Patrick Bernard
We generalize to higher dimension results of Birkhoff and Mather on the existence of orbits wandering in regions of instability of twist maps. This generalization is strongly inspired by the one already proposed by Mather. However, its advantage is that it really contains most of the results of Birkhoff and Mather on twist maps.
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We consider a natural Lagrangian system defined on a complete Riemannian manifold being subjected to the action of a non-stationary force field with potential $U(q,t) = f(t)V(q)$. It is assumed that the factor $f(t)$ tends to $\infty$ as $t\to \pm\infty$ and vanishes at a unique point $t_{0}\in \mathbb{R}$. Let $X_{+}$, $X_{-}$ denote the sets of isolated critical points of $V(x)$ at which $U(x,t)$ as a function of $x$ attains its maximum for any fixed $t> t_{0}$ and $t<t_{0}$, respectively. Under nondegeneracy conditions on points of $X_{\pm}$ we apply Newton-Kantorovich type method to study …
We consider the 3-body problem of celestial mechanics in Euclidean, elliptic, and hyperbolic spaces, and study how the Lagrangian (equilateral) relative equilibria bifurcate when the Gaussian curvature varies. We thus prove the existence of new classes of orbits. In particular, we find some families of isosceles triangles, which occur in elliptic space.
We study the problem of existence of orbits connecting stationary points for the nonlinear heat and strongly damped wave equations being at resonance at infinity. The main difficulty lies in the fact that the problems may have no solutions for general nonlinearity. To address this question we introduce geometrical assumptions for the nonlinear term and use them to prove index formulas expressing the Conley index of associated semiflows. We also prove that the geometrical assumptions are generalizations of the well known Landesman- Lazer and strong resonance conditions. Obtained index formulas …
Motivated by the problem of global stability of thermodynamical equilibria in non-equilibrium thermodynamics formulated in a recent paper [12], we introduce some mechanisms for constructing semi-infinite orbits of contact Hamiltonian systems connecting two Legendrian graphs from the viewpoint of Aubry-Mather theory and weak KAM theory.
We propose a method for computation of stable and unstable sets associated to hyperbolic equilibria of nonautonomous ODEs and for computation of specific type of connecting orbits in nonautonomous singular ODEs. We apply the method to a certain a singular nonautonomous real Ginzburg-Landau type equation, which that arises from the problem of formation of spots in the Swift-Hohenberg equation.
Inspired by the classical Poincaré criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the variational properties of periodic solutions of a non-autonomous Lagrangian on a finite dimensional Riemannian manifold. We establish a general criterion for a priori detecting the linear instability of a periodic orbit on a Riemannian manifold for a (maybe not Legendre convex) non-autonomous Lagrangian simply by looking at the parity of the spectral index, which is the right substitute of the Mors…
An efficient algorithm for locating and continuing connecting orbits
本文提出一种改进的连接轨道定位与延拓算法，包含新的不变子空间延拓方法，后者具有独立应用价值。
In this paper we present a computer-assisted procedure for proving the existence of transverse heteroclinic orbits connecting hyperbolic equilibria of polynomial vector fields. The idea is to compute high-order Taylor approximations of local charts on the (un)stable manifolds by using the Parameterization Method and to use Chebyshev series to parameterize the orbit in between, which solves a boundary value problem. The existence of a heteroclinic orbit can then be established by setting up an appropriate fixed-point problem amenable to computer-assisted analysis. The fixed point problem simult…
We consider an open connected set $Ω$ and a smooth potential $U$ which is positive in $Ω$ and vanishes on $\partialΩ$. We study the existence of orbits of the mechanical system \[ \ddot{u}=U_x(u), \] that connect different components of $\partialΩ$ and lie on the zero level of the energy. We allow that $\partialΩ$ contains a finite number of critical points of $U$. The case of symmetric potential is also considered.
