Family Of Quasi-periodic Solutions Research Articles

Quasi-periodic solutions for class of Hamiltonian partial differential equations with fixed constant potential

We consider Hamiltonian partial differential equations u tt +|∂ x u|+σu = f(u); x ∈ T; t ∈ ℝ; with periodic boundary conditions, where f(u) is a real-analytic function of the form f(u) = u5 + o(u5) near u = 0; σ ∈ (0; 1) is a fixed constant, and T = ℝ/2πℤ: A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with π ∈ (0; 1)ℚ: The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.

Islands of stability and recurrence times in AdS

We study the stability of anti-de Sitter (AdS) spacetime to spherically symmetric perturbations of a real scalar field in general relativity. Further, we work within the context of the "two time framework" (TTF) approximation, which describes the leading nonlinear effects for small amplitude perturbations, and is therefore suitable for studying the weakly turbulent instability of AdS---including both collapsing and non-collapsing solutions. We have previously identified a class of quasi-periodic (QP) solutions to the TTF equations, and in this work we analyze their stability. We show that there exist several families of QP solutions that are stable to linear order, and we argue that these solutions represent islands of stability in TTF. We extract the eigenmodes of small oscillations about QP solutions, and we use them to predict approximate recurrence times for generic non-collapsing initial data in the full (non-TTF) system. Alternatively, when sufficient energy is driven to high-frequency modes, as occurs for initial data far from a QP solution, the TTF description breaks down as an approximation to the full system. Depending on the higher order dynamics of the full system, this often signals an imminent collapse to a black hole.

Double Hopf Bifurcations and Chaos of a Nonlinear Vibration System

A double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized by nonsemisimple 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 chaoses via a period-doubling route. It is shown that all the solutions and chaotic motions are obtained under positive damping.

Quasi-periodic solutions of a linear functional-differential equation

We use the Kolmogorov-Arnol'd-Moser method to establish the existence of an n-parameter family of quasi-periodic solutions of a system of linear functional-differential equations.

Time-Reversibility and Particle Sedimentation

This paper studies an ordinary differential equation (ODE) model, called the Stokeslet model, and describes sedimentation of small clusters of particles in a highly viscous fluid. This model has a trivial solution in which the n particles arrange themselves at the vertices of a regular n-sided polygon. When $n = 3$, Hocking [J. Fluid Mech., 20 (1964), pp. 129–139] and Caflisch et al. [Phys. Fluids, 31 (1988), pp. 3175–3179] prove the existence of periodic motion (in the frame moving with the center of gravity in the cluster) in which the particles form an isosceles triangle. The study of periodic and quasiperiodic solutions of the Stokeslet model is continued, with emphasis on the spatial and time reversal symmetry of the model (time reversibility is due to infinite viscosity and spatial (dihedral) symmetry is due to the assumption of identical particles and the symmetry of the trivial solution). For three particles, the existence of a second family of periodic solutions and a family of quasiperiodic solutions is proved. It is also indicated how the methods generalize to the case of n particles.

Hopf Bifurcation in the Presence of Spherical Symmetry: Analytical Results

This paper considers a one-parameter family of vector fields equivariant under the orthogonal group $O(3)$, with an invariant fixed point. Hopf bifurcation of this family is studied assuming $O(3)$ acts on the eigenspaces belonging to each purely imaginary eigenvalue as an $l = 2$ representation. The dynamics are then reduced to a ten-dimensional center manifold, and the normal form of the vector field is explicitly given up to fifth order. The five different types of bifurcating periodic solutions, predicted geometrically by Golubitsky and Stewart, are derived analytically: a family of axisymmetric solutions, two types of rotating waves (one rotating at twice the speed of the other), a family of standing waves and a family of tetrahedral waves. The stability conditions are given for all these solutions. These stabilities depend on the three coefficients appearing at cubic order in the normal form and on one combination of three coefficients occurring at fifth order. All solutions can be stable except for the fastest family of rotating waves. The slower rotating waves and the axisymmetric solutions may be simultaneously stable. Finally it is shown that a family of quasiperiodic solutions may bifurcate directly from the invariant fixed point together with the periodic solutions.

Isoenergetic families of quasi-periodic solutions near the equilateral solution of the three-body problem

The tensor\(C_k^i = R_{jkl}^i \dot q^j \dot q^l \) appearing in the equation of geodesic deviation is computed for the equilateral solution of the general three-body problem. The eigenvalues and eigenvectors ofCki are determined. It is found that at least one of the eigenvalues is negative, irrespective of the masses of the bodies. This implies that the equilateral solution is not stable. The eigenvectors with positive eigenvalues generate isoenergetic 1-parameter families of quasi-periodic solutions in the neighborhood of the equilateral solution. The relation between the 1-parameter families constructed here and those known from the literature is discussed.