M. Farber, T. Kappeler, J. Latschev, E. Zehnder
We find conditions which guarantee that a given flow on a closed smooth manifold admits a smooth Lyapunov one-form lying in a prescribed de Rham cohomology class. These conditions are formulated in terms of Schwartzman's asymptotic cycles of the flow.
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One knows that the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is open and dense in the analytic topology. In this paper, we construct cocycles with positive Lyapunov exponent which can be approximated by ones with zero Lyapunov exponent in the space of ${\cal C}^ l$ ($1 \le l\le \infty$) smooth quasi-periodic cocycles. As a consequence, the set of quasi-periodic Schrödinger cocycles with positive Lyapunov exponent is not ${\cal C}^ l$ open.
We prove the Hölder continuity of the Lyapunov exponent for quasi-periodic Schrödinger cocycles with a $C^2$ cos-type potential and any fixed Liouvillean frequency, provided the coupling constant is sufficiently large. Moreover, the Hölder exponent is independent of the frequency and the coupling constant.
This paper deals with asymptotic stability of a class of dynamical systems in terms of smooth Lyapunov pairs. We point out that well known converse Lyapunov results for differential inclusions cannot be applied to this class of dynamical systems. Following an abstract approach we put an assumption on the trajectories of the dynamical systems which demands for any trajectory the existence of a neighboring trajectory such that their difference grows linearly in time and distance of the starting points. Under this assumption, we prove the existence of a $C^\infty$-smooth Lyapunov pair. We also sh…
We give necessary and sufficient conditions for the existence of smooth Lyapunov 1-forms for the flow of a smooth vector field in terms of the behavior of certain locally finite invariant measures. The main statement generalizes a result of Schwartzman, whereas the methods are adapted from work of Sullivan.
A rigorous proof of a theorem on the coexistence of smooth Lyapunov function and smooth planar dynamical system with one arbitrary limit cycle is given, combining with a novel decomposition of the dynamical system from the perspective of mechanics. We base on this dynamic structure incorporating several efforts of this dynamic structure on fixed points, limit cycles and chaos, as well as on relevant known results, such as Schoenflies theorem, Riemann mapping theorem, boundary correspondence theorem and differential geometry theory, to prove this coexistence. We divide our procedure into three …
Discontinuity of Lyapunov exponents vs Entropy for smooth surface diffeomorphisms
研究光滑曲面微分同胚中Lyapunov指数与熵的关系，通过简化论证展示最大熵测度附近指数的连续性性质。
Motivated by models that arise in controlled ship maneuvering, we analyze Hopf bifurcations in systems with piecewise smooth nonlinear part. In particular, we derive explicit formulas for the generalization of the first Lyapunov coefficient to this setting. This generically determines the direction of branching (super- versus sub-criticality), but in general this differs from any fixed smoothing of the vector field. We focus on nonsmooth nonlinearities of the form $u_i|u_j|$, but our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part.…
In this paper we first construct smooth Morse-Lyapunov functions of attractors for nonsmooth dynamical systems. Then we prove that all open attractor neighborhoods of an attractor have the same homotopy type. Based on this basic fact we finally introduce the concept of critical group for Morse sets of an attractor and establish Morse inequalities and equations.
Entropy Continuity of Lyapunov Exponents for Non-flat 1-dimensional Maps
该研究证明了非平坦一维映射中李雅普诺夫指数的熵连续性，并展示了其在熵上的统一可积性。
In the context of mechanical Lagrangian dynamics, we prove a new Lyapunov instability criterion for a non strict local minimum equilibrium point of a smooth potential where the sufficient condition for instability is the existence of a smooth solution of a certain linear PDE derived from the mechanical Lagrangian governing the dynamics. In the presence of a magnetostatic field, we also give an additional sufficient condition for the motion of a charged particle to be Lyapunov unstable.
We study the regularity of the Lyapunov exponent for quasi-periodic cocycles $(T_ω, A)$ where $T_ω$ is an irrational rotation $x\to x+ 2πω$ on $\SS^1$ and $A\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$, $0\le l\le \infty$. For any fixed $l=0, 1, 2, \cdots, \infty$ and any fixed $ω$ of bounded-type, we construct $D_{l}\in {\cal C}^l(\SS^1, SL(2,\mathbb{R}))$ such that the Lyapunov exponent is not continuous at $D_{l}$ in ${\cal C}^l$-topology. We also construct such examples in a smaller Schrödinger class.
