of synchronization (unidirectional or mutual) of two identical systems being in two different coexisting states. The effects of quenching the oscillations are shown. The quenching seems very promising for design of some advanced signal processing.     

On Lyapunov stability of impulsive Takagi–Sugeno fuzzy systems

We analyze the Lyapunov stability of impulsive Takagi–Sugeno fuzzy systems. Using the direct Lyapunov method, we establish sufficient conditions for the stability of these systems. We show that these conditions can be expressed in terms of a system of linear matrix inequalities. As an example, we consider an impulsive fuzzy control in a two-species “predator–prey” model. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 481–494, October–December, 2008.     

A new perturbation procedure for limit cycle analysis in three-dimensional nonlinear autonomous dynamical systems

By introducing a new parametric transformation and a suitable nonlinear frequency expansion, the modified Lindstedt–Poincaré method is extended to derive analytical approximations for limit cycles in three-dimensional nonlinear autonomous dynamical systems. By considering two typical examples, it can be seen that the results of the present method are in good agreement with those obtained numerically even if the control parameter is moderately large. Moreover, the present prediction is considerably more accurate than some published results obtained by the multiple time scales method and the normal form method.     

Lyapunov’s first and second instability theorems for Caputo fractional-order systems

As well as proving stability, Lyapunov functions can also be used to prove instability, for which there are two well-known theorems: Lyapunov’s first and second instability theorems, in the integer-order (ordinary differential equation) case. However, these instability theorems for Caputo fractional-order systems remain blank, due to the long lack of general results on continuation of solution and Caputo fractional derivative of Lyapunov functions along trajectories. In this paper, based on recent advances in these two aspects, the Lyapunov’s first and second instability theorems for Caputo fractional-order systems are presented with proofs and then illustrated by examples.

     

Robust H ∞ output tracking control for fuzzy networked systems with stochastic sampling and multiplicative noise

In this paper, the problem of robust sampled-data H _?? output tracking control is investigated for a class of fuzzy networked systems with stochastic sampling, multiplicative noise and time-varying norm-bounded uncertainties. For the sake of technical simplicity, only two different sampling periods are considered, their occurrence probabilities are given constants and satisfy Bernoulli distribution, and they can be extended to the case with multiple stochastic sampling periods. By using an input-delay method, the probabilistic system is transformed into a stochastic continuous time-delay system. A?new linear matrix inequality(LMI)-based procedure is proposed for designing state-feedback controllers, which would guarantee that the closed-loop networked system with stochastic sampling tracks the output of a given reference model well in the sense of H _??. Conservatism is reduced by taking the probability into account. Both network-induced delays and packet dropouts have been considered. Finally, an illustrative example is given to show the usefulness and effectiveness of the proposed H _?? output tracking design.     

Approximate stationary solution and stochastic stability for a class of differential equations with parametric colored noise

This paper aims to study a class of differential equations with parametric Gaussian colored noise. We present the general framework to get the solvability conditions of the approximate stationary probability density function, which is determined by the Fokker-Planck-Kolmogorov (FPK) equations. These equations are derived using the stochastic averaging method and the operator theory with the perturbation technique. An illustrative example is proposed to demonstrate the procedure of our proposed method. The analytical expression of approximate stationary probability density function is obtained. Numerical simulation is carried out to verify the analytical results and excellent agreement can be easily found. The FPK equation for the probability density function of order ε ⁰ is used to examine the almost-sure stability for the amplitude process. Finally, the stability in probability of the amplitude process is investigated by Lin and Cai’s method.     

Compatibility of local and global stability conditions for some discrete population models

In this paper, we investigate the stability of the model that has been proposed to study the growth of Bobwhite Quail Populations of Northern Wisconsin and prove that local stability implies global stability. We will prove the main results by using a suitable Lyapunov function and for illustration we apply the results on Hassell and Smith models. Also, we will show that for different values of the parameters, the population will exhibit variations in dynamics. Some illustrative examples and graphs are included to demonstrate the validity and applicability of the results.     

Stabilization of center systems via immersion and invariance

This paper considers the stabilization problem of nonlinear systems with center manifold (center systems). A new method based on (system) immersion and (manifold) invariance (I&I) is introduced to stabilize the center systems. One of the key steps is to define a target dynamics whose order should be strictly smaller than that of the system to be controlled. For the center systems, we prove that the order of the target dynamics can be equal to that of the corresponding reduced dynamics on their center manifolds. Constructing solution is given for the target dynamics of the quadratic center system with a transcritical or a Hopf control bifurcation. Illustrating examples with simulations are respectively presented to validate the proposed stabilization scheme. Supported by the National Natural Science Foundation of China (Grant No. 60674024).     