Connecting Orbits in Cooperative McKean-Vlasov SDEs
本文提出了一种在合作型麦肯-沃拉斯科夫随机微分方程中建立随机顺序下单调连接轨道的方法，揭示了相关不变测度的不稳定性。
For flows, the singular cycles connecting saddle periodic orbit and saddle equilibrium can poten- tially result in the so-called singular horseshoe, which means the existence of a non-uniformly hyperbolic chaotic invariant set. However, it is very hard to find a specific dynamical system that exhibits such singular cycles in general. In this paper, the existence of the singular cycles involved in saddle periodic orbits is studied by two types of piecewise affine systems: one is the piecewise affine system having an admissible saddle point with only real eigenvalues and an admissible saddle per…
We investigate the existence of Arnold diffusion-type orbits for systems obtained by iterating in any order the time-one maps of a family of Tonelli Hamiltonians. Such systems are known as 'polysystems' or 'iterated function systems'. When specialized to families of twist maps on the cylinder, our results are similar to those obtained by Moeckel [20] and Le Calvez [15]. Our approach is based on weak KAM theory and is close to the one used by Bernard in [3] to study the case of a single Tonelli Hamiltonian.
Computing Invariant Manifolds and Connecting Orbits in the Circular Restricted Three Body Problem
利用边值公式与数值延拓方法，高效计算CR3BP中周期轨道的不稳定流形，并检测到异宿连接和同宿连接，为航天任务设计提供潜在轨道。
Continuation of connecting orbits in 3D-ODEs: (II) Cycle-to-cycle connections
本文提出了一种使用投影边界条件和伴随变分方程的特征函数，将点对周期轨道的数值延拓方法扩展到周期轨道到周期轨道连接，并在AUTO中实现，通过种群动力学示例验证了方法的有效性。
We consider the bifurcation diagram in a suitable parameter plane of a quadratic vector field in $\mathbb{R}^3$ that features a homoclinic flip bifurcation of the most complicated type. This codimension-two bifurcation is characterized by a change of orientability of associated two-dimensional manifolds and generates infinite families of secondary bifurcations. We show that curves of secondary $n$-homoclinic bifurcations accumulate on a curve of a heteroclinic bifurcation involving infinity. We present an adaptation of the technique known as Lin's method that enables us to compute such connect…
Continuation of connecting orbits in 3D-ODEs: (I) Point-to-cycle connections
提出使用投影边界条件进行三维自治常微分方程中点到周期轨道连接轨道的数值延拓新方法，通过伴随变分方程的特征函数避免单值矩阵计算，并在AUTO中实现。
We present a new existence mechanism, based on symplectic topology, for orbits of Hamiltonian flows connecting a pair of disjoint subsets in the phase space. The method involves function theory on symplectic manifolds combined with rigidity of Lagrangian submanifolds. Applications include superconductivity channels in nearly integrable systems and dynamics near a perturbed unstable equilibrium.
We make explicit the geometric content of Mel'nikov's method for detecting heteroclinic points between transversally hyperbolic periodic orbits. After developing the general theory of intersections for pairs of family of Lagrangian submanifolds constrained to live in an auxiliary family of submanifolds, we explain how the heteroclinic orbits are detected by the zeros of the Mel'nikov 1 -form. This 1 -form admits an integral expression, which is non-convergent in general. Finally, we discuss different solutions to this convergence problem.
In this paper, we prove Morse index theorem of Lagrangian system with any self-adjoint boundary conditions. Based on it, we give some nontrivial estimation on the difference of Morse indices. As an application, we get a new criterion for the stability problem of brake periodic orbit.
Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points
本文提出计算机辅助证明方法计算定态相对指标，以推进 Morse-Floer 同调的可计算性，并通过三个偏微分方程示例验证其有效性，进而证明连接轨道与行波的存在性定理。
We use a result of J. Mather on the existence of connecting orbits for compositions of monotone twist maps of the cylinder to prove the existence of connecting geodesics on the unit tangent bundle $ST^2$ of the 2-torus in regions without invariant tori.
