Stable Invariant Manifolds Research Articles

Yosida distance and existence of invariant manifolds in the infinite-dimensional dynamical systems

We consider the existence of invariant manifolds to evolution equations u ′ ( t ) = A u ( t ) u’(t)=Au(t) , A : D ( A ) ⊂ X → X A:D(A)\subset \mathbb {X}\to \mathbb {X} near its equilibrium A ( 0 ) = 0 A(0)=0 under the assumption that its proto-derivative ∂ A ( x ) \partial A(x) exists and is continuous in x ∈ D ( A ) x\in D(A) in the sense of Yosida distance. Yosida distance between two (unbounded) linear operators U U and V V in a Banach space X \mathbb {X} is defined as d Y ( U , V ) ≔ lim sup μ → + ∞ ‖ U μ − V μ ‖ d_Y(U,V)≔\limsup _{\mu \to +\infty } \| U_\mu -V_\mu \| , where U μ U_\mu and V μ V_\mu are the Yosida approximations of U U and V V , respectively. We show that the above-mentioned equation has local stable and unstable invariant manifolds near an exponentially dichotomous equilibrium if the proto-derivative of ∂ A \partial A is continuous in the sense of Yosida distance. The Yosida distance approach allows us to generalize the well-known results with possible applications to larger classes of partial differential equations and functional differential equations. The obtained results seem to be new.

On Morse–Smale diffeomorphisms on simply connected manifolds

We study relations between a structure of non-wandering set of a Morse–Smale diffeomorphism f and its carrying closed manifold Mn. We prove that if f has no any saddle periodic points with one-dimensional unstable manifolds, and for any periodic point σ of Morse index (n−1) its unstable manifolds do not intersect stable invariant manifolds of saddle periodic points different from σ, then Mn is simply connected. This fact does not follow from Morse inequalities that give only restrictions on homology groups of Mn.

Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

A dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of intersection points of the respective one-dimensional stable manifold with a chosen planar section and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document}.

Invariant manifolds for stochastic delayed partial differential equations of parabolic type

The aim of this paper is to prove the existence and smoothness of stable and unstable invariant manifolds for a stochastic delayed partial differential equation of parabolic type. The stochastic delayed partial differential equation is firstly transformed into a random delayed partial differential equation by a conjugation, which is then recast into a Hilbert space. For the auxiliary equation, the variation of constants formula holds and we show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov–Perron method. Subsequently, we prove the smoothness of these invariant manifolds under appropriate spectral gap condition by carefully investigating the smoothness of auxiliary equation, after which, we obtain the invariant manifolds of the original equation by projection and inverse transformation. Eventually, we illustrate the obtained theoretical results by their application to a stochastic single-species population model.

On the reduction of the topological classification of gradient-like flows problem to the classification of polar flows I. A. Saraev

In this paper we consider a class G(Mn) of gradient-like flows on connected closed manifolds of dimension n≥4 such that for any flow ft∈G(Mn) stable and unstable invariant manifolds of saddle equilibria do not intersect invariant manifolds of other saddle equilibria. It is known that the ambient manifold of any flow from the class G(Mn) can be splitted into connected summ of the sphere Sn , gft≥0 copies of direct products Sn−1×S1 , and a simply connected manifold which is not homeomorphic to the sphere. The number gft is determined only by the number of nodal equilibria and the number of saddle equilibria such that one of their invariant manifolds has the dimension (n−1) (we call such equilibria trivial saddles). A simply connected manifold which is not homeomorphic to the sphere presents in the splitting if and only if the set of saddle equilibria contains points with unstable manifolds of dimension i∈{2,…,n−2} (we call such equilibria non-trivial saddles). Moreover, the complete topological classification was obtained for flows from the class G(Mn) without non-trivial saddles. In this paper we prove that for any flow ft∈G(Mn) the carrier manifold can be splitted into a connected sum along pairwise disjoint smoothly embedded spheres (separating spheres) that do not contain equilibrium states of the flow ft and transversally intersect its trajectories. The restriction of the flow ft to the complements to these spheres uniquely (up to topological equivalence and numbering) defines a finite set of flows ft1,…,ftl defined on the components of a connected sum. Moreover, for any j∈1,…,l , the set of saddle equilibria of the flow ftj consists either only of trivial saddles or only of of non-trivial ones and then the flow ftj is polar. We introduce the notion of consistent topological equivalence for flows ft1,…ftj and show that flows ft,f′t∈G(Mn) are topologically equivalent if and only if for each of these flows the set of separating spheres exists that defines consistently topologically equivalent flows on the components of the connected sum.

