Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields

We study bifurcations of two types of homoclinic orbits—a homoclinic orbit with resonant eigenvalues and an inclination-flip homoclinic orbit. For the former, we prove thatN-homoclinic orbits (N3) never bifurcate from the original homoclinic orbit. This answers a problem raised by Chow-Deng-Fiedler (J. Dynam. Diff. Eq. 2, 177–244, 1990). For the latter, we investigate mainlyN-homoclinic orbits andN-periodic orbits forN=1, 2 and determine whether they bifurcate or not under an additional condition on the eigenvalues of the linearized vector field around the equilibrium point.     

Controlling homoclinic orbits

In this paper we analyze various control-theoretic aspects of a nonlinear control system possessing homoclinic or heteroclinic orbits. In particular, we show that for a certain class of nonlinear control system possessing homoclinic orbits, a control can be found such that the system exhibits arbitrarily long periods in a neighborhood of the homoclinic. We then apply these ideas to bursting phenomena in the near wall region of a turbulent boundary layer. Our analysis is based on a recently developed finite-dimensional model of this region due to Aubry, Holmes, Lumley, and Stone.The research of A.M. Bloch was partially supported by the U.S. Army Research Office through MSI at Cornell University and by NSF Grant DMS-8701576 and AFOSR Grant AFOSR-ISSA-87-0077, J.E. Marsden's research was partially supported by DOE Contract DE-ATO3-88ER-12097 and MSI at Cornell University and by AFOSR Contract No. 88-NA-321.     

Revealing the evolution, the stability, and the escapes of families of resonant periodic orbits in Hamiltonian systems

We investigate the evolution of families of periodic orbits in a bisymmetrical potential made up of a two-dimensional harmonic oscillator with only one quartic perturbing term, in a number of resonant cases. Our main objective is to compute sufficiently and accurately the position and the period of the periodic orbits. For the derivation of the above quantities (position and period) we deploy in each resonance case semi-numerical methods. The comparison of our semi-numerical results with those obtained by numerical integration of the equations of motion indicates that in every case the relative error is always less than 1 %, and therefore, the agreement is more than sufficient. Thus, we claim that semi-numerical methods are very effective tools for computing periodic orbits. We also study in detail the case when the energy of the orbits is larger than the escape energy. In this case, the periodic orbits in almost all resonance families become unstable and eventually escape from the system. Our target is to calculate the escape period and the escape position of the periodic orbits and also to monitor their evolution with respect to the value of the energy.     

Sequences of orbits and the boundaries of the basin of attraction for two double heteroclinic orbits

The boundaries of the basin of attraction are usually assumed to be rather elementary for Hamiltonian systems with autonomous perturbations. In the case of one saddle point, the sequences of orbits before capture are unique for each basin. However, we show that for two saddle points each with double heteroclinic orbits, there is an infinite number of different sequences of nearly homoclinic orbits before capture depending on the four heteroclinic parameters. The probabilities of capture are independent of the capture sequence.     

Connecting orbits in perturbed systems

We apply Newton’s method and continuation techniques to determine heteroclinic connections in perturbed non-autonomous differential equations which do not exist for the underlying unperturbed system. This approach is particularly useful in a higher-dimensional context, where the numerical computation of invariant manifolds is very expensive. A detailed discussion of a four-dimensional model is presented, which describes a pendulum coupled to a harmonic oscillator.     

Wave-modulated orbits in rate-and-state friction

A frictional spring-block system has been widely used historically as a model to display some of the features of two slabs in sliding frictional contact. Putelat et al. (2008) [7] demonstrated that equations governing the sliding of two slabs could be approximated by spring-block equations, and studied relaxation oscillations for two slabs driven by uniform relative motion at their outer surfaces, employing this approximation. The present work revisits this problem. The equations of motion are first formulated exactly, with full allowance for wave reflections. Since the sliding is restricted to be independent of position on the interface, this leads to a set of differential-difference equations in the time domain. Formal but systematic asymptotic expansions reduce the equations to differential equations. Truncation of the differential system at the lowest non-trivial order reproduces a classical spring-block system, but with a slightly different “equivalent mass” than was obtained in the earlier work. Retention of the next term gives a new system, of higher order, that contains also some explicit effects of wave reflections. The smooth periodic orbits that result from the spring-block system in the regime of instability of steady sliding are “decorated” by an oscillation whose period is related to the travel time of the waves across the slabs. The approximating differential system reproduces this effect with reasonable accuracy when the mean sliding velocity is not too far from the critical velocity for the steady state. The differential system also displays a period-doubling bifurcation as the mean sliding velocity is increased, corresponding to similar behaviour of the exact differential-difference system.     

The geometry of the orbits of artificial satellites

A novel approach to the study of the orbits of artificial satellites is presented. Emphasis is placed upon the basic geometry and other aspects of satellite motion which are of first importance to satellite engineering. The motion of the orbital plane as a rigid body is introduced and a non-elliptical orbit motion in this plane is defined. The plane orbit so defined possesses the very desirable feature of representing a succession of satellite positions and hence reveals the true motion of the satellite. An analytical treatment yields a completely general second order theory of earth satellite motion which is suitable for engineering purposes. In the latter development, particular attention is paid to the apsidal motion of the orbit and the concomitant resonance effects at the critical orbit inclination. The basic nonlinear features of the apsidal motion, which have not been recognized in earlier theories, are incorporated in the analytical development so as to produce a theory valid at all angles of inclination of the orbit.     