We compare Lagrangian thimbles for the potential of a Landau-Ginzburg model to the Morse theory of its real part. We explore Landau-Ginzburg models defined using Lie theory, constructing their real Lagrangian thimbles explicitly and comparing them to the stable and unstable manifolds of the real gradient flow.
We consider how breakdown of the quasistatic approximation for attractors can lead to rate-induced tipping, where a qualitative change in tracking/tipping behaviour of trajectories can be characterised in terms of a critical rate. Associated with rate-induced tipping (where tracking of a branch of quasistatic attractors breaks down) we find a new phenomenon for attractors that are not simply equilibria: partial tipping of the pullback attractor where certain phases of the periodic attractor tip and others track the quasistatic attractor. For a specific model system with a parameter shift betwe…
In this paper we provide a sufficient condition for the linear instability of a periodic orbit for a free period Lagrangian system on a Riemannian manifold. The main result establish a general criterion for the linear instability of a maybe degenerate) periodic orbit admitting a orbit cylinder in terms to the parity of a suitable spectral index encoding the functional and symplectic property of the problem.
In this paper we study the existence of periodic orbits with prescribed energy levels of convex Lagrangian systems on complete Riemannian manifolds. We extend the existence results of Contreras by developing a modified minimax principal to a class of Lagrangian systems on noncompact Riemannian manifolds, namely the so called $\lsh$ Lagrangian systems. In particular, we prove that for almost every $k\in(0,c_u(L))$ the exact magnetic flow associated to a $\lsh$ Lagrangian has a contractible periodic orbit with energy $k$. We also discuss the existence and non-existence of closed geodesics on the…
Periodic orbits of reversible Lagrangian systems without self-intersections and Mañé genericity
本文证明了经典类型拉格朗日系统在紧致流形上，固定能量的素周期轨道既不自相交也不相交，基于马恩é典型的凸哈密顿量结果。
Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have quadratic growth in the velocities or in the momenta. Such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions. In this note we show how these results can be extended to the classical setting of Tonelli Lagrangians (Lagrangians which are $C^2$-convex and superlinear in the velocities), or to Hamiltonians which are superlinear in the momenta and have …
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*\T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in \cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*\T^2$. This solves a problem of Arnold in \cite{A}.
We are concerned with conservative systems $\ddot{q}=\nabla V(q), \; q\in\mathbb{R}^N$ for a general class of potentials $V\in C^1(\mathbb{R}^N)$. Assuming that a given sublevel set $\{V\leq c\}$ splits in the disjoint union of two closed subsets $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$, for some $c\in\mathbb{R}$, we establish the existence of bounded solutions $q_c$ to the above system with energy equal to $-c$ whose trajectories connect $\mathcal{V}^c_-$ and $\mathcal{V}^c_+$. The solutions are obtained through an energy constrained variational method, whenever mild coerciveness properties ar…
In this paper, we show that, for scalar reaction-diffusion equations $u_t=u_{xx}+f(x,u,u_x)$ on the circle $S^1$, the Morse-Smale property is generic with respect to the non-linearity $f$. In \cite{CR}, Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In \cite{JR}, we have shown that, generically with respect to the non-linearity $f$, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showi…
Optimal Low-Thrust Orbit Transfers Connecting Gateway with Earth and Moon
针对地球、月球与 Gateway 间的双向最短时间低推力转移，提出统一正反向传播与多弧段最优控制框架，并通过隐式协态变换和间接启发式方法高效求解。
We present the multicomponent functionalized free energies that characterize the low-energy packings of amphiphilic molecules within a membrane through a correspondence to connecting orbits within a reduced dynamical system. To each connecting orbits we associate a manifold of low energy membrane-type configurations parameterized by a large class of admissible interfaces. The normal coercivity of the manifolds is established through criteria depending solely on the structure of the associated connecting orbit. We present a class of examples that arise naturally from geometric singular perturba…