Model-Based Policy Iterations for Nonlinear Systems via Controlled Hamiltonian Dynamics

The infinite-horizon optimal control problem for nonlinear systems is studied. In the context of model-based, iterative learning strategies we propose an alternative definition and construction of the <i>temporal difference error</i> arising in Policy Iteration strategies. In such architectures the error is computed via the evolution of the Hamiltonian function (or, possibly, of its integral) along the trajectories of the closed-loop system. Herein the temporal difference error is instead obtained via two subsequent steps: first the dynamics of the underlying costate variable in the Hamiltonian system is steered by means of a (virtual) control input in such a way that the stable invariant manifold becomes externally attractive. Then, the <i>distance-from-invariance</i> of the manifold, induced by approximate solutions, yields a natural candidate measure for the <i>policy evaluation</i> step. The <i>policy improvement</i> phase is then performed by means of standard gradient descent methods that allows to correctly update the weights of the underlying functional approximator. The above architecture then yields an <i>iterative (episodic) learning</i> scheme based on a scalar, constant <i>reward</i> at each iteration, the value of which is insensitive to the length of the episode, as in the original spirit of Reinforcement Learning strategies for discrete-time systems. Finally, the theory is validated by means of a numerical simulation involving an automatic flight control problem.

Boxing-in of a blender in a Hénon-like family

IntroductionThe extension of the Smale horseshoe construction for diffeomorphisms in the plane to those in spaces of at least dimension three may result in a hyperbolic invariant set referred to as a blender. The defining property of a blender is that it has a stable or unstable invariant manifold that appears to have a dimension larger than expected. In this study, we consider a Hénon-like family in ℝ3, which is the only explicitly given example of a system known to feature a blender in a certain range of a parameter (corresponding to an expansion or contraction rate). More specifically, as part of its hyperbolic set, this family has a pair of saddle fixed points with one-dimensional stable or unstable manifolds. When there is a blender, the closure of these manifolds cannot be avoided by one-dimensional curves coming from an appropriate direction. This property has been checked for the Hénon-like family by the method of computing extremely long pieces of global one-dimensional manifolds to determine the parameter range over which a blender exists.MethodsIn this study, we take the complimentary and local point of view of constructing an actual three-dimensional box (a parallelopiped) that acts as an outer cover of the hyperbolic set. The successive forward or backward images of this box form a nested sequence of sub-boxes that contains the hyperbolic set, as well as its respective local invariant manifold.ResultsThis constitutes a three-dimensional horseshoe that, in contrast to the idealized affine construction, is quite general and features sub-boxes with curved edges. The initial box is defined in a parameter-dependent way, and this allows us to characterize properties of the hyperbolic set intuitively.DiscussionIn particular, we trace relevant edges of sub-boxes as a function of the parameter to provide additional geometric insight into when the hyperbolic set may or may not be a blender.

Optimal transfers from Moon to [formula omitted] halo orbit of the Earth-Moon system

In this paper, optimal solutions are investigated for a transfer from a parking orbit around the Moon to a halo orbit around L2 of the Earth-Moon system. The transfers are executed by applying a single maneuver and exploiting the stable invariant manifold of the hyperbolic parking solution at arrival. In this regard, an optimization problem is proposed where both the orbital characteristics of a parking solution around the Moon (its Keplerian elements) and the characteristics of a transfer trajectory (guided by the stable manifold of the arrival Halo orbit) are considered as variables. The problem involved in the single maneuver transfer is solved using a nonlinear programming method (NLP), which aims to minimize the cost of ΔV within the framework of the Earth-Moon system using the circular restricted three-body problem. The feasibility of this kind of transfer for a Cubesat is shown in this paper through results with low ΔV combined with suitable times of flight.

