This new version revises and supersedes the original submissions. It shows that all the examples constructed have divergent Birkhoff normal form at the origin. Moreover, it gives in all degrees of freedom larger or equal to 2 explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form
A new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom is introduced. We thus obtain explicit real entire Hamiltonians on $\R^{2d}$, $d\geq 4$, that have a Lyapunov unstable elliptic equilibrium with an arbitrary chosen frequency vector whose coordinates are not all of the same sign. For non-resonant frequency vectors, our examples all have divergent Birkhoff normal form at the equilibrium.
On $\R^4$, we give explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form.
Asymptotic and Lyapunov stability of Poisson equilibria
本文主要包括三个方面的结果，它们都与Poisson动力系统平衡点的非线性稳定性有关。首先，我们证明了一个能量Casimir型充分条件的稳定性，使用的功能，不一定是保守的流，并考虑到某些渐近稳定的行为，可能会发生在泊松类别。这种方法是适应泊松系统通过一个减少的过程，我们在例子中表明，我们提出的那种稳定性是适当的，当处理一些约束系统的平衡点的稳定性。最后，讨论了连续Casimir函数在稳定性研究中的应用与Patrick { \it et al.}
For the curved n-body problem in S^3, we show that a regular polygonal configuration for n masses on a geodesic is an equilibrium configuration if and only if n is odd and the masses are equal. The equilibrium configuration is associated with a one-parameter family (depending on the angular velocity) of relative equilibria, which take place on S^1 embedded in S^2. We then study the stability of the associated relative equilibria on two invariant manifolds, T^*((§^1)^n\D) and T^*((§^2)^n\D). We show that they are Lyapunov stable on S^1, they are Lyapunov stable on S^2 if the absolute value of a…
We prove that for every smooth compact manifold $M$ and any $r \ge 1$, whenever there is an open domain in $\mathrm{Diff}^r(M)$ exhibiting a persistent homoclinic tangency related to a basic set with a sectionally dissipative periodic saddle, topologically generic diffeomorphisms in this domain have Lyapunov unstable Milnor attractors. This implies, in particular, that the instability of Milnor attractors is locally topologically generic in $C^1$ if $\mathrm{dim}\,M \ge 3$ and in $C^2$ if $\mathrm{dim}\,M = 2$. Moreover, it follows from the results of C. Bonatti, L. J. Díaz and E. R. Pujals th…
For a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits.
Further Results on Lyapunov-Like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems
我们给出了非线性系统收敛于不稳定平衡点的几个特征。我们目前的贡献是三方面的。首先，我们提出了一个简单的代数条件，建立局部收敛的非平凡解的非线性系统的不稳定平衡。这些条件是基于早期的工作（A.N. Gorban，I.Yu. Tyukin，E. Steur和H. Nijmeijer，SIAM Journal on Control and Optimization，Vol.51，No.3，2013），并且可以被视为李雅普诺夫第一方法的扩展，因为它们适用于其中对应的雅可比矩阵具有一个零特征值的系统。其次，我们表明，对于一个相关的子类系统，在右手边的方程的时间函数的激励的持久性的系统动力学确保存在一个吸引子盆地，通过这个盆地的解决方案在向前的时间收敛到原点指数。最后，我们证明了条件发展（A.N. Gorban，I.Yu. Tyukin，E. Steur和H. Nijmeijer，SIAM Journal on Control and Optimization，第51卷，第3期，2013年）可能非常紧凑。
Double averaged planar restricted elliptic three-body problem has a two-parametric family of stable equilibria. We show that these equilibria are stable in the linear approximation as equilibria of the double averaged spatial restricted elliptic three-body problem. They are Lyapunov stable for all values of parameters but, possibly, parameters from some finite set of analytic curves.
