ON THE STABILITY OF PERIODIC SOLUTIONS OF PIECEWISE SMOOTH PERIODIC DIFFERENTIAL EQUATIONS

In this paper,we address the stability of periodic solutions of piecewise smooth periodic differential equations.By studying the Poincarémap,we give a sufficient condition to judge the stability of a periodic solution.We also present examples of some applications.

SMALE HORSESHOES AND CHAOS IN DISCRETIZED PERTURBED NLS SYSTEMS （Ⅰ）-POINCARE MAP

The existence of Smale horseshoes for a certain discretized perturbed nonlinear Schroedinger （NLS） equations was established by using n-dimensional versions of the Conley-Moser conditions. As a result, the discretized perturbed NLS system is shown to possess an invadant set A on which the dynamics is topologically conjugate to a shift on four symbols.

Reduced differential transform and Sumudu transform methods for solving fractional financial models of awareness

In that paper,we new study has been carried out on previous studies of one of the most important mathematical models that describe the global economic movement,and that is described as a non-linear fractional financial model of awareness,where the studies are represented at the steps following:One:The schematic of the model is suggested.Two:The disease-free equilibrium point(DFE)and the stability of the equilibrium point are discussed.Three:The stability of the model is fulfilled by drawing the Lyapunov exponents and Poincare map.Fourth:The existence of uniformly stable solutions have discussed.Five:The Caputo is described as the fractional derivative.Six:Fractional optimal control for NFFMA is discussed by clarifying the fractional optimal control through drawing before and after control.Seven:Reduced differential transform method(RDTM)and Sumudu Decomposition Method(SDM)are used to take the resolution of an NFFMA.Finally,we display that SDM and RDTM are highly identical.

CHAOTIC MOTION OF AN ELASTIC CIRCULAR PLATE

The chaotic motion of a harmonically forced circular plate is studied in the paper. The virtual displacement principle is used to derive the dynamic equation of motion, with the effect of large deflection of plate taken into account. By means of Garlerkin approach and Melnikov function method, the critical condition for chaotic motion is obtained. A demonstrative example is discussed through the Poincare mapping, phase portrait and time history.

Generating one-,two-,three-and four-scroll attractors from a novel four-dimensional smooth autonomous chaotic system

A new four-dimensional quadratic smooth autonomous chaotic system is presented in this paper, which can exhibit periodic orbit and chaos under the conditions on the system parameters. Importantly, the system can generate one-, two-, three- and four-scroll chaotic attractors with appropriate choices of parameters. Interestingly, all the attractors are generated only by changing a single parameter. The dynamic analysis approach in the paper involves time series, phase portraits, Poincare maps, a bifurcation diagram, and Lyapunov exponents, to investigate some basic dynamical behaviours of the proposed four-dimensional system.

Global positive periodic solutions of age-dependent competing systems

This paper is to investigate positive periodic solutions of a biological system composed of two competing species. The existence and uniqueness of nonnegative solutions to the model for a set of given vital rates and initial distribution are treated and the contractive property of the solutions explored. Based on these results, some simple conditions for the global existence of positive periodic orbits are established by means of Horn＇s asymptotic fixed point theorem.

Optimal and Memristor-Based Control of A Nonlinear Fractional Tumor-Immune Model

In this article,the reduced differential transform method is introduced to solve the nonlinear fractional model of Tumor-Immune.The fractional derivatives are described in the Caputo sense.The solutions derived using this method are easy and very accurate.The model is given by its signal flow diagram.Moreover,a simulation of the system by the Simulink of MATLAB is given.The disease-free equilibrium and stability of the equilibrium point are calculated.Formulation of a fractional optimal control for the cancer model is calculated.In addition,to control the system,we propose a novel modification of its model.This modification is based on converting the model to a memristive one,which is a first time in the literature that such idea is used to control this type of diseases.Also,we study the system’s stability via the Lyapunov exponents and Poincare maps before and after control.Fractional order differential equations(FDEs)are commonly utilized to model systems that have memory,and exist in several physical phenomena,models in thermoelasticity field,and biological paradigms.FDEs have been utilized to model the realistic biphasic decline manner of elastic systems and infection of diseases with a slower rate of change.FDEs are more useful than integer-order in modeling sophisticated models that contain physical phenomena.