Breakdown of homoclinic orbits to L3 in the RPC3BP (I). Complex singularities and the inner equation

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, L1,..,L5, called the Lagrange points.The Lagrange point L3 is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper and its sequel [10], we provide an asymptotic formula for the distance between the one dimensional stable and unstable invariant manifolds of L3 when the ratio between the masses of the primaries μ is small. It implies that L3 cannot have one-round homoclinic orbits.If the mass ratio μ is small, the hyperbolic eigenvalues are weaker than the elliptic ones by factor of order μ. This implies that the distance between the invariant manifolds is exponentially small with respect to μ and, therefore, the classical Poincaré–Melnikov method cannot be applied.In this first paper, we approximate the RPC3BP by an averaged integrable Hamiltonian system which possesses a saddle center with a homoclinic orbit and we analyze the complex singularities of its time parameterization. We also derive and study the inner equation associated to the original perturbed problem. The difference between certain solutions of the inner equation gives the leading term of the distance between the stable and unstable manifolds of L3.In the sequel [10] we complete the proof of the asymptotic formula for the distance between the invariant manifolds.

Invariant manifolds of homoclinic orbits and the dynamical consequences of a super-homoclinic: A case study in [formula omitted] with [formula omitted]-symmetry and integral of motion

We consider a Z2-equivariant flow in R4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered.

Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

Oscillatory motions and parabolic manifolds at infinity in the planar circular restricted three body problem

Consider the Restricted Planar Circular 3 Body Problem. If the trajectory of the body of zero mass is defined for all time, it can have the following four types of asymptotic motion when time tends to infinity forward or backward in time: bounded, parabolic (goes to infinity with asymptotic zero velocity), hyperbolic (goes to infinity with asymptotic positive velocity) or oscillatory (the position of the body is unbounded but goes back to a compact region of phase space for a sequence of arbitrarily large times). We consider realistic mass ratio for the Sun-Jupiter pair and Jacobi constant which allows the massless body to cross Jupiter's orbit. This is a non-perturbative regime. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it.The proof relies on analyzing the stable and unstable invariant manifolds of infinity and their intersections. We construct orbits shadowing these invariant manifolds by the method of correctly aligned windows. The proof is computer assisted.

Evolution maps and symmetry

We describe in detail how the symmetries of equivariance and reversibility induce corresponding properties for the stable and unstable invariant manifolds and for the stable and unstable foliations for any sufficiently small Lipschitz perturbation of a linear hyperbolic dynamics. This requires establishing first a faithful correspondence between the reversibility and equivariance properties of the original dynamics and corresponding properties for its evolution map.

Mission Design of an Aperture-Synthetic Interferometer System for Space-Based Exoplanet Exploration

In recent years, exoplanet detection has become the technological frontier in the field of astronomy, because it provides evidence of the origin of life and the future human habitable exoplanet. Deploying several satellites to form an aperture-synthetic interferometer system in space may help discover “another Earth” via interferometry and midinfrared broadband spectroscopy. This paper analyzes a space-based exoplanet exploration mission in terms of the scientific background, mission profile, trajectory design, and orbital maintenance. First, the system architecture and working principle of the interferometer system are briefly introduced. Secondly, the mission orbit and corresponding transfer trajectories are discussed. The halo orbit near the Sun-Earth L2 (SEL2) orbit is chosen as the candidate mission orbit. The low-energy transfer via stable invariant manifold with multiple perigees is designed, and the proper launch windows are presented. A speed increment less than 10 m/s is imposed for each transfer to achieve the insertion of the halo orbit. Finally, the tangent targeting method (TTM) is applied for high-precision formation maintenance with the whole velocity increments of less than 5 × 1 0 − 4 m / s for each spacecraft when the error bound is 0.1 m. The overall fuel budget during the mission period is evaluated and compared. The design in this paper will provide technical support and reliable reference for future exoplanet exploration missions.

Stable invariant manifolds with application to control problems

&lt;p style='text-indent:20px;'&gt;In this article we develop the theory of stable invariant manifolds for evolution equations with application to control problem. We will construct invariant subspaces for linear equations which can be extended to the non-linear equations in the neighbourhood of the equilibrium with help of perturbation theory. Here will be considered both cases of the discrete and continuous spectrum of the generator associated with resolving semi-group. The example of global invariant manifold will be presented for Burgers equation.&lt;/p&gt;

Near-Earth Asteroid Surveillance Constellation in the Sun-Venus Three-Body System

