In this paper, we prove existence of symmetric homoclinic orbits for the suspension bridge equation u⁗+βu″+eu−1=0 for all parameter values β∈0.5,1.9. For each β, a parameterization of the stable ...manifold is computed and the symmetric homoclinic orbits are obtained by solving a projected boundary value problem using Chebyshev series. The proof is computer-assisted and combines the uniform contraction theorem and the radii polynomial approach, which provides an efficient means of determining a set, centered at a numerical approximation of a solution, on which a Newton-like operator is a contraction.
In this article, we propose a symplectic algorithm for the stable manifolds of the Hamilton-Jacobi equations combined with an iterative procedure in N. Sakamoto and A. J. van der Schaft, "Analytical ...approximation methods for the stabilizing solution of the Hamilton-Jacobi equation," IEEE Trans. Autom. Control , 2008. Our algorithm includes two key aspects. The first one is to prove a precise estimate for radius of convergence and the errors of local approximate stable manifolds. The second one is to extend the local approximate stable manifolds to larger ones by symplectic algorithms, which have better long-time behaviors than general-purpose schemes. Our approach avoids the case of divergence of the iterative sequence of approximate stable manifolds and reduces the computation cost. We illustrate the effectiveness of the algorithm by an optimal control problem with exponential nonlinearity.
Strong NLS soliton–defect interactions GOODMAN, Roy H; HOLMES, Philip J; WEINSTEIN, Michael I
Physica. D,
06/2004, Letnik:
192, Številka:
3
Journal Article
Recenzirano
Odprti dostop
We consider the interaction of a nonlinear Schrödinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes ...under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et al. Phys. Rev. E 49 (1994) 3400, we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schrödinger simulations, and comment on differences due to the finite-dimensional ansatz.
•The perigee distance threshold for aerobraking is analysed.•NEA capture strategies using Earth flybys with and without aerobraking are proposed.•The dynamical model of aerobraking in the Sun-Earth ...rotating system is investigated.•A global optimization is carried out, based on a detailed design procedure of the two NEA capture strategies.
Since the Sun-Earth libration points L1 and L2 are regarded as ideal locations for space science missions and candidate gateways for future crewed interplanetary missions, capturing near-Earth asteroids (NEAs) around the Sun-Earth L1/L2 points has generated significant interest. Therefore, this paper proposes the concept of coupling together a flyby of the Earth and then capturing small NEAs onto Sun–Earth L1/L2 periodic orbits. In this capture strategy, the Sun-Earth circular restricted three-body problem (CRTBP) is used to calculate target Lypaunov orbits and their invariant manifolds. A periapsis map is then employed to determine the required perigee of the Earth flyby. Moreover, depending on the perigee distance of the flyby, Earth flybys with and without aerobraking are investigated to design a transfer trajectory capturing a small NEA from its initial orbit to the stable manifolds associated with Sun-Earth L1/L2 periodic orbits. Finally, a global optimization is carried out, based on a detailed design procedure for NEA capture using an Earth flyby. Results show that the NEA capture strategies using an Earth flyby with and without aerobraking both have the potential to be of lower cost in terms of energy requirements than a direct NEA capture strategy without the Earth flyby. Moreover, NEA capture with an Earth flyby also has the potential for a shorter flight time compared to the NEA capture strategy without the Earth flyby.
This work describes a method for approximating a branch of stable or unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic ...dynamical systems. We approximate the branch of invariant manifolds by polynomials and develop a-posteriori theorems which provide mathematically rigorous bounds on the truncation error. The hypotheses of these theorems are formulated in terms of certain inequalities which are checked via a finite number of calculations on a digital computer. By exploiting the analytic category we are able to obtain mathematically rigorous bounds on the jets of the manifolds, as well as on the derivatives of the manifolds with respect to the parameter. A number of example computations are given.
We obtain in a simpler manner versions of the Grobman-Hartman theorem and of the stable manifold theorem for a sequence of maps on a Banach space, which corresponds to consider a nonautonomous ...dynamics with discrete time. The proofs are made short by using a suspension to an infinite-dimensional space that makes the dynamics autonomous (and uniformly hyperbolic when originally it was nonuniformly hyperbolic).
For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can potentially result in the so-called singular horseshoe, which means the existence of a non-uniformly ...hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involving saddle periodic orbits is studied by two types of piecewise smooth systems: One is the piecewise smooth systems having an admissible saddle point with only real eigenvalues and an admissible saddle periodic orbit, and the other is the piecewise smooth systems having an admissible saddle-focus and an admissible saddle periodic orbit. Several kinds of sufficient conditions are obtained for the existence of only one heteroclinic cycle or only two heteroclinic cycles in the two types of piecewise smooth systems, respectively. In addition, some examples are presented to illustrate the results.
PDF
