Elucidating the escape dynamics of the four hill potential

This paper treads discontinuous bifurcation in piecewise smooth systems of Filippov type. These bifurcations occur when a fixed point or a periodic orbit crosses with the border between two regions of smooth behavior. A detailed analysis of generalization Poincaré map and monodromy matrix which are related shows that subfamily of system with invariant cone-like objects is foliated by periodic orbits and determines its stability. In addition, we introduce a theoretical framework for analyzing 3D perturbed nonlinear piecewise smooth systems and give necessary conditions so that different types of bifurcations occur. The analysis identifies criteria for the existence of a novel bifurcation based on sensitively the location of the return map. Moreover, the piecewise smooth Melnikov function and sufficient conditions of the existence of the periodic orbits for nonlinear perturbed system are explicitly obtained.     

Poincaré maps modeling and local orbital stability analysis of discontinuous piecewise affine periodically driven systems

This paper presents a methodology to study the local stability of periodic orbits (orbital stability) in switched discontinuous piecewise affine (DPWA) periodically driven multiple-input multiple-output (MIMO) systems. The switched system of interest has a bilinear state space representation where the controller inputs are binary signals taking values in the set {0,1}. These systems are characterized by a set of affine differential equations together with switching rules to commute between them. These switching rules are described by switching functions that are periodic in time and linear in state. The methodology is based on obtaining a discrete time model (Poincaré map), its steady state operation points, and its Jacobian matrix. This provides a powerful tool for studying their stability and to predict some kind of instability phenomena that the system can undergo like subharmonic oscillations. The proposed approach is applied to a power electronic circuit which toggles among six different system equations with five switching boundaries within a switching cycle. This work was supported by the Spanish Ministerio de Educación y Ciencia under Grant TEC-2004-05608-C02-02.     

<Emphasis Type="Italic">J</Emphasis><Subscript>2</Subscript> invariant relative orbits via differential correction algorithm

This paper describes a practical method for finding the invariant orbits in J ₂ relative dynamics. Working with the Hamiltonian model of the relative motion including the J ₂ perturbation, the effective differential correction algorithm for finding periodic orbits in three-body problem is extended to formation flying of Earth’s orbiters. Rather than using orbital elements, the analysis is done directly in physical space, which makes a direct connection with physical requirements. The asymptotic behavior of the invariant orbit is indicated by its stable and unstable manifolds. The period of the relative orbits is proved numerically to be slightly different from the ascending node period of the leader satellite, and a preliminary explanation for this phenomenon is presented. Then the compatibility between J ₂ invariant orbit and desired relative geometry is considered, and the design procedure for the initial values of the compatible configuration is proposed. The influences of measure errors on the invariant orbit are also investigated by the Monte–Carlo simulation. The project supported by the Innovation Foundation of Beihang University for Ph.D. Graduates, and the National Natural Science Foundation of China (60535010).     

Vibro-impact dynamics near a strong resonance point

Two typical vibratory systems with impact are considered, one of which is a two-degree-of-freedom vibratory system impacting an unconstrained rigid body, the other impacting a rigid amplitude stop. Such models play an important role in the studies of dynamics of mechanical systems with repeated impacts. Two-parameter bifurcations of fixed points in the vibro-impact systems, associated with 1:4 strong resonance, are analyzed by using the center manifold and normal form method for maps. The single-impact periodic motion and Poincaré map of the vibro-impact systems are derived analytically. Stability and local bifurcations of a single-impact periodic motion are analyzed by using the Poincaré map. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional one, and the normal form map for 1:4 resonance is obtained. Local behavior of two vibro-impact systems, near the bifurcation points for 1:4 resonance, are studied. Near the bifurcation point for 1:4 strong resonance there exist a Neimark–Sacker bifurcation of period one single-impact motion and a tangent (fold) bifurcation of period 4 four-impact motion, etc. The results from simulation show some interesting features of dynamics of the vibro-impact systems: namely, the “heteroclinic” circle formed by coinciding stable and unstable separatrices of saddles, T _in, T _on and T _out type tangent (fold) bifurcations, quasi-periodic impact orbits associated with period four four-impact and period eight eight-impact motions, etc. Different routes of period 4 four-impact motion to chaos are obtained by numerical simulation, in which the vibro-impact systems exhibit very complicated quasi-periodic impact motions. The project supported by National Natural Science Foundation of China (50475109, 10572055), Natural Science Foundation of Gansu Province Government of China (3ZS061-A25-043(key item)). The English text was polished by Keren Wang.     

Analysis of atypical orbits in one-dimensional piecewise-linear discontinuous maps

In this paper, boundary regions of 1-D linear piecewise-smooth discontinuous maps are examined analytically. It is shown that, under certain parameter conditions, maps exhibit atypical orbits like a continuum of periodic orbits and quasi-periodic orbits. Further, we have derived the conditions under which such phenomenon occurs. The paper also illustrates that there exists a specific parameter region where as the parameter is varied, there is a transition from stable to unstable periodic orbits. Moreover, we have derived an expression for the value of parameter at which this transition from stable to unstable periodic orbits occurs. Additionally, the dynamics concerning this value of parameter is also given.