We consider white noise perturbations of a nonlinear dynamical system in the neighborhood of an unstable critical point with linearization given by a Jordan block of full dimension. For the associated exit problem, we study the joint limiting behavior of the exit location and exit time, in the vanishing noise limit. The exit typically happens near one of two special deterministic points associated with the eigendirection, and we obtain several more terms in the expansion for the exit point. The leading correction term is deterministic and logarithmic in the noise magnitude, while the random re…
We found a dichotomy involving the unstable Lyapunov exponent of a special Anosov endomorphism of the torus induced by the conjugacy with the linearization. In fact, either every unstable leaf meets on a set of zero measure the set for which is defined such unstable Lyapunov exponent or the endomorphims is smoothly conjugated with its linearization. Also we are able to characterize the absolute continuity of the intermediate foliation for a class of volume preserving special Anosov endomorphisms of $\mathbb{T}^3.$
We consider families of systems of two-dimensional ordinary differential equations with the origin $0$ as a non-hyperbolic equilibrium. For any number $s \in (-\infty, +\infty)$ we show that it is possible to choose a parameter in these equations such that the stability index $σ(0)$ is precisely $σ(0)=s$. In contrast to that, for a hyperbolic equilibrium $x$ it is known that either $σ(x)=-\infty$ or $σ(x)=+\infty$. Furthermore, we discuss a system with an equilibrium that is locally unstable but globally attracting, highlighting some subtle differences between the local and non-local stability…
Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$ u_t = u_{xx} + f(x,u,u_x)\,, $$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria $v_t=0$ are assumed to be hyperbolic. Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball St…
We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in ${\mathbb R}^d$. This condition can be used to stabilize weakly unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed linear and nonlinear maps in one- and two-dimensional spaces.
We study the phase space structure and the orbital diffusion from the vicinity of the vertical Lyapunov periodic orbits around the unstable Lagrangian points L1,2 in a 3D barred galaxy model. By perturbing the initial conditions of these periodic orbits, we detected the following five types of orbital structures in the 4D spaces of section: (i) Ring-like structures, sticky for large time intervals to the unstable invariant manifolds of the simple and double unstable vertical Lyapunov periodic orbits. (ii) 2D tori belonging to quasi-periodic orbits around stable periodic orbits existing in the …
We consider the two-body problem on surfaces of constant non-zero curvature and classify the relative equilibria and their stability. On the hyperbolic plane, for each q>0 we show there are two relative equilibria where the masses are separated by a distance q. One of these is geometrically of elliptic type and the other of hyperbolic type. The hyperbolic ones are always unstable, while the elliptic ones are stable when sufficiently close, but unstable when far apart. On the sphere of positive curvature, if the masses are different, there is a unique relative equilibrium (RE) for every angular…
This paper demonstrates input-to-state stability (ISS) of the SIR model of infectious diseases with respect to the disease-free equilibrium and the endemic equilibrium. Lyapunov functions are constructed to verify that both equilibria are individually robust with respect to perturbation of newborn/immigration rate which determines the eventual state of populations in epidemics. The construction and analysis are geometric and global in the space of the populations. In addition to the establishment of ISS, this paper shows how explicitly the constructed level sets reflect the flow of trajectorie…
We derive a saturated feedback control, which locally stabilizes a linear reaction-diffusion equation. In contrast to most other works on this topic, we do not assume the Lyapunov stability of the uncontrolled system and consider general unstable systems. Using Lyapunov methods, we provide estimates for the region of attraction for the closed-loop system, given in terms of linear and bilinear matrix inequalities. We show that our results can be used with distributed as well as scalar boundary control, and with different types of saturations. The efficiency of the proposed method is demonstrate…
In this paper, we apply a Lyapunov functional approach to Lotka-Volterra systems with infinite delays and feedback controls and establish that the feedback controls have no influence on the attractivity properties of a saturated equilibrium. This improves previous results by the authors and others, where, while feedback controls were used mostly to change the position of a unique saturated equilibrium, additional conditions involving the controls had to be assumed in order to preserve its global attractivity. The situation of partial extinction is further analysed, for which the original syste…
We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schr{ö}dinger and Manakov equations. We provid…
