Complex projective synchronization in coupled chaotic complex dynamical systems

In previous papers, the projective factors are always chosen as real numbers, real matrices, or even real-valued functions, which means the coupled systems evolve in the same or inverse direction simultaneously. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, the projective synchronization with respect to a complex factor, called complex projective synchronization (CPS), should be taken into consideration. In this paper, based on Lyapunov stability theory, three typical chaotic complex dynamical systems are considered and the corresponding controllers are designed to achieve the complex projective synchronization. Further, an adaptive control method is adopted to design a universal controller for partially linear systems. Numerical examples are provided to show the effectiveness of the proposed method.     

Lie transformation method for dynamical systems having chaotic behavior

The paper persents recent developments in a singular perturbation method, known as the Lie transformation method for the analysis of nonlinear dynamical systems having chaotic behavior. A general approximate solution for a system of first-order differential equations having algebraic nonlinearities is introduced. Past applications to simple dynamical nonlinear models have shown that this method yields highly accurate solutions of the systems. In the present paper the capability of this method is extended to the analysis of dynamical systems having chaotic behavior: indeed, the presence of small divisors in the general expression of the solution suggests a modification of the method that is necessary in order to analyze nonlinear systems having chaotic behavior (indeed, even non-simple-harmonic behavior). For the case of Hamiltonian systems this is consistent with the KAM (Kolmogorov-Arnold-Moser) theory, which gives the limits of integrability for such systems; in contrast to the KAM theory, the present formulation is not limited to conservative systems. Applications to a classic aeroelastic problem (panel flutter) are also included.     

Analytical solutions of one-dimensional discrete dynamical systems with chaotic behaviour

This paper studies exact analytical solutions in closed form of the difference equation

Particle filtering for chaotic dynamical systems using future right-singular vectors

Nonlinear Dynamics - In this paper, we combine tools from the study of chaotic dynamical systems with nonlinear non-Gaussian data assimilation algorithms to produce novel particle filtering...     

The fastest,simplified method of Lyapunov exponents spectrum estimation for continuous-time dynamical systems

Nonlinear Dynamics - Classical method of Lyapunov exponents spectrum estimation for a n-th-order continuous-time, smooth dynamical system involves Gram–Schmidt orthonormalization and...     

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations,attractors of a dynamical system will drift in the phase space,which readily leads to colliding and mixing with each other,so it is very difficult to identify irregular signals evolving from arbitrary initial states.Here,periodic attractors from the simple cell mapping method are further iterated by a specific Poincare’ map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations.The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure.From the positions and the variations of attractors in the phase space,the action mechanism of bounded noise excitation is studied in detail.Several numerical examples are employed to illustrate the present procedure.It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.     

基于大偏差理论非高斯随机动力系统离出行为研究

本文介绍了大偏差理论的基本思想及其在非高斯随机动力系统的离出问题研究中的应用.依据不同的非高斯噪声类型,本文分别评述了随机混合系统、指数轻跳跃过程和α稳定Lévy噪声驱动的随机动力系统的离出问题的主要研究方法和近期研究进展.针对随机混合系统,本文介绍了利用随机微分方程对其进行近似的拟稳态扩散近似方法,计算拟势和最优离出路径的WKB近似方法与细致平衡条件的研究,以及求解随机混合系统的简化版本(即生灭过程)的离出问题的研究进展.对于指数轻跳跃过程驱动的随机动力系统,本文介绍了其大偏差原理和中度偏差原理的泛函极值问题的建立,拟势概念的定义和平均离出时间的估计.针对具有α稳定Lévy噪声的随机动力系统,本文介绍了计算平均首次离出时间和离出概率的理论和数值方法,计算最优离出路径的Onsager-Machlup理论、机器学习方法、最大似然法和数据驱动方法.最后,给出了非高斯随机动力系统的离出现象相关的一些开放性问题.     

Non-existence of Shilnikov chaos in continuous-time systems

In this paper,a non-existence condition for homoclinic and heteroclinic orbits in n-dimensional,continuous-time,and smooth systems is obtained.Based on this result and an elementary example,it can be conjectured that there is a fourth kind of chaos in polynomial ordinary differential equation(ODE) systems characterized by the nonexistence of homoclinic and heteroclinic orbits.     