We prove a finite smooth version of the entropic continuity of Lyapunov exponents of Buzzi-Crovisier-Sarig for $C^\infty$ surface diffeomorphisms [9]. As a consequence we show that any $C^r$, $r > 1$, smooth surface diffeomorphism $f$ with $h_{top}(f) > \frac{1}{r} \limsup_n \frac{1}{n} \log^+ \|df^n\|$ admits a measure of maximal entropy. We also prove the $C^r$ continuity of the topological entropy at $f$.
Control Lyapunov function is a central tool in stabilization. It generalizes an abstract energy function -- a Lyapunov function -- to the case of controlled systems. It is a known fact that most control Lyapunov functions are non-smooth -- so is the case in non-holonomic systems, like wheeled robots and cars. Frameworks for stabilization using non-smooth control Lyapunov functions exist, like Dini aiming and steepest descent. This work generalizes the related results to the stochastic case. As the groundwork, sampled control scheme is chosen in which control actions are computed at discrete mo…
Efficient computation of stationary measures and the Lyapunov Landscape for families random dynamical systems with smooth additive noise
本文提出一种高效方法，用于计算具有光滑加性噪声的随机动力系统的平稳测度及李雅普诺夫景观，研究噪声诱导秩序的转变现象。
Lyapunov 1-forms on orbifolds
本文研究了轨道上由向量场生成的Lyapunov 1形式的存在性，并通过拓扑条件保证其在特定上同调类中的存在。
We explore some properties of Lyapunov exponents of measures preserved by smooth maps of the interval, and study the behaviour of the Lyapunov exponents under topological conjugacy.
In the present paper, the reducibility is derived for linear wave equation of finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition. Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent.
Solutions of a smooth first order dynamic equation can be made Lyapunov stable at will by the choice of an appropriate time-dependent Riemannian metric.
We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into one-dimensional bundles.
We show that the time-1 map of an Anosov flow, whose strong-unstable foliation is $C^2$ smooth and minimal, is $C^2$ close to a diffeomorphism having positive central Lyapunov exponent Lebesgue almost everywhere and a unique physical measure with full basin, which is $C^r$ stably ergodic. Our method is perturbative and does not rely on preservation of a smooth measure.
We describe, in the general setting of closed cone fields, the set of causal functions which can be approximated by smooth Lyapunov. We derive several consequences on causality theory. Dans le contexte général des champs de cones fermés, on décrit l'ensemble des fonctions causales qui peuvent être approchées par des fonctions de Lyapunov lisses. On en déduit quelques conséquence en théorie de la causalité.
Normal distribution of Lyapunov exponents of periodic orbits for expanding circle maps
研究显示，扩张圆映射中周期点的Lyapunov指数分布近似正态，误差随周期增长而减小，验证了中心极限定理在有限周期轨道中的应用。
We consider a smooth expanding map g on the circle of degree 2. It is known that the Lyapunov exponent of g with respect to the unique invariant measure that is absolutely continuous with respect to the Lebesgue measure is positive and less than or equal to log 2 which, in addition, is less than or equal to the Lyapunov exponent of g with respect to the measure of maximal entropy. Moreover, the equalities only occur simultaneously. We show that these are the only restrictions on the Lyapunov exponents considered above for smooth expanding maps of degree 2.
Characterizing symmetric spaces by their Lyapunov spectra
本文通过构造水平测度及其 Lyapunov 谱和水平维数，证明闭负曲率局部对称空间可由测地流周期轨道的 Lyapunov 谱在等距意义下唯一刻画，并推广到光滑 Anosov 流的轨道等价刚性。
On Lyapunov exponents properties of special Anosov endomorphisms on $\mathbb{T}^d$
本文比较 $\mathbb{T}^d$（$d\geq 3$）上特殊 Anosov 自同态及其线性化的 Lyapunov 指数，刻画弱不稳定叶状结构的绝对连续性，并证明高维存在保守却不 Lipschitz 共轭于线性化的光滑例子。
We consider volume-preserving perturbations of the time-one map of the geodesic flow of a compact surface with negative curvature. We show that if the Liouville measure has Lebesgue disintegration along the center foliation then the perturbation is itself the time-one map of a smooth volume-preserving flow, and that otherwise the disintegration is necessarily atomic.
