Recently, a flexible and stable algorithm was introduced for the computation of 2D unstable manifolds of periodic solutions to systems of ordinary differential equations. The main idea of this approach is to represent orbits in this manifold as the solutions of an appropriate boundary value problem. The boundary value problem is under determined and a one parameter family of solutions can be found by means of arclength continuation. This family of orbits covers a piece of the manifold. The quality of this covering depends on the way the boundary value problem is discretised, as do the tractability and accuracy of the computation. In this paper, we describe an implementation of the orbit continuation algorithm which relies on multiple shooting and Newton-Krylov continuation. We show that the number of time integrations necessary for each continuation step scales only with the number of shooting intervals but not with the number of degrees of freedom of the dynamical system. The number of shooting intervals is chosen based on linear stability analysis to keep the conditioning of the boundary value problem in check. We demonstrate our algorithm with two test systems: a low-order model of shear flow and a well-resolved simulation of turbulent plane Couette flow.
We study the structure of the smooth manifold which is defined as the intersection of a stable manifold and an unstable manifold for an invariant Morse-Smale function.
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of stochastic partial differential equations. We first show the existence of Lipschitz continuous stable and unstable manifolds by the Lyapunov-Perron's method. Then, we prove the smoothness of these invariant manifolds.
We outline a method for controlling the location of stable and unstable manifolds in the following sense. From a known location of the stable and unstable manifolds in a steady two-dimensional flow, the primary segments of the manifolds are to be moved to a user-specified time-varying location which is near the steady location. We determine the nonautonomous perturbation to the vector field required to achieve this control, and give a theoretical bound for the error in the manifolds resulting from applying this control. The efficacy of the control strategy is illustrated via a numerical exampl…
Anosov families were introduced by A. Fisher and P. Arnoux motivated by generalizing the notion of Anosov diffeomorphism defined on a compact Riemannian manifold. In addition to presenting several properties and examples of Anosov families, in this paper we build local stable and local manifolds for such families.
We consider compact sets which are invariant and partially hyperbolic under the dynamics of a diffeomorphism of a manifold. We prove that such a set K is contained in a locally invariant center submanifold if and only if each strong stable and strong unstable leaf intersects K at exactly one point.
State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of a…
We present a method for establishing invariant manifolds for saddle--center fixed points. The method is based on cone conditions, suitably formulated to allow for application in computer assisted proofs, and does not require rigorous integration of the vector field in order to prove the existence of the invariant manifolds. We apply our method to the restricted three body problem and show that for a given choice of the mass parameter, there exists a homoclinic orbit to one of the libration points.
In this paper we study high order expansions of chart maps for local finite dimensional unstable manifolds of hyperbolic equilibrium solutions of scalar parabolic partial differential equations. Our approach is based on studying an infinitesimal invariance equation for the chart map that recovers the dynamics on the manifold in terms of a simple conjugacy. We develop formal series solutions for the invariance equation and efficient numerical methods for computing the series coefficients to any desired finite order. We show, under mild non-resonance conditions, that the formal series expansion …
SRB measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectory-driven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We app…
This article is a sequel to [M.Z.Z.1] aimed at completing the characterization of the pathwise local structure of solutions of semilinear stochastic evolution equations (see's) and stochastic partial differential equations (spde's) near stationary solutions. Stationary solution are viewed as random points in the infinite-dimensional state space, and the characterization is expressed in terms of the almost sure long-time behavior of trajectories of the equation in relation to the stationary solution. More specifically, we establish local stable manifold theorems for semilinear see's and spde's …
First we prove a general spectral theorem for the linear Navier-Stokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in $H^\ell$ ($\ell = 0,1,2, ...$). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zero-viscosity limits ($ν\ra 0^+$) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the center-s…
Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limit spaces are homeomorphic if and only if the substitutions associated with f and g are weakly equivalent. This, and related results, are applied to one-dimensional substitution tiling spaces, one-dimensional unstable manifolds of hyperbolic sets, and inverse limits of tent maps w…
This note presents a non-rigorous study of the linear response for an SRB (or `natural physical') measure $ρ$ of a diffeomorphism $f$ in the presence of tangencies of the stable and unstable manifolds of $ρ$. We propose that generically, if $ρ$ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of $ρ(ϕ)$ with respect to $f$ along $X$ is convergent: $$ Ψ(z)=\sum_{n=0}^\infty z^n\intρ(dx)\,X(x)\cdot\nabla_x(ϕ\circ f^n)\qq…
