In this article, we consider the dynamics in a neighborhood of a quasi-periodic torus which is invariant by a Hamiltonian flow, we discuss several notions of stability and we prove several results of instability when the frequency of the invariant torus is resonant.

2015-01-05math.DS

Optimal stability and instability for near-linear Hamiltonians

In this paper, we will prove a very general result of stability for perturbations of linear integrable Hamiltonian systems, and we will construct an example of instability showing that both our result and our example are optimal. Moreover, in the same spirit as the notion of KAM stable integrable Hamiltonians, we will introduce a notion of effectively stable integrable Hamiltonians, conjecture a characterization of these Hamiltonians and show that our result prove this conjecture in the linear case.

2011-05-18math.DS

Effect of the time delay on the stability and instability of the logistic map

A proper discretization of the logistic differential equation, which is preserving these two distinct equilibrium solutions and their unstability and stability, suggest that we need to examine the time delay of the logistic map. According to Murray, the effect of delay in models is "usually" to increase the potential for instability. However the word "usually" is really ambiguous. In this paper, we mathematically formulate and prove the two conjectures about stability and instability.

2009-08-27math.DS

A note on micro-instability for Hamiltonian systems close to integrable

In this note, we consider the dynamics associated to an epsilon-perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of "micro-diffusion": under generic assumptions on h and f , there exists an orbit of the system for which the drift of its action variables is at least of order square root of epsilon, after a time of order the inverse of square root of epsilon. The assumptions, which are essentially minimal, are that there exists a resonant point for h and that the corresponding averaged perturbation …

2015-01-09math.DS

An example of instability in high-dimensional Hamiltonian systems

In this article, we use a mechanism introduced by Herman, Marco and Sauzin to show that if a perturbation of a quasi-convex integrable Hamiltonian system is not too small with respect to the number of degrees of freedom, then the classical exponential stability estimates do not hold. Indeed, we construct an unstable solution whose drifting time is polynomial with respect to the inverse of the size of the perturbation. A different example was already given by Bourgain and Kaloshin, with a linear time of drift but with a perturbation which is larger than ours. As a consequence, we obtain a bette…

2010-11-08math.DS

Completely integrable Hamiltonian systems with weak Lyapunov instability or isochrony

The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no non-constant solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium…

2014-02-01math.DS

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Weak instability of Hamiltonian equilibria

This is an expository paper on Lyapunov stability of equilibria of autonomous Hamiltonian systems. Our aim is to clarify the concept of weak instability, namely instability without non-constant motions which have the equilibrium as limit point as time goes to minus infinity. This is done by means of some examples. In particular, we show that a weakly unstable equilibrium point can be stable for the linearized vector field.

2012-10-04math.CAmath.DS

具有两个自由度的KAM稳定哈密顿系统中的李雅普诺夫不稳定性

Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom

本文证明存在任意接近可积KAM非退化解析哈密顿函数的Gevrey光滑哈密顿函数，其具有频率向量满足$\omega_1\omega_2<0$的李雅普诺夫不稳定但KAM稳定的椭圆平衡点，并给出近可积情形下的类似例子。

2020-05-18math.DS

Hyperbolicity and Stability for Hamiltonian flows

We prove that a Hamiltonian star system, defined on a 2d-dimensional symplectic manifold M, is Anosov. As a consequence we obtain the proof of the stability conjecture for Hamiltonians. This generalizes the 4-dimensional results in [6].

2012-04-05math.DS

Improved exponential stability for near-integrable quasi-convex Hamiltonians

In this article, we improve previous results on exponential stability for analytic and Gevrey perturbations of quasi-convex integrable Hamiltonian systems. In particular, this provides a sharper upper bound on the speed of Arnold diffusion which we believe to be optimal.

