Stability of relative equilibria with singular momentum values in simple mechanical systems

The Transversal Relative Equilibria of a Hamiltonian System with Symmetry

We show that, given a certain transversality condition, the set of relative equilibria $\mcl E$ near $p_e\in\mcl E$ of a Hamiltonian system with symmetry is locally Whitney-stratified by the conjugacy classes of the isotropy subgroups (under the product of the coadjoint and adjoint actions) of the momentum-generator pairs $(μ,ξ)$ of the relative equilibria. The dimension of the stratum of the conjugacy class (K) is $\dim G+2\dim Z(K)-\dim K$, where Z(K) is the center of K, and transverse to this stratum $\mcl E$ is locally diffeomorphic to the commuting pairs of the Lie algebra of $K/Z(K)$. Th…

On homoclinic orbits to center manifolds of elliptic-hyperbolic equilibria in Hamiltonian systems

We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure. This leads to the study of a singularity which inherits certain structure from the Hamiltonian nature of the system. Under non-degeneracy assumptions, we classify the possible Morse indices of this singularity, permitting a local description of the set of homoclinic orbits. We also consider the case of time-reversible Hamiltonian systems.

Symmetry-breaking singular controller design for Bogdanov-Takens bifurcations with an application to Chua system

We provide a complete symmetry-breaking bifurcation control for equivariant smooth differential systems with Bogdanov-Takens singularities. Controller coefficient space is partitioned by critical controller sets into different connected regions. The connected regions provide a classification for all qualitatively different dynamics of the controlled system. Hence, a state feedback controller design with four small controller coefficients is proposed for an efficient and full singular symmetry-breaking control. Our approach works well for nonlinear control systems with both controllable and unc…

Sturm attractors for quasilinear parabolic equations with singular coefficients

The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovsk'y and Fiedler (1986), generalized by Fiedler and Rocha (1996) and later for quasilinear equa- tions by the author (2017). In particular, we construct heteroclinic connections between hyperbolic equilibria, stating necessary and sufficient conditions for heteroclinics to occur. Such conditions can be computed through a permutation of the equilibria. Lastly, an example is c…

Almost Global Convergence in Singular Perturbations of Strongly Monotone Systems

This paper deals with global convergence to equilibria, and in particular Hirsch's generic convergence theorem for strongly monotone systems, for singular perturbations of monotone systems.

Relative equilibria in the 3-dimensional curved n-body problem

We consider the 3-dimensional gravitational $n$-body problem, $n\ge 2$, in spaces of constant Gaussian curvature $κ\ne 0$, i.e.\ on spheres ${\mathbb S}_κ^3$, for $κ>0$, and on hyperbolic manifolds ${\mathbb H}_κ^3$, for $κ<0$. Our goal is to define and study relative equilibria, which are orbits whose mutual distances remain constant in time. We also briefly discuss the issue of singularities in order to avoid impossible configurations. We derive the equations of motion and define six classes of relative equilibria, which follow naturally from the geometric properties of ${\mathbb S}_κ^3$ and…

On relative normal modes

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative perodic orbits near a relative equilibrium in a Hamiltonian system with continuous symmetries. In particular we prove that under appropriate hypotheses there exist relative periodic orbits near relative equilibria even when these relative equilibria are singular points of the corresponding moment map, i.e. when the reduced spaces are singular.

Linear instability of relative equilibria for $n$-body problems in the plane

Following Smale, we study simple symmetric mechanical systems of $n$ point particles in the plane. In particular, we address the question of the linear and spectral stability properties of relative equilibria, which are special solutions of the equations of motion. Our main result is a sufficient condition to detect spectral (hence linear) instability. Namely, we prove that if the Morse index of an equilibrium point with even nullity is odd, then the associated relative equilibrium is spectrally unstable. The proof is based on some refined formulæ for computing the spectral flow. As a notable …

Molecular chains interacting by Lennard-Jones and Coulomb forces

We study equations for the mechanical movement of chains of identical particles in the plane interacting with their nearest-neighbors by bond stretching and by van der Waals and Coulomb forces. We find collinear and circular equilibria as minimizers of the energy potential for chains with Neumann and periodic boundary conditions. We prove global bifurcation of periodic brake orbits from these equilibria applying the global Rabinowitz alternative. These results are complemented with numeric computations for ranges of parameters that include carbon atoms among other molecules.

Focus-focus singularities in classical mechanics

In this paper the local singularities of integrable Hamiltonian systems with two degrees of freedom are studied. The topological obstruction to the existence of focus-focus singularity with given complexity was found. It has been showed that only simple focus-focus singularities can appear in a typical mechanical system. The model examples of mechanical systems with complex focus-focus singularity are given.

Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms

In this paper we introduce universal asymptotic unfolding normal forms for nonlinear singular systems. Next, we propose an approach to find the parameters of a parametric singular system that they play the role of universal unfolding parameters. These parameters effectively influence the local dynamics of the system. We propose a systematic approach to locate local bifurcations in terms of these parameters. Here, we apply the proposed approach on Hopf-zero singularities whose the first few low degree terms are incompressible. In this direction, we obtain novel orbital and parametric normal for…

深度线性网络中的熵正则化

Entropic Regularization in the Deep Linear Network

本文研究了深度线性网络中熵正则化对自由能平衡点和梯度流的影响，并分析了熵在不同几何空间中的凹性特性。

平面N体问题中椭圆相对平衡态的定量线性稳定性分析

Quantitative Linear Stability Analysis of Elliptic Relative Equilibria in the Planar N-Body Problem

本文针对平面N体问题中的椭圆相对平衡态，利用迹公式建立非退化上界，从而在完整偏心率范围内给出其线性稳定性的定量估计。

Stability and bifurcations of symmetric tops

We study the stability and bifurcation of relative equilibria of a particle on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$ invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is to reduce the number of degrees of freedom at singular values of the $SO(2)\times SO(2)$ momentum map and study the stability of the equilibria of the reduced systems as a function of spin. The result is an elementary analysis of the fast/slow transition in the Lagrange and Kirchhoff tops. More generally, since an $SO(2)\times SO(2)$ invariant potential on $SO(3)$ ca…

Singularly perturbed dynamics of the tippedisk

The tippedisk is a mathematical-mechanical archetype for a peculiar friction-induced instability phenomenon leading to the inversion of an unbalanced spinning disk, being reminiscent to (but different from) the well-known inversion of the tippetop. A reduced model of the tippedisk, in the form of a three-dimensional ordinary differential equation, has been derived recently, followed by a preliminary local stability analysis of stationary spinning solutions. In the current paper, a global analysis of the reduced system is pursued using the framework of singular perturbation theory. It is shown …

Dynamic and asymptotic behavior of singularities of certain weak KAM solutions on the torus

For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristics semiflows associated with certain Hamilton-Jacobi equations, and build the relation between the $ω$-limit set of this semiflow and the projected Aubry set.