Coupled trajectory and attitude stability of displaced orbits

Coupled trajectory and attitude stability of displaced solar orbits is studied by using sailcraft with a kind of two-folding construction with two unequal rectangular plates forming a right angle. Three-dimensional coupled trajectory and attitude equations are developed for the coupled dynamical system, and the results show that all three types of displaced solar orbits widely referenced can be achieved through selecting an appropriate size of the two-folding sail. An anal- ysis of the corresponding linear stability of the trajectory and attitude coupled system is carried out, and both trajectory and attitude linearly stable orbits are found to exist in a small range of parameters, whose non-linear stability is then examined via numerical simulations. Finally, passively stable orbits are found to have weak stability, and such passive means of station-keeping are attractive and useful in practice because of its simplicity.     

Optimization of spacecraft transfers between distant elliptical orbits

Optimization is made of the trajectories, controls, and the parameters of a low-thrust constant-power engine with energy storage of a spacecraft executing the maneuver of synchronous considerable change in the semimajor axis, eccentricity, and angle of an elliptical orbit in a spherical gravitational field. The gain in payload mass due to the energy storage is estimated. The optimal control law and the optimal ratio for the masses of the propulsion system are found     

Halo轨道的ASRE非线性留位控制方法

针对Halo轨道的留位控制问题,提出一种跟踪名义参考轨道的非线性最优跟踪控制策略.首先得到日-地系统第二平动点附近的有控制力作用下的轨道动力学方程,其次利用微分修正法得到要跟踪的名义参考轨道,然后将ASRE非线性最优跟踪控制方法与精细积分法相结合,针对Halo轨道的留位设计非线性控制器,最后利用数值算例验证了本文方法的...     

Utilizing true periodic orbits in chaos-based cryptography

Nonlinear Dynamics - Digital realizations of chaos-based cryptosystems suffer from lack of a reliable method for implementation. The common choice for implementation is to use fixed or...     

Relative periodic orbits in plane Poiseuille flow

A branch of relative periodic orbits is found in plane Poiseuille flow in a periodic domain at Reynolds numbers ranging from Re=3000$\mathit{Re}=3000$ to Re=5000$\mathit{Re}=5000$. These solutions consist in sinuous quasi-streamwise streaks periodically forced by quasi-streamwise vortices in a self-sustained process. The streaks and the vortices are located in the bulk of the flow. Only the amplitude, but not the shape, of the averaged velocity components does change as the Reynolds number is increased from 3000 to 5000. We conjecture that these solutions could therefore be related to large- and very large-scale structures observed in the bulk of fully developed turbulent channel flows.     

A method for classifying orbits near asteroids

A method for classifying orbits near asteroids under a polyhedral gravitational field is presented, and may serve as a valuable reference for spacecraft orbit design for asteroid exploration. The orbital dynamics near aster- oids are very complex. According to the variation in orbit characteristics after being affected by gravitational perturbation during the periapsis passage, orbits near an as- teroid can be classified into 9 categories： （1） surrounding- to-surrounding, （2） surrounding-to-surface, （3） surrounding- to-infinity, （4） infinity-to-infinity, （5） infinity-to-surface, （6） infinity-to-surrounding, （7） surface-to-surface, （8） surface- to-surrounding, and （9） surface-to- infinity. Assume that the orbital elements are constant near the periapsis, the gravitation potential is expanded into a harmonic series. Then, the influence of the gravitational perturbation on the orbit is studied analytically. The styles of orbits are dependent on the argument of periapsis, the periapsis radius, and the periapsis velocity. Given the argument of periapsis, the orbital energy before and after perturbation can be derived according to the periapsis radius and the periapsis velocity. Simulations have been performed for orbits in the gravitational field of 216 Kleopatra. The numerical results are well consistent with analytic predictions.     

N-pulse homoclinic orbits in perturbations of resonant hamiltonian systems

In this paper we develop an analytical method to detect orbits doubly asymptotic to slow manifolds in perturbations of integrable, two-degree-of-freedom resonant Hamiltonian systems. Our energy-phase method applies to both Hamiltonian and dissipative perturbations and reveals families of multi-pulse solutions which are not amenable to Melnikov-type methods. As an example, we study a two-mode approximation of the nonlinear, nonplanar oscillations of a parametrically forced inextensional beam. In this problem we find unusually complicated mechanisms for chaotic motions and verify their existence numerically.     

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.

     

Asymptotically stable periodic orbits of a coupled electromechanical system

In this paper an electromechanical system is analyzed. The existence and asymptotic stability of a periodic orbit are obtained in a mathematically rigorous way as well as an expansion of the period by using an adequate small parameter. For the analytical results the main tool used is the regular perturbation theory. Some results, such as the growing of the period according to some powers of the parameters and the relation 2:1 between the period of the cart, which is a part of the electromechanical system, and the period of the current, are compatible with earlier numerical findings.