     

Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1：2 resonance case

A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed. The project supported by the National Natural Science Foundation of China (10572055, 50475109) and the Natural Science Foundation of Gansu Province Government of China (3ZS051-A25-030(key item)) The English text was polished by Keren Wang.     

Intermittency as a codimension-three phenomenon

We analyze the interaction of three Hopf modes and show that locally a bifurcation gives rise to intermittency between three periodic solutions. This phenomenon can occur naturally in three-parameter families. Consider a vector fieldf with an equilibrium and suppose that the linearization off about this equilibrium has three rationally independent complex conjugate pairs of eigenvalues on the imaginary axis. As the parameters are varied, generically three branches of periodic solutions bifurcate from the steady-state solution. Using Birkhoff normal form, we can approximatef close to the bifurcation point by a vector field commuting with the symmetry group of the three-torus. The resulting system decouples into phase amplitude equations. The main part of the analysis concentrates on the amplitude equations in R³ that commute with an action ofZ ₂+Z ₂+Z ₂. Under certain conditions, there exists an asymptotically stable heteroclinic cycle. A similar example of such a phenomenon can be found in recent work by Guckenheimer and Holmes. The heteroclinic cycle connects three fixed points in the amplitude equations that correspond to three periodic orbits of the vector field in Birkhoff normal form. We can considerf as being an arbitrarily small perturbation of such a vector field. For this perturbation, the heteroclinic cycle disappears, but an invariant region where it was is still stable. Thus, we show that nearby solutions will still cycle around among the three periodic orbits.     

Averaging Theory of Arbitrary Order for Piecewise Smooth Differential Systems and Its Application

The averaging theory for studying periodic orbits of smooth differential systems has a long history. Whereas the averaging theory for piecewise smooth differential systems appeared only in recent years, where the unperturbed systems are smooth. When the unperturbed systems are only piecewise smooth, there is not an existing averaging theory to study existence of periodic orbits of their perturbed systems. Here we establish such a theory for one dimensional perturbed piecewise smooth periodic differential equations. Then we show how to transform planar perturbed piecewise smooth differential systems to one dimensional piecewise smooth periodic differential equations when the unperturbed planar piecewise smooth differential systems have a family of periodic orbits. Finally as application of our theory we study limit cycle bifurcation of planar piecewise differential systems which are perturbation of a \(\Sigma \)-center.     

Forced resonance orbit analysis of binary asteroid system with consideration of solar radiation pressure

The solar radiation pressure is one of the major perturbations to orbits in the study of binary asteroid system, since asteroids have relatively weak gravity fields. In this paper, based on the idea of treating the solar radiation pressure as periodic external excitation, one novel family of orbits due to primary resonance and another novel family of orbits due to both primary resonance and internal resonance have been found by the classical perturbation method. The two types of steady-state orbits due to external resonance with different area-to-mass ratios have been determined and discussed by the frequency–response equations analytically. Four binary asteroid systems, 283 Emma-S/2003 (283) 1, 22 Kalliope-Linus, 2006 Polonskaya-S/2005 (2006) 1 and 4029 Bridges have been taken as examples to show the validity of the proposed mechanism in the explanation of orbits formation due to resonance. The multiple shooting method is applied to obtain the resonance orbits after numerical iterations.

     

An application of Curie’s principle to elastoplastic dynamics

This paper deals with the stability and the dynamics of a harmonically excited elastic–perfectly plastic unsymmetrical oscillator. Stability of the periodic orbits is analytically investigated with a perturbation approach. The occurrence of ratcheting effect is discussed for this system, and is related to the loss of symmetry of the periodic orbit in the phase space. Curie’s principle of symmetry is numerically verified for the symmetrical system with positive damping. Therefore, the observation of ratcheting phenomenon is necessarily associated to a breaking of symmetry in the constitutive behaviour, or in the forcing term. However, the generalized version of Curie’s principle has to be considered when a negative damping is introduced.     

Dynamics of a mass–spring–pendulum system with vastly different frequencies

We investigate the dynamics of a simple pendulum coupled to a horizontal mass?Cspring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass?Cspring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and ??/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.     