Exponential stability in the mean square for stochastic neural networks with mixed time-delays and Markovian jumping parameters

In this paper, the stability analysis problem is considered for a class of stochastic neural networks with mixed time-delays and Markovian jumping parameters. The mixed delays include discrete and distributed time-delays, and the jumping parameters are generated from a continuous-time discrete-state homogeneous Markov process. The aim of this paper is to establish some criteria under which the delayed stochastic neural networks are exponentially stable in the mean square. By constructing suitable Lyapunov functionals, several stability conditions are derived on the basis of inequality techniques and the stochastic analysis. An example is also provided in the end of this paper to demonstrate the usefulness of the proposed criteria.     

A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems

This paper discusses the synchronization and anti-synchronization of new uncertain unified chaotic systems (UUCS). Based on the idea of active control, a novel active Pinning control strategy is presented, which only needs a state of new UUCS. The proposed controller can achieve synchronization between a response system and a drive system, and ensure the synchronized robust stability of new UUCS. Numerical simulations of new UUCS show that the controller can make chaotic systems achieve synchronization or anti-synchronization in a quite short period and both are of good robust stability.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Stochastic stability for nonlinear systems driven by Lévy noise

This paper is to investigate the stochastic stability for nonlinear systems with Lévy process based on Lyapunov exponents. A method of equivalent linearization is proposed to reduce and simplify the original systems. And the mean square responses are carried out to verify the effectiveness of the proposed approach. Then the Lyapunov exponents will be defined and derived to explore the stochastic stability, and two examples are presented to demonstrate the procedure of equivalent linearization and stochastic stability is considered for these two special examples. The results show that the technique of equivalent linearization can be used to study nonlinear systems excited by Lévy noise.     

Local and global exponential output synchronization of complex delayed dynamical networks

This paper proposes a new complex dynamical network model with output coupling. This model is totally different from some existing complex network models. It is well known that the node state is difficult to be observed or measured, even the node state cannot be observed or measured at all. Moreover, sometimes only part states are needed to make synchronization to come true. For these phenomena, the output synchronization is investigated in this paper. Several criteria on local and global exponential output synchronization are derived for the proposed network model. Finally, numerical simulations are given to illustrate the effectiveness of the derived results.     

A symbolic algorithm for the automatic computation of multitone-input harmonic balance equations for nonlinear systems

A symbolic algorithm is developed for the automatic generation of harmonic balance equations for multitone input for a class of nonlinear differential systems with polynomial nonlinearities. Generalized expressions are derived for the construction of balance equations for a defined multitone signal form. Procedures are described for determining combinations for a given output frequency from the desired set obtained from box truncated spectra and their permutations to automate symbolic algorithm. An application of method is demonstrated using the well-known Duffing–Van der Pol equation. Then the obtained analytical results are compared with numerical simulations to show the accuracy of the approach. The computation times for both the numerical solutions of equations versus the number of frequency components and the symbolic generation of the equations versus the power of nonlinearity are also investigated.     

Estimation of the largest Lyapunov exponent from?the?perturbation vector and its derivative dot product

Nowadays, the largest Lyapunov exponent (LLE) is employed in many different areas of the scientific research. Thus, there is still need to elaborate fast and simple methods of LLE calculation. The new method of the LLE estimation is presented in this paper. The method (LLEDP) applies the perturbation vector and its derivative dot product to calculate Lyapunov exponent in the direction of disturbance. Value of this exponent is the LLE. The theoretical improvement was introduced. Results of the numerical simulations were shown and compared with the Stefański method (Stefański in Chaos Solitons Fractals 11(15):2443–2451, 2000; Stefański and Kapitaniak in Chaos Solitons Fractals 15:233–244, 2003; Stefański et al. in Chaos Solitons Fractals 23:1651–1659, 2005; Stefański in J. Theor. Appl. Mech. 46:665–678, 2008). Investigations of the Duffing oscillators with external excitation and three Duffing oscillators coupled in ring scheme without external excitation were made with use of the presented method. Fast time LLE estimation results convergence was shown.