Dense Phenomena for Ergodic Schrödinger Operators: I. Spectrum, Integrated Density of States, and Lyapunov Exponent
本文证明对紧致度量空间上同胚的遍历轨道采样得到的薛定谔算子，存在稠密采样函数集使谱间隙有限、态密度积分光滑且李雅普诺夫指数光滑为正，并肯定回答Walters的非一致上循环存在性问题。
Dynamics, Cohomology and Topology
本文研究了带 Lyapunov 约束的光滑 Morse-Smale 向量场，揭示了上同调的加法结构可检测静止点，乘法结构可检测瞬子，并基于带角结构流形解释了这一拓扑现象。
We study one-parameter families of quasi-periodically forced monotone interval maps and provide sufficient conditions for the existence of a parameter at which the respective system possesses a non-uniformly hyperbolic attractor. This is equivalent to the existence of a sink-source orbit, that is, an orbit with positive Lyapunov exponent both forwards and backwards in time. The attractor itself is a non-continuous invariant graph with negative Lyapunov exponent, often referred to as "SNA". In contrast to former results in this direction, our conditions are C^2-open in the fibre maps. By applyi…
For a non-conformal repeller $Λ$ of a $C^{1+α}$ map $f$ preserving an ergodic measure $μ$ of positive entropy, this paper shows that the Lyapunov dimension of $μ$ can be approximated gradually by the Carathéodory singular dimension of a sequence of horseshoes. For a $C^{1+α}$ diffeomorphism $f$ preserving a hyperbolic ergodic measure $μ$ of positive entropy, if $(f, μ)$ has only two Lyapunov exponents $λ_u(μ)>0>λ_s(μ)$, then the Hausdorff or lower box or upper box dimension of $μ$ can be approximated by the corresponding dimension of the horseshoes $\{Λ_n\}$. The same statement holds true if $…
Differential topology of the spaces of asymptotically stable vector fields and Lyapunov functions
该研究探究了n维欧氏空间上光滑渐近稳定向量场空间与对应Lyapunov函数空间的拓扑性质，证明了不同维度下的连通性与弱同伦型结论，拓展了微分拓扑在动力系统中的应用并给出多个相关推论。
In this paper we find conditions which guarantee that a given flow $Φ$ on a compact metric space $X$ admits a Lyapunov one-form $ω$ lying in a prescribed Čech cohomology class $ξ\in \check H^1(X;\R)$. These conditions are formulated in terms of the restriction of $ξ$ to the chain recurrent set of $Φ$. The result of the paper may be viewed as a generalization of a well-known theorem of C. Conley about the existence of Lyapunov functions.
Extremal Lyapunov exponents in random and simultaneous dynamics
针对光滑紧测度空间上含至少一个度严格大于1映射的有限映射集合生成的随机动力学，基于Kingman遍历定理（Birkhoff遍历定理的推广）推导得到刻画典型轨道不稳定性的极值李雅普诺夫指数表达式。
We consider an ergodic invariant measure $μ$ for a smooth action of $Z^k$, $k \ge 2$, on a $(k+1)$-dimensional manifold or for a locally free smooth action of $R^k$, $k \ge 2$ on a $(2k+1)$-dimensional manifold. We prove that if $μ$ is hyperbolic with the Lyapunov hyperplanes in general position and if one element of the action has positive entropy, then $μ$ is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.
Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise
本文针对由非线性乘性分数噪声驱动的抛物型SPDE，通过插值理论构造合适函数空间并利用离散化Lyapunov-Perron型算子的唯一不动点，证明了其局部稳定流形的存在性与光滑性，从而推进了Lu-Schmalfuß猜想的研究。
We consider actions of Z^k, k \ge 2, by Anosov diffeomorphisms which are uniformly quasiconformal on each coarse Lyapunov distribution. These actions generalize Cartan actions for which coarse Lyapunov distributions are one-dimensional. We show that, under certain non-resonance assumptions on the Lyapunov exponents, a finite cover of such an action is smoothly conjugate to an action by toral automorphisms.
Normal form method of center-focus problem in piecewise-smooth systems and algorithm design
研究分段光滑单值系统中区分中心与焦点、确定焦点阶数及退化霍普夫分岔的范式方法，证明李雅普诺夫常数与范式系数系列代数等价，据此设计算法计算范式系数系列并构建稳定切换信号。
Stochastic Saddle Avoidance Beyond Unit Excitation and Smoothness: A Pathwise Lyapunov-Perron Framework
本文提出路径式李雅普诺夫-佩龙框架，在不满足单位激励假设的常见随机优化场景下，证明随机递推的严格鞍点规避结果，确立相关算法收敛到原目标函数的局部极小值点。
Curvature Rigidity Through Level Sets of Lyapunov Exponents in Geodesic Flows
本文通过不变测度研究Lyapunov指数水平集，证明零Lyapunov指数水平集全测度蕴含流形平坦、水平集全测度蕴含截面曲率常数，并给出无共轭点流形上阻碍光滑共轭的新判据，统一证明了已有刚性结果。
We study the effects of IID random perturbations of amplitude $ε> 0$ on the asymptotic dynamics of one-parameter families $\{f_a : S^1 \to S^1, a \in [0,1]\}$ of smooth multimodal maps which "predominantly expanding", i.e., $|f'_a| \gg 1$ away from small neighborhoods of the critical set $\{ f'_a = 0 \}$. We obtain, for any $ε> 0$, a \emph{checkable, finite-time} criterion on the parameter $a$ for random perturbations of the map $f_a$ to exhibit (i) a unique stationary measure, and (ii) a positive Lyapunov exponent comparable to $\int_{S^1} \log |f_a'| \, dx$. This stands in contrast with the …
We extend the concept of Lyapunov 1-forms for the case of diffu- sion processes to study its asymptotic behavior. We give some examples and a condition for the existence of these objects.
We derive a series summation formula for the average logarithm norm of the action of a matrix on the projective space. This formula is shown to be useful to evaluate some Lyapunov exponents of random $\SL$-matrix cocycles, which include a special class for which H. Furstenberg had provided an explicit integral formula.
We prove global smooth classification results for TNS totally Anosov Z^k actions on general compact manifolds, under each one of the following conditions: joint integrability, resonance-free or Lyapunov pinching condition. Unlike the previous results, we do not require any uniform quasiconformality or pinching condition of action elements on coarse Lyapunov distributions, nor do we have any restriction on the dimension of coarse Lyapunov distributions. The main novelty is in proving a new standard form of the derivative cocycle for any TNS totally Anosov Z^k action on general manifold. A main …
An integral formula is given representing the generalized principal Lyapunov estimate for random linear parabolic PDEs. As an application, an upper estimate of the exponent is obtained.
Positive Lyapunov Exponent in the Hopf Normal Form with Additive Noise
本研究证明了加性白噪声扰动下Hopf分岔规范形在强剪切强度下Lyapunov指数为正，借助Furstenberg-Khasminskii公式下的Lyapunov指数连续性结论完善了相关简化情形的研究结果。
We investigate the Lyapunov Exponents of a variation of Hodge structure which has $G_2$ as geometric monodromy group, and discuss formulas for the sum of positive Lyapunov Exponents of variations of Hodge structures of any weight.
Oseledets regularity functions quantify the deviation of the growth associated with a dynamical system along its Lyapunov bundles from the corresponding uniform exponential growth. Precise degree of regularity of these functions is unknown. We show that for every invariant Lyapunov bundle of a volume preserving Anosov flow on a closed smooth Riemannian manifold, the corresponding Oseledets regularity functions are in $L^p(m)$, for some $p > 0$, where $m$ is the probability measure defined by the volume form. We prove an analogous result for essentially bounded cocycles over volume preserving A…
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