Bifurcation and chaos analysis for aeroelastic airfoil with freeplay structural nonlinearity in pitch

The dynamics character of a two degree-of-freedom aeroelastic airfoil with combined freeplay and cubic stiffness nonlinearities in pitch submitted to supersonic and hypersonic flow has been gaining significant attention. The Poincare mapping method and Floquet theory are adopted to analyse the limit cycle oscillation flutter and chaotic motion of this system. The result shows that the limit cycle oscillation flutter can be accurately predicted by the Floquet multiplier. The phase trajectories of both the pitch and plunge motion are obtained and the results show that the plunge motion is much more complex than the pitch motion. It is also proved that initial conditions have important influences on the dynamics character of the airfoil system. In a certain range of airspeed and with the same system parameters, the stable limit cycle oscillation, chaotic and multi-periodic motions can be detected under different initial conditions. The figure of the Poincare section also approves the previous conclusion.

Passive walker that can walk down steps:simulations and experiments

A planar passive walking model with straight legs and round feet was discussed. This model can walk down steps, both on stairs with even steps and with random steps. Simulations showed that models with small moments of inertia can navigate large height steps. Period-doubling has been observed when the space between steps grows. This period-doubling has been validated by experiments, and the results of experiments were coincident with the simulation.

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincare map of the system is constructed. Using the Poincare map and the Gram Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.

A kind of noise-induced transition to noisy chaos in stochastically perturbed dynamical system

We investigate a kind of noise-induced transition to noisy chaos in dynamical systems. Due to similar phenomenological structures of stable hyperbolic attractors excited by various physical realizations from a given stationary random process, a specific Poincar6 map is established for stochastically perturbed quasi-Hamiltonian system. Based on this kind of map, various point sets in the Poincar6＇s cross-section and dynamical transitions can be analyzed. Results from the customary Duffing oscillator show that, the point sets in the Poincare＇s global cross-section will be highly compressed in one direction, and extend slowly along the deterministic period-doubling bifurcation trail in another direction when the strength of the harmonic excitation is fixed while the strength of the stochastic excitation is slowly increased. This kind of transition is called the noise-induced point-overspreading route to noisy chaos.

Topological horseshoe in nonlinear Bloch system

This paper demonstrates rigorous chaotic dynamics in nonlinear Bloch system by virtue of topological horseshoe and numerical method. It considers a properly chosen cross section and the corresponding Poincare map, and shows the existence of horseshoe in the Poincare map. In this way, a rigorous verification of chaos in the nonlinear Bloch system is presented.

Novel four-dimensional autonomous chaotic system generating one-,two-,three- and four-wing attractors

In this paper, we propose a novel four-dimensional autonomous chaotic system. Of particular interest is that this novel system can generate one-, two, three- and four-wing chaotic attractors with the variation of a single parameter, and the multi-wing type of the chaotic attractors can be displayed in all directions. The system is simple with a large positive Lyapunov exponent and can exhibit some interesting and complicated dynamical behaviours. Basic dynamical properties of the four-dimensional chaotic system, such as equilibrium points, the Poincare map, the bifurcation diagram and the Lyapunov exponents are investigated by using either theoretical analysis or numerical method. Finally, a circuit is designed for the implementation of the multi-wing chaotic attractors. The electronic workbench observations axe in good agreement with the numerical simulation results.

THE STUDY ON THE CHAOTIC MOTION OF A NONLINEAR DYNAMIC SYSTEM

In this paper, the system of the forced vibration -λ 1T+λ 2T 2+λ 3T 3=ε(g cos ωt-ε′) is discussed, which contains square and cubic items. The critical condition that the system enters chaotic states is given by the Melnikov method. By Poincaré map, phase portrait and time_displacement history diagram, whether the chaos occurs is determined.

Chaotic motion of the dynamical system under both additive and multiplicative noise excitations

With both additive and multiplicative noise excitations, the effect on the chaotic behaviour of the dynamical system is investigated in this paper. The random Melnikov theorem with the mean-square criterion that applies to a type of dynamical systems is analysed in order to obtain the conditions for the possible occurrence of chaos. As an example, for the Duffing system, we deduce its concrete expression for the threshold of multiplicative noise amplitude for the rising of chaos, and by combining figures, we discuss the influences of the amplitude, intensity and frequency of both bounded noises on the dynamical behaviour of the Duffing system separately. Finally, numerical simulations are illustrated to verify the theoretical analysis according to the largest Lyapunov exponent and Poincaré map.