Morse index theorem for heteroclinic, homoclinic and halfclinic orbits of Lagrangian systems
本文提出了一种更通用的Lagrangian系统中异宿、同宿和半clinic轨道莫尔斯指数定理，并通过具体模型计算了相关指数。
These notes were prepared in occasion of a mini-course given by the author at the "CIMPA Research School - Hamiltonian and Lagrangian Dynamics" (10-19 March 2015 - Salto, Uruguay). The talks were meant as an introduction to the problem of finding periodic orbits of prescribed energy for autonomous Tonelli Lagrangian systems on the twisted cotangent bundle of a closed manifold. In the first part of the lecture notes, we put together in a general theorem old and new results on the subject. In the second part, we focus on an important class of examples: magnetic flows on surfaces. For such system…
It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter $β$ and on the eccentricity $e$ of the orbit. We consider only the circular case ($e = 0$) but under the action of a broader family of singular potentials: $α$-homogeneous potentials, for $α\in (0,2)$, and the logarithmic one. It turns out indeed that the Lagrangian circular orbit persists also in this more general setting. We discover a region of linear stability expressed in terms of the homogeneity parameter $α$ and the mass parameter $β$, th…
On the set of asymptotic homologies of orbits on invariant Lagrangian graphs
本文证明了不变拉格朗日图上回复轨道渐近同调集包含于真锥中，并在三维环面下证明了有理渐近同调蕴含周期轨道上的 Mather 测度，进而部分解答了 Carneiro-Ruggiero 关于超临界能级下不存在 Hedlund 拉格朗日环面的猜想。
We introduce and study a non uniform hyperbolicity condition for complex rational maps, that does not involve a growth condition. We call this condition Backward Contraction. We show this condition is weaker than the Collet-Eckmann condition, and than the summability condition with exponent~1. Our main result is a connecting lemma for Backward Contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurston's algorithm and some rig…
This paper introduces techniques of symplectic topology to the study of homoclinic orbits in Hamiltonian systems. The main result is a strong generalization of homoclinic existence results due to Sere and to Coti-Zelati, Ekeland and Sere, which were obtained by variational methods. Our existence result uses a modification of a construction due to Mohnke (originally in the context of Legendrian chords), and an energy--capacity inequality of Chekanov. In essence, we show the existence of a homoclinic orbit by showing a certain Lagrangian embedding cannot exist. We consider a (possibly time depen…
Non-minimality and instability of brake orbits for natural Lagrangians on Riemannian manifolds
研究了自然拉格朗日系统在黎曼流形上制动轨道的非最小性和不稳定性，通过莫塞指数分析揭示了其稳定性条件及退化特性。
Assume that the Aubry set of the time-periodic positive definite Lagrangian $L$ consists of one hyperbolic 1-periodic orbit. We provide an upper bound estimate of the rate of convergence of the family of new Lax-Oleinik type operators associated with $L$ introduced by the authors in \cite{W-Y}. In addition, we construct an example where the Aubry set of a time-independent positive definite Lagrangian system consists of one hyperbolic periodic orbit and the rate of convergence of the Lax-Oleinik semigroup cannot be better than $O(\frac{1}{t})$.
We investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We utilize wrapped Floer cohomology to define the spectral invariant of an admissible Lagrangian submanifold within a Weinstein domain. We show that the pseudo-metric derived from this spectral invariant is a valid $Ham$-invariant metric. Furthermore, we establish that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wra…
We prove that, on a closed surface, a Lagrangian system defined by a Tonelli Lagrangian $L$ possesses a periodic orbit that is a local minimizer of the free-period action functional on every energy level belonging to the low range of energies $(e_0(L),c_{\mathrm{u}}(L))$. We also prove that almost every energy level in $(e_0(L),c_{\mathrm{u}}(L))$ possesses infinitely many periodic orbits. These statements extend two results, respectively due to Taimanov and Abbondandolo-Macarini-Mazzucchelli-Paternain, valid for the special case of electromagnetic Lagrangians.