Dynamics of Relative Phases: Generalised Multibreathers

For small Hamiltonian perturbation of a Hamiltonian systemof arbitrary number of degrees of freedom with anormally non-degenerate submanifold of periodic orbits we construct a nearbysubmanifold and an `effective Hamiltonian' on it such that the differencebetween the two Hamiltonian vector fields is small. The effective Hamiltonianis independent of one coordinate, the `overall phase', and hence thecorresponding action is preserved. Unlike standard averaging approaches,critical points of our effective Hamiltonian subject to given actioncorrespond to exact periodic solutions. We prove there has to be at least acertain number of these critical points given by global topological principles.The linearisation of the effective Hamiltonian about critical points is provedto give the linearised dynamics for the full system to leading order in theperturbation. Hence in the case of distinct eigenvalues which move at non-zerospeed with ,the linear stability type of the periodic orbit can be read offfrom the effective Hamiltonian. Our principal application is to networks ofoscillators or rotors where many such submanifolds of periodic orbits occurat the uncoupled limit – simply excite a number N 2 of the units inrational frequency ratio and put the others on equilibria, subject to anon-resonance condition. The resulting exact periodic solutions for weakcoupling are known as multibreathers. We call the approximate solutions givenby the effective Hamiltonian dynamics, `generalised multibreathers'. Theycorrespond to solutions which look periodic on a short time scale but therelative phases of the excited units may evolve slowly. Extensions aresketched to travelling breathers and energy exchange between degrees offreedom.     

Bifurcation and chaos of biochemical reaction model with impulsive perturbations

In this paper, a biochemical model with the impulsive perturbations is considered. By using the Floquet theorem for the impulsive equation and small-amplitude perturbation skills, we see that the boundary-periodic solution ([(x)\tilde](t),0)(\tilde{x}(t),0) is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. By numerical simulation, we can show that the system presents rich dynamics, including periodic solutions, quasi-periodic oscillations, period doubling cascades, periodic halving cascades, symmetry bifurcations, and chaos.     

Subharmonic Melnikov method of six-dimensional nonlinear systems and application to a laminated composite piezoelectric rectangular plate

In this paper, the bifurcations of subharmonic orbits are investigated for six-dimensional non-autonomous nonlinear systems using the improved subharmonic Melnikov method. The unperturbed system is composed of three independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The key problem at hand is the determination of the sufficient conditions on some of the periodic orbits for the unperturbed system to generate the subharmonic orbits after the periodic perturbations. Using the periodic transformations and the Poincaré map, an improved subharmonic Melnikov method is presented. Two theorems are obtained and can be used to analyze the subharmonic dynamic responses of six-dimensional non-autonomous nonlinear systems. The subharmonic Melnikov method is directly utilized to investigate the subharmonic orbits of the six-dimensional non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Using the subharmonic Melnikov method, the bifurcation function of the subharmonic orbit is obtained. Numerical simulations are used to verify the analytical predictions. The results of the numerical simulation also indicate the existence of the subharmonic orbits for the laminated composite piezoelectric rectangular plate.     

Chaotic thresholds for the piecewise linear discontinuous system with multiple well potentials

In this paper we investigate the bifurcations and the chaos of a piecewise linear discontinuous (PWLD) system based upon a rig-coupled SD oscillator, which can be smooth or discontinuous (SD) depending on the value of a system parameter, proposed in [18], showing the equilibrium bifurcations and the transitions between single, double and triple well dynamics for smooth regions. All solutions of the perturbed PWLD system, including equilibria, periodic orbits and homoclinic-like and heteroclinic-like orbits, are obtained and also the chaotic solutions are given analytically for this system. This allows us to employ the Melnikov method to detect the chaotic criterion analytically from the breaking of the homoclinic-like and heteroclinic-like orbits in the presence of viscous damping and an external harmonic driving force. The results presented here in this paper show the complicated dynamics for PWLD system of the subharmonic solutions, chaotic solutions and the coexistence of multiple solutions for the single well system, double well system and the triple well dynamics.     

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.     

Networks of periodic orbits in the circular restricted three-body problem with first order post-Newtonian terms

The motivation of this article is to numerically investigate the orbital dynamics of the planar post-Newtonian circular restricted problem of three bodies. By numerically integrating several large sets of initial conditions of orbits we obtain the basins of escape. Additionally, we determine the influence of the transition parameter on the orbital structure of the system, as well as on the families of simple symmetric periodic orbits. The networks and the stability of the symmetric periodic orbits are revealed, while the corresponding critical periodic solutions are also identified. The parametric evolution of the horizontal and the vertical stability of the periodic orbits are also monitored, as a function of the transition parameter.

     

Chaotic Coexistence in a Top-predator Mediated Competitive Exclusive web

A basic food web of 4 species is considered, of which there is a bottom prey X, two predators Y, Z on X, and a top predatorW only on Y. The study concerns with how one type of chaotic coexistence arises. It is shown that under the situation that without the top-predator W, competitor Z goes to extinction, without Z the XYW locks in a periodic cycle, yet with all species, the noncompetitiveZ can derive the dynamics from periodic orbits to chaos. The dynamics can be captured analytically by 1-dimensional unimodal and multimodal maps and symbolic shift maps.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.