2010-11-08math.DS

Improved exponential stability for near-integrable quasi-convex Hamiltonians

2010-04-09math.DS

Effective stability for slow time-dependent near-integrable Hamiltonians and application

The aim of this note is to prove a result of effective stability for a non-autonomous perturbation of an integrable Hamiltonian system, provided that the perturbation depends slowly on time. Then we use this result to clarify and extend a stability result of Giorgilli and Zehnder for a mechanical system with an arbitrary time-dependent potential.

2013-03-20math.DS

Exact stability and instability regions for two-dimensional linear autonomous multi-order systems of fractional-order differential equations

Necessary and sufficient conditions are explored for the asymptotic stability and instability of linear two-dimensional autonomous systems of fractional-order differential equations with Caputo derivatives. Fractional-order-dependent and fractional-order-independent stability and instability properties are fully characterized, in terms of the main diagonal elements of the systems' matrix, as well as its determinant.

2021-02-27math.CAmath.DS

带时滞和变系数的某些非线性系统的解范数的双边界及有界性/稳定性和不稳定性

Bilateral Bounds for Norms of Solutions and Boundedness/Stability and Instability of Some Nonlinear Systems with Delays and Variable Coefficients

本文提出一种通过构建两个标量对应系统来评估带时滞和变系数的向量非线性系统解范数双边界的新方法，并据此导出了新的有界性、稳定性和不稳定性判据。

2024-08-22math.DS

The stability of equilibrium solutions of periodic Hamiltonian systems in the case of degeneracy

In this paper we are concerned with the stability of equilibrium solutions of periodic Hamiltonian systems with one degree of freedom in the case of degeneracy, which means that the characteristic exponents of the linearized system have zero real part, and the high order terms must be considered to solve the stability problem. For almost all degenerate cases, sufficient conditions for the stability and instability are obtained.

2017-05-30math.DS

Optimal stability and instability results for a class of nearly integrable Hamiltonian systems

We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) $O(μ)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_d = O((1/ μ) \log (1/ μ))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d$ is optimal as a consequence of a general stability result proved via classical perturbation theory.

2002-03-19math.DSmath.FA

Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians

This paper is a sequel to "Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians", in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrable Gevrey Hamiltonians at a quasi-periodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we derive normal forms with a polynomially small remainder. As applications, we obtain a polynomially large upper bound on the stability time for the evolution of the action variables and a polynomial…

2012-12-06math.DS

Stability of a 2-dimensional Mathieu-type system with quasiperiodic coefficients

In the following we consider a 2-dimensional system of ODE's containing quasiperiodic terms. The system is proposed as an extension of Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a loss of stability. The 2-d system has two `natural' frequencies when the time dependent terms are switched off, and it is internally driven by quasiperiodic terms in the same frequencies. Stability charts in the parameter space are generated first using numerical simulations and Floquet theory. While some instability regions are easy to antici…

2012-03-12math.DS

Stability and instability results in a model of Fermi acceleration

We consider the static wall approximation to the dynamics of a particle bouncing on a periodically oscillating infinitely heavy plate while subject to a potential force. We assume the case of a potential given by a power of the particle's height and sinusoidal motions of the plate. We find that for powers smaller than 1 the set of escaping orbits has full Hausdorff dimension for all motions and obtain existence of elliptic island of period 2 for arbitrarily high energies for a full-measure set of motions. Moreover we obtain conditions on the potential to ensure that the total (Lebesgue) measur…

2008-03-07math.DS

Stability and instability of standing-wave solutions to one dimensional quadratic-cubic Klein-Gordon equations

We study the stability of standing-waves solutions to a scalar non-linear Klein-Gordon equation in dimension one with a quadratic-cubic non-linearity. Orbits are obtained by applying the semigroup generated by the negative complex unit multiplication on a critical point of the energy constrained to the charge.