Topological horseshoe analysis and field-programmable gate array implementation of a fractional-order four-wing chaotic attractor

We present a fractional-order three-dimensional chaotic system, which can generate four-wing chaotic attractor. Dy- namics of the fractional-order system is investigated by numerical simulations. To rigorously verify the chaos properties of this system, the existence of horseshoe in the four-wing attractor is presented. Firstly, a Poincar6 section is selected properly, and a first-return Poincar6 map is established. Then, a one-dimensional tensile horseshoe is discovered, which verifies the chaos existence of the system in mathematical view. Finally, the fractional-order chaotic attractor is imple- mented physically with a field-programmable gate array （FPGA） chip, which is useful in further engineering applications of information encryption and secure communications.

Analysis of limit cycle oscillations of a typical airfoil section with freeplay

A typical airfoil section system with freeplay is investigated in the paper. The classic quasi-steady flow model is applied to calculate the aerodynamics, and a piecewise-stiffness model is adopted to characterize the non- linearity of the airfoil section＇s freeplay. There are two crit- ical speeds in the system, i.e., a lower critical speed, above which the system might generate limit cycle oscillation, and an upper critical one, above which the system will flutter. Then a Poincar6 map is constructed for the limit cycle os- cillations by using piecewise-linear solutions with and with- out contact in the system. Through analysis of the Poincar6 map, a series of equations which can determine the frequen- cies of period-1 limit cycle oscillations at any flight veloc- ity are derived. Finally, these analytic results are compared to the results of numerical simulations, and a good agree- ment is found. The effects of freeplay value and contact stiffness ratio on the limit cycle oscillation are also analyzed through numerical simulations of the original system. More- over, there exist multi-periods limit cycle oscillations and even complicated ＂chaotic＂ oscillations may occur, which are usually found in smooth nonlinear dynamic systems.

Nonlinear Coupled Motions for a Given Two-Point Tension Mooring System

The nonlinear behaviors of plane coupled motions for a given two-point tension mooring system, are discussed in the present paper. For a cylinder moored by two taut lines under the action of gravity, buoyance and forces due to wave-current and mooring lines, a mathematical model of motions with three degrees of freedom is established. The steady solution and stability are analyzed. By integrating the equations of motions, history, phase map and Poincare map are obtained. The Liapunov exponents are also computed. The numerical results show that: the horizontal movement will increase, and stability will also increase as the steady force increases. The amplitude of responses will decrease as time-dependent forces decrease. Because of the geometric nonlinearity, there exist many windows bifurcating to pseudo-periodic or multi-periodic solution. The bifurcating patterns may be different. The behaviors are very complex. Under wave excitation alone, the motions are nonsymmetrical but still symmetrical statistically.

Chaos Existence in Surface Discharge of Tracking Test

Tracking tests for different polymer materials were carried out to investigate the chaotic behavior of surface discharge. The discharge sequences measured during the discharge process were analyzed for finding the evidence of chaos existence. Four kinds of nonlinear analysis methods were adopted： estimating the largest Lyapunov exponent, calculating the fractal dimension with increasing the embedding dimension, drawing the recurrence plots, and plotting the Poincare maps. It is found that the largest Lyapunov exponent of the discharge is positive, and the plot of fractal dimension, as a function of embedding dimension, will saturate at a value. The recur- rence plots show the chaotic frame-work patterns, and the Poincar6 maps also have the chaotic characteristics. The results indicate that the chaotic behavior does exist in the discharge currents of the tracking test.

TWISTED BIFURCATIONS AND STABILITY OF HOMOCLINIC LOOP WITH HIGHER DIMENSIONS

By using the linear independent solutions of the linear variational equation along the homoclinic loop as the demanded local coordinates to construct the Poincaré map,the bifurcations of twisted homoclinic loop for higher dimensional systems are studied.Under the nonresonant and resonant conditions,the existence,number and existence regions of the 1-homoclinic loop,1-periodic orbit,2-homoclinic loop,2-periodic orbit and 2-fold 2-periodic orbit were obtained.Particularly,the asymptotic repressions of related bifurcation surfaces were also given.Moreover, the stability of homoclinic loop for higher dimensional systems and nontwisted homoclinic loop for planar systems were studied.