Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies

ACKNOWLEDGMENTS The authors express their gratitude to the anonymous reviewers for their valuable comments and suggestions that help to improve the paper. This work was supported by the National Natural Science Foundation of China (Grants No. 11202085, No. 21276115, No. 11302087, No. 11302086, and No. 11402226), the Natural Science Foundation of Jiangsu Province (Grant No. BK20130479), and the Research Foundation for Advanced Talents of Jiangsu University (Grant No. 11JDG075 ).Peer reviewe

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

. In this paper, a double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by non-semisimple 1:1 resonance. Based on normal form theory, it is shown that two phase-locked periodic solutions may bifurcate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two dimensional torus. Further cascading bifurcations from the quasi-periodic motions lead to two chaos via period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping. Key words: double pendulum system, double Hopf bifurcation, stability, chaos y Permanent address: Department of Mechanics, Tianjin University, P. R. China, 300072. z The author to receive correspondence and proofs. Q. BI AND P. YU DOUBLE HOPF BIFURCATION 2 1. Introduction Hopf and general..

Symbolic Computation of Normal Forms for Semi-Simple Cases

This paper presents a method and computer programs for computing the normal forms of ordinary differential equations whose Jacobian matrix evaluated at an equilibrium involves semi-simple eigenvalues. The method can be used to deal with systems which are not necessarily described on a center manifold. An iterative procedure is developed for finding the closed-form expressions of the normal forms and associated nonlinear transformations. Computer programs using a symbolic computer language Maple are developed to facilitate the application of the method. The programs can be conveniently executed on a main frame, a workstation or a PC machine without any interaction. A number of examples are presented to demonstrate the applicability of the method and the computation eciency of the Maple programs

Non-Smooth Dynamic Behaviors as well as the Generation Mechanisms in a Modified Filippov-Type Chua’s Circuit with a Low-Frequency External Excitation

The main purpose of this paper is to study point-cycle type bistability as well as induced periodic bursting oscillations by taking a modified Filippov-type Chua’s circuit system with a low-frequency external excitation as an example. Two different kinds of bistable structures in the fast subsystem are obtained via conventional bifurcation analyses; meanwhile, nonconventional bifurcations are also employed to explain the nonsmooth structures in the bistability. In the following numerical investigations, dynamic evolutions of the full system are presented by regarding the excitation amplitude and frequency as analysis parameters. As a consequence, we can find that the classification method for periodic bursting oscillations in smooth systems is not completely applicable when nonconventional bifurcations such as the sliding bifurcations and persistence bifurcation are involved; in addition, it should be pointed out that the emergence of the bursting oscillation does not completely depend on bifurcations under the point-cycle bistable structure in this paper. It is predicted that there may be other unrevealed slow–fast transition mechanisms worthy of further study

Quasi-Matrix and Quasi-Inverse-Matrix Projective Synchronization for Delayed and Disturbed Fractional Order Neural Network

This paper is concerned with the quasi-matrix and quasi-inverse-matrix projective synchronization between two nonidentical delayed fractional order neural networks subjected to external disturbances. First, the definitions of quasi-matrix and quasi-inverse-matrix projective synchronization are given, respectively. Then, in order to realize two types of synchronization for delayed and disturbed fractional order neural networks, two sufficient conditions are established and proved by constructing appropriate Lyapunov function in combination with some fractional order differential inequalities. And their estimated synchronization error bound is obtained, which can be reduced to the required standard as small as what we need by selecting appropriate control parameters. Because of the generality of the proposed synchronization, choosing different projective matrix and controllers, the two synchronization types can be reduced to some common synchronization types for delayed fractional order neural networks, like quasi-complete synchronization, quasi-antisynchronization, quasi-projective synchronization, quasi-inverse projective synchronization, quasi-modified projective synchronization, quasi-inverse-modified projective synchronization, and so on. Finally, as applications, two numerical examples with simulations are employed to illustrate the efficiency and feasibility of the new synchronization analysis

Approximation to Hadamard Derivative via the Finite Part Integral

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique

Novel bursting patterns induced by hysteresis loops in a one-degree-of-freedom nonlinear oscillator with parametric and external excitations

This paper investigates the novel bursting patterns induced by different types of hysteresis loops in a parametrically and externally driven one-degree-of-freedom nonlinear oscillator (abbreviated as the PEDODOFNO) that can be represented the dynamics of a density perturbation in plasma device model. The bursting behavior named “fold/turning point-turning point/fold” form via “fold/turning point” hysteresis loop is investigated using the slow–fast analysis method in detail. Moreover, the dynamical transitions of the novel bursting, i.e., bursting of “fold/turning point-turning point/extreme point” form via “fold/turning point” hysteresis loop, bursting of “fold/fold-fold/extreme point” form via “fold/fold” hysteresis loop, bursting of “fold/fold-fold/extreme point-extreme point/turning point-turning point/fold” form via “fold/fold-extreme point/turning point” hysteresis loop, bursting of “fold/fold-fold/extreme point-extreme point/fold-fold/fold” form via “fold/fold-extreme point/fold” hysteresis loop, bursting of “fold/fold-fold/turning point-turning point/fold” form via “fold/fold” hysteresis loop, and bursting of “fold/fold” form via “fold/fold” hysteresis loop with a hypocritical cycle, are also studied. In these bursting patterns, the excited states are not the oscillations of the trajectory vibrating in the stable limit cycles conventionally, but the oscillations behaving in the hypocritical cycles that are produced by the procedures of the trajectory running steadily to the control range of the stable equilibria. The hypocritical cycles disappear by two approaches: the first is colliding with the saddle branches and the second is the control parameter reaching the maxima value. The two approaches result in the occurrence of the turning point and the extreme point, and then, the different types of hysteresis loops related to fold bifurcation, the turning point and the extreme point are created, which lead to the appearance of the distinctive hysteresis loops