We address a numerical methodology for the computation of coarse-grained stable and unstable manifolds of saddle equilibria/stationary states of multiscale/stochastic systems for which a "good" macroscopic description in the form of Ordinary (ODEs) and/or Partial differential equations (PDEs) does not explicitly/ analytically exists in a closed form. Thus, the assumption is that we have a detailed microscopic simulator of a complex system in the form of Monte-Carlo, Brownian dynamics, Agent-based models e.t.c. (or a black-box large-scale discrete time simulator) but due to the inherent complex…
This paper presents the space-split sensitivity or the S3 algorithm to transform Ruelle's linear response formula into a well-conditioned ergodic-averaging computation. We prove a decomposition of Ruelle's formula that is differentiable on the unstable manifold, which we assume to be one-dimensional. This decomposition of Ruelle's formula ensures that one of the resulting terms, the stable contribution, can be computed using a regularized tangent equation, similar to in a non-chaotic system. The remaining term, known as the unstable contribution, is regularized and converted into an efficientl…
It is well-known that stable and unstable manifolds strongly influence fluid motion in unsteady flows. These emanate from hyperbolic trajectories, with the structures moving nonautonomously in time. The local directions of emanation at each instance in time is the focus of this article. Within a nearly autonomous setting, it is shown that these time-varying directions can be characterised through the accumulated effect of velocity shear. Connections to Oseledets spaces and projection operators in exponential dichotomies are established. Availability of data for both infinite and finite time-in…
Let $M$ be a locally compact metric space endowed with a continuous flow $ϕ: M \times \mathbb{R} \longrightarrow M$. Frequently an attractor $K$ for $ϕ$ exists which is of interest, not only in itself but also the dynamics in its basin of attraction $\mathcal{A}(K)$. In this paper the class of {\sl attractors with no external explosions}, which is intermediate between the well known {\sl stable attractors} and the extremely wild {\sl unstable attractors}, is studied. We are mainly interested in their cohomological properties, as well as in the strong relations which exist between their shape (…
It is known that the asymptotic invariant manifolds around an unstable periodic orbit in conservative systems can be represented by convergent series (Cherry 1926, Moser 1956, 1958, Giorgilli 2001). The unstable and stable manifolds intersect at an infinity of homoclinic points, generating a complicated homoclinic tangle. In the case of simple mappings it was found (Da Silva Ritter et al. 1987) that the domain of convergence of the formal series extends to infinity along the invariant manifolds. This allows in practice to study the homoclinic tangle using only series. However in the case of Ha…
A well-known lemma by John Franks asserts that one obtains any perturbation of the derivative of a diffeomorphism along a periodic orbit by a $C^1$-perturbation of the whole diffeomorphism on a small neighbourhood of the orbit. However, one does not control where the invariant manifolds of the orbit are, after perturbation. We show that if the perturbated derivative is obtained by an isotopy along which some strong stable/unstable manifolds of some dimensions exist, then the Franks perturbation can be done preserving the corresponding stable/unstable semi-local manifolds. This is a general per…
We consider a hyperbolic automorphism $A\colon\mathbb T^3\to\mathbb T^3$ of the 3-torus whose 2-dimensional unstable distribution splits into weak and strong unstable subbundles. We unfold $A$ into two one-parameter families of Anosov diffeomorphisms --- a conservative family and a dissipative one. For diffeomorphisms in these families we numerically calculate the strong unstable manifold of the fixed point. Our calculations strongly suggest that the strong unstable manifold is dense in $\mathbb T^3$. Further, we calculate push-forwards of the Lebesgue measure on a local strong unstable manifo…
We present a Melnikov method to analyze two-dimensional stable or unstable manifolds associated with a saddle point in three-dimensional non-volume preserving autonomous systems. The time-varying perturbed locations of such manifolds is obtained under very general, non-volume preserving and with arbitrary time-dependence, perturbations. In unperturbed situations with a two-dimensional heteroclinic manifold, we adapt our theory to quantify the splitting into a stable and unstable manifold, and thereby obtain a Melnikov function characterizing the time-varying locations of transverse intersectio…
Transversality of stable and unstable manifolds of hyperbolic periodic trajectories is proved for monotone cyclic systems with negative feedback. Such systems in general are not in the category of monotone dynamical systems in the sense of Hirsch. Our main tool utilized in the proofs is the so-called cone of high rank. We further show that stable and unstable manifolds between a hyperbolic equilibrium and a hyperbolic periodic trajectory, or between two hyperbolic equilibria with different dimensional unstable manifolds also intersect transversely.
For a partiallyhyperbolic diffeomorphism on a 3-manifold, we show that any invariant foliation tangent to the center-unstable (or center-stable) bundle has no compact leaves.