2022-09-09math.APmath.DSmath.FA

Transients in quasi-controllable systems. Overshooting, stability and instability

Families of regimes for control systems are studied possessing the so called quasi-controllability property that is similar to the Kalman controllability property. A new approach is proposed to estimate the degree of transients overshooting in quasi-controllable systems. This approach is conceptually related with the principle of bounded regimes absence in the absolute stability problem. Its essence is in obtaining of constructive a priori bounds for degree of overshooting in terms of the so called quasi-controllability measure. It is shown that relations between stability, asymptotic stabilit…

2009-09-24math.DSmath.OC

Stability analysis and μ-synthesis control of brake systems

The concept of friction-induced brake vibrations, commonly known as judder, is investigated. Judder vibration is based on the class of geometrically induced or kinematic constraint instability. After presenting the modal coupling mechanism and the associated dynamic model, a stability analysis as well as a sensitivity analysis have been conducted in order to identify physical parameters for a brake design avoiding friction-induced judder instability. Next, in order to reduce the size of the instability regions in relation to possible system parameter combinations, robust stability via μ-synthe…

2008-01-23math.DSphysics.class-ph

Nekhoroshev type stability results for Hamiltonian systems with an additional transversal component

We prove exponential stability theorems of Nekhoroshev type for motion in the neighbourhood of an elliptic fixed point in Hamiltonian systems having an additional transverse component of arbitrary dimension.

2012-01-18math-phmath.DSmath.MP

False Asymptotic Instability Behavior at Iterated Functions with Lyapunov Stability in Nonlinear Time Series

Empirically defining some constant probabilistic orbits of f(x) and g(x) iterated high-order functions, the stability of these functions in possible entangled interaction dynamics of the environment through its orbit's connectivity (open sets) provides the formation of an exponential dynamic fixed point b = S(n+1) as a metric space (topological property) between both iterated functions for short time lengths. However, the presence of a dynamic fixed point f(g(x)) (x+1) can identify a convergence at iterations for larger time lengths of b (asymptotic stability in Lyapunov sense). Qualitative (Q…

2019-11-13cs.SYeess.SYmath.AT

Stability transitions for axisymmetric relative equilibria of Euclidean symmetric Hamiltonian systems

In the presence of noncompact symmetry, the stability of relative equilibria under momentum-preserving perturbations does not generally imply robust stability under momentum-changing perturbations. For axisymmetric relative equilibria of Hamiltonian systems with Euclidean symmetry, we investigate different mechanisms of stability: stability by energy-momentum confinement, KAM, and Nekhoroshev stability, and we explain the transitions between these. We apply our results to the Kirchhoff model for the motion of an axisymmetric underwater vehicle, and we numerically study dissipation induced inst…

2008-01-28math-phmath.DSmath.MP

对鲁棒渐近稳定的耗散哈密顿微分代数系统的稳定性半径的刻画

Characterization of stability radii for robustly asymptotically stable dissipative Hamiltonian differential-algebraic systems

研究耗散哈密顿微分代数系统的鲁棒渐近稳定性条件及在结构保持扰动下稳定性丧失的精确条件和界限

2026-05-12cs.NAmath.DSmath.NA

哈密顿泊松积分器的一些稳定性性质

Some stability properties of Hamiltonian Poisson integrators

本文阐述可积与不可积系统中泊松积分器修正哈密顿量的重要性，以洛特卡-沃尔泰拉动力学为例，展示5维不可积洛特卡-沃尔泰拉系统并通过数值研究其哈密顿泊松积分器的稳定性性质。

2025-11-18cs.NAmath-phmath.DG

Instability of $H^1$-stable peakons in the Camassa-Holm equation

It is well-known that peakons in the Camassa-Holm equation are $H^1$-orbitally stable thanks to the presence of conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise $C^1$ perturbations to peakons grow in time in spite of their stability in the $H^1$-norm. We also show that the linearized stability analysis near peakons contradicts the $H^1$-orbital stability result, hence passage from linear to nonlinear theory is false in $H^1$.

2019-03-22math-phmath.APmath.DS