Existence of fast traveling waves for some parabolic equations: A dynamical systems approach

We study semilinear elliptic equationsu + cu _x =f(u,u) and ² u + cu _x =f(u,u, ² u) in infinite cylinders (x,y) × ⁿ⁺¹ using methods from dynamical systems theory. We construct invariant manifolds, which contain the set of bounded solutions and then study a singular limitc, where the equations change type from elliptic to parabolic. In particular we show that on the invariant manifolds, the elliptic equation generates a smooth dynamical system, which converges to the dynamical system generated by the parabolic limit equation. Our results imply the existence of fast traveling waves for equations like a viscous reactive 2d-Burgers equation or the Cahn-Hillard equation in infinite strips.     

Impulsive control of chaotic attractors in nonlinear chaotic systems

Based on the study from both domestic and abroad, an impulsive control scheme on chaotic attractors in one kind of chaotic system is presented. By applying impulsive control theory of the universal equation, the asymptotically stable condition of impulsive control on chaotic attractors in such kind of nonlinear chaotic system has been deduced, andwith it, the upper bond of the impulse interval for asymptotically stable control was given.Numerical results are presented, which are considered with important reference value for control of chaotic attractors.     

Mother-infant interaction and quality of child's attachment: a nonlinear dynamical systems approach

The traditional classification of infant attachment described three distinct types (Ainsworth et al. 1978): Secure (B), Insecure-avoidant (A), and Insecure-resistant (C). Research shows that the quality of infant attachment reflects the child's history of interaction with their primary caregiver and, therefore, maternal sensitivity and appropriateness of maternal responses during the first year of life has been found to predict infant attachment. In this study Nonlinear Dynamic Systems (NDS) approach was applied to broaden the study of maternal sensitivity into the overall temporal organization of mother-infant relationship exchanges. The study focuses on understanding the differences between secure and insecure attached children by applying NDS in two temporal scales: real time and a developmental scale, with the notions of 'flexibility' and 'self-organization', respectively. Infants, classified as securely or insecurely attached at 15 months, had free-play situations with their mothers, at 6 and 12 months of age, videotaped and coded in real time. Results showed that at 6 months dyads from the B group, compared to the non-B group, showed higher flexibility through several NDS indices derived from the State-Space Grid method (SSG). The dyads at 12 months did not show differences in those indices. Moreover, B group showed self-organization by decreasing the number of attractors, from 6 to 12 months of infant's age, in contrast with A and C groups that either showed less self-organization, by increasing the number of attractors, or stayed basically as they were at 6 months. Furthermore, the B group showed an increase in the proportion of attractors with higher values from time 1 to time 2, in contrast to the non-B groups. Findings provide some grounds for using a SSG approach to deepen the construct of maternal sensitivity in dyadic terms.     

Stability analysis of switched fractional-order continuous-time systems

Nonlinear Dynamics - In this paper, a class of switched fractional-order continuous-time systems with order $$0<alpha <1$$ is investigated. First, an interesting property of...     

About the boundedness of 3-D continuous-time quadratic systems

In this paper, we generalize all the existing results in the current literature for the upper bound of a general 3-D quadratic continuous-time system. In particular, we find large regions in the bifurcation parameter space of this system where it is bounded.     

Conductance in discrete dynamical systems

We will cover results related to the study of the conductance on digraphs arising from discrete dynamical systems. Several definitions of conductance are known and we present a study which gives a comparison between them in order to choose the appropriate definition to applications. The study makes a strong use of symbolic dynamics. As an application, we analyze the mechanical system of the nonlinear pendulum.     

System of equations for describing dynamical problems associated with the mechanics of lung parenchyma

In [1], which is an expanded version of the paper [2], the equations of conservation of mass and momentum are shown to be valid for dynamical problems of lung parenchyma. This system of equations is now closed by means of the heat flux equation. As in [1, 2], the heat flux equation is obtained on the basis of the methods of the mechanics of heterogeneous media [3] and anatomical data on the structure of lung parenchyma.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 21–29, May–June, 1988.     

Second-order consensus seeking in directed networks of multi-agent dynamical systems via generalized linear local interaction protocols

This paper focuses on the analytical study of final consensus convergence state of multi-agent dynamical systems by using a kind of generalized linear local interaction protocols. All the agents in the fixed directed network topology are governed by double-integrator dynamics. Almost all the existing linear local interaction consensus protocols can be considered as special cases of the present paper. By combining the algebraic graph theory and the matrix theory, some necessary and sufficient conditions are derived for reaching the second-order consensus. Moreover, the finial consensus convergence states of all agents are also be analytically determined. According to the obtained results, it is found that both the linear gains and the eigenvalues of the Laplacian matrix associated with the directed network topology play key roles in reaching consensus. Finally, the effectiveness and correctness of our theoretical findings are demonstrated by some numerical examples.