Ergodic closing lemmas and invariant Lagrangians
受Mañé引理启发，研究高维哈密顿系统中$C^\infty$闭包引理，证明在Floer条件下不变拉格朗日子流形可产生统计可控的周期轨道，并给出零截面情形的收敛结果。
A simple chaos indicator based on the Lagrangian descriptor difference of neighboring orbits
本文提出了一种基于邻近轨道拉格朗日描述符差值的简单混沌指示器，通过对比实验验证其在检测混沌行为方面的有效性。
Complementary to existing applications of Lagrangian descriptors as an exploratory method, we use Lagrangian descriptors to find invariant manifolds in a system where some invariant structures have already been identified. In this case we use the parametrisation of a periodic orbit to construct a Lagrangian descriptor that will be locally minimised on its invariant manifolds. The procedure is applicable (but not limited) to systems with highly unstable periodic orbits, such as the isokinetic Chesnavich CH4+ model subject to a Hamiltonian isokinetic theromostat. Aside from its low computational…
Connecting orbits for delay differential equations with unimodal feedback
本文研究具单峰反馈的延迟微分方程，刻画不动点与周期轨道不稳定集的异宿连接结构，以Mackey--Glass型方程为例证明了不动点及邻近周期轨道的不稳定集的连接性质。
Coherent boundaries of Lagrangian vortices in fluid flows have recently been identified as closed orbits of line fields associated with the Cauchy-Green strain tensor. Here we develop a fully automated procedure for the detection of such closed orbits in large-scale velocity data sets. We illustrate the power of our method on ocean surface velocities derived from satellite altimetry.
Morse Index, Stability and Spectral Bounds for Periodic Orbits in Reversible Lagrangian Systems
本文针对可逆拉格朗日系统，利用莫尔斯指标与狄利克雷指标给出周期轨道弗洛凯乘子的定量谱界，证明指标一刹车轨道的横向双曲性，并展示精细界的尖锐性。
In this paper, we explore the dynamics of a Hamiltonian system after a double van der Waals potential energy surface degenerates into a single well. The energy of the system is increased from the bottom of the potential well up to the dissociation energy, which occurs when the system becomes open. In particular, we study the bifurcations of the basic families of periodic orbits of this system as the energy increases using Lagrangian descriptors and Poincaré maps. We investigate the capability of Lagrangian descriptors to find periodic orbits of bifurcating families for the case of resonant, sa…
Symmetries are ubiquitous in a wide range of nonlinear systems. Particularly in systems whose dynamics are determined by a Lagrangian or Hamiltonian function. For hybrid systems which possess a continuous-time dynamics determined by a Lagrangian function, with a cyclic variable, the degrees of freedom for the corresponding hybrid Lagrangian system can be reduced by means of a method known as \textit{hybrid Routhian reduction}. In this paper we study sufficient conditions for the existence of periodic orbits in hybrid Routhian systems which also exhibit time-reversal symmetry. Likewise, we expl…
We prove several new results concerning action minimizing periodic orbits of Tonelli Lagrangian systems on an oriented closed surface $M$. More specifically, we show that for every energy larger than the maximal energy of a constant orbit and smaller than or equal to the Mañé critical value of the universal abelian cover, the Lagrangian system admits a minimal boundary, i.e. a global minimizer of the Lagrangian action on the space of smooth boundaries of open sets of $M$. We also extend the celebrated graph theorem of Mather in this context: in the tangent bundle $TM$, the union of the support…
Minimization and Hyperbolicity
本文研究了时间依赖的lagrangian系统中严格局部极小化轨道与对应lagrangian流双曲性性质之间的关系。
In 1956, Bott in his celebrated paper on closed geodesics and Sturm intersection theory, proved an Index Iteration Formula for closed geodesics on Riemannian manifolds. Some years later, Ekeland improved this formula in the case of convex Hamiltonians and, in 1999, Long generalized the Bott iteration formula by putting in its natural symplectic context and constructing a very effective Index Theory. The literature about this formula is quite broad and the dynamical implications in the Hamiltonian world (e.g. existence, multiplicity, linear stability etc.) are enormous. Motivated by the recent …
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