In this paper we study a certain regularity property of Axiom A flows over basic sets related to diameters of balls in Bowen's metric, which we call regular distortion along unstable manifolds. The motivation to investigate the latter comes from the study of spectral properties of Ruelle transfer operators. We prove that if the bottom of the spectrum of the tangent map of the flow over unstable manifolds is point-wisely pinched and integrable, then the flow has regular distortion along unstable manifolds over the basic set. In the process, under the same conditions, we show that locally the fl…
A blender-horseshoe is a locally maximal transitive hyperbolic set that appears in dimension at least three carrying a distinctive geometrical property: its local stable manifold "behaves" as a manifold of topological dimension greater than the expected one (the dimension of the stable bundle). This property persists under perturbations turning this kind of dynamics an important piece in the global description of robust non-hyperbolic systems. In this paper, we consider a parameterized family of center-unstable Hénon-like of endomorphisms in dimension three and show how blender-horseshoes natu…
When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, families of unstable periodic orbits break up into whiskered tori, with most tori persisting into the perturbed system. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) regularize the equations of motion; and 5) globalize these manifolds. Our methodology impro…
It was shown that in robustly transitive, partially hyperbolic diffeomorphisms on three dimensional closed manifolds, the strong stable or unstable foliation is minimal. In this article, we prove ``almost all'' leaves of both stable and unstable foliations are dense in the whole manifold.
Let ($M$, $Ω$) be a smooth symplectic manifold and $f:M\rightarrow M$ be a symplectic diffeomorphism of class $C^l$ ($l\geq 3$). Let $N$ be a compact submanifold of $M$ which is boundaryless and normally hyperbolic for $f$. We suppose that $N$ is controllable and that its stable and unstable bundles are trivial. We consider a $C^1$-submanifold $\d$ of $M$ whose dimension is equal to the dimension of a fiber of the unstable bundle of $T_NM$. We suppose that $\d$ transversely intersects the stable manifold of $N$. Then, we prove that for all $\varepsilon>0$, and for $n$ $\in$ $\N$ large enough, …
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…
Given a locally maximal compact invariant hyperbolic set $Λ$ for a $C^1$ flow or diffeomorphism on a Riemann manifold with $C^1$ unstable laminations, we construct an invariant continuous bundle of tangent vectors to local unstable manifolds that locally approximates $Λ$ in a certain way.
Let a system of differential equations possess a saddle-node periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases where the splitting of such a homoclinic connection causes the Blue Sky Catastrophe, including the onset of complex dynamics. The birth of an invariant torus or a Klein bottle is also described.
The stable and unstable manifolds of an invariant set of a piecewise-smooth map are themselves piecewise-smooth. Consequently, as parameters of a piecewise-smooth map are varied, an invariant set can develop a homoclinic connection when its stable manifold intersects a non-differentiable point of its unstable manifold (or vice-versa). This is a codimension-one bifurcation analogous to a homoclinic tangency of a smooth map, referred to here as a homoclinic corner. This paper presents an unfolding of generic homoclinic corners for saddle fixed points of planar piecewise-smooth continuous maps. I…
We present an argument for proving the existence of local stable and unstable manifolds in a general abstract setting and under very weak hyperbolicity conditions.
Accurate approximations of the change of system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the sensitivity of chaotic dynamical systems, which enables a better understanding of chaotic phenomena. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting regularizing Ruelle's original expression together with measur…
We provide a new approach to stable ergodicity of systems with dominated splittings, based on a geometrical analysis of global stable and unstable manifolds of hyperbolic points. Our method suggests that the lack of uniform size of Pesin's local stable and unstable manifolds - a notorious problem in the theory of non-uniform hyperbolicity - is often less severe than it appeas to be.
In this article we will describe a new construction for Gibbs measures for hyperbolic attractors generalizing the original construction of Sinai, Bowen and Ruelle of SRB measures. The classical construction of the SRB measure is based on pushing forward the normalized volume on a piece of unstable manifold. By modifying the density at each step appropriately we show that the resulting measure is a prescribed Gibbs measure. This contrasts with, and complements, the construction of Climenhaga-Pesin-Zelerowicz who replace the volume on the unstable manifold by a fixed reference measure. Moreover,…
We study a class of skew products with overlaps in fibers and show that in this case the unstable manifolds really depend on prehistories, even for perturbations of the original maps. We also give several results about the Hausdorff dimension of the fibers of the respective locally maximal invariant set, by using the inverse pressure, the thickness of Cantor sets and some bounds for the preimage counting function.
A well-known lemma by John Franks asserts that