The threat of potential hazardous near-Earth asteroid (PHA) impact on Earth is increasingly attracting public attention. Monitoring and early warning of those PHAs are the premise of planetary defense. In this paper, we proposed a novel concept of surveillance constellation of heterogeneous wide-field near-Earth asteroid (NEA) surveyors (CROWN), in which six space-based surveyors are loosely deployed in Venus-like orbits to detect the NEAs along the direction of the sunlight. First, the concept and overall design of the NEA surveillance constellation are discussed. Second, the transfer and deployment trajectory of the surveyors are investigated based on the Sun-Venus three-body system. The Sun-Venus libration orbit is taken as the parking orbit, and its stable invariant manifolds are used to reduce the deployment fuel consumption. Next, the detection performance of the CROWN was evaluated considering constraints of apparent visual magnitude and field of view. The NEA orbit determination (OD) using the CROWN was studied and verified. Simulation results show that the CROWN can be deployed with a total velocity increment of approximately 300 m/s. During the 5 years of observation, 99.8% of PHAs can be detected and the OD precision is better than a single-surveyor system. This paper can provide a reference for the construction of future asteroid defense system.

An algorithm for computing phase space structures in chemical reaction dynamics using Voronoi tessellation

An efficient sampling algorithm is proposed for computing reactive islands, which consist of intersections of reactive trajectories and a Poincaré surface of section (PSOS), and are characterized by the number of times the reactive trajectories intersect the PSOS before/after reactions take place. The boundaries of reactive islands are series of intersections of cylindrical stable and unstable invariant manifolds emanated from a normally hyperbolic invariant manifold located, typically, nearby a rank-one potential energy saddle, which forms “reactivity boundaries”. The reactive islands structures enable us to predict the fate of trajectories. The proposed algorithm estimates regions of reactive islands on the PSOS as a Voronoi diagram constructed from the set of labeled intersections of computed molecular dynamics trajectories. The algorithm iteratively refines the boundaries of the estimated reactive islands efficiently, by computing trajectories of which initial condition is sampled from the estimated boundaries of reactive islands in the Voronoi diagram of the previous iteration step. The efficiency of the proposed algorithm is scrutinized, compared with random sampling algorithm and using a two-degrees of freedom Hamiltonian system of a double well potential, and further possible extensions of the proposed algorithm are addressed.

Hyperbolicity of delay equations via cocycles

We characterize the existence of an exponential dichotomy for a nonautonomous linear delay equation via the hyperbolicity of an appropriate cocycle. An important advantage of this approach is that the base is compact under mild additional assumptions. Moreover, we give a few applications of the equivalence of the two notions of hyperbolicity. In particular, we consider the robustness and the admissibility of the equation, and we obtain stable and unstable invariant manifolds.

Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

Low-fuel transfers from Mars to quasi-satellite orbits around Phobos exploiting manifolds of tori

Quasi-satellite orbits (QSOs) are considered by JAXA’s MMX mission, in which CNES is involved, for the scientific observation of the Martian moon Phobos prior to landing and sample return operations. These periodic orbits, originally defined in the Mars-Phobos circular restricted three-body problem, generically lose periodicity once the eccentricity of Phobos’ orbit is taken into account. In this case, QSOs are replaced by quasi-periodic tori. Recent work on MMX project includes, among many others, station-keeping strategies around QSOs exploiting invariant tori in the elliptic Hill problem. In the present paper, a resonant QSO will be first computed in the Mars-Phobos circular restricted three-body problem. Then, by continuation on the eccentricity of the secondary, this QSO will be converted into a periodic resonant QSO in the Mars-Phobos elliptic restricted three-body problem. The latter problem is more precise than the Hill problem at far distance from the secondary and thus more adapted to handle Mars-Phobos transfers. Notice that considering resonant orbits enables to preserve the periodicity of the QSOs when the eccentricity is nonzero. Then, a family of invariant tori surrounding the resonant QSO in the Elliptic Restricted Three-Body Problem will be obtained by continuation on the frequency. In the next step, stable invariant manifolds emanating from the tori will be computed. Finally, two-impulse transfers between a parking orbit around Mars and the tori surrounding the QSO will be presented with the objective to minimize the total velocity increment. Interesting transfer trajectories will be shown allowing for a ballistic approach to Phobos. These trajectories lead to a reduction in the total velocity increment in comparison with classical Mars to QSOs transfers. Here, instead of targeting directly the resonant QSO, the spacecraft will reach first an invariant manifold before coasting along this manifold until encountering a torus surrounding the QSO. Thus, the spacecraft will reach a quasi-periodic QSO inside the torus that is close to the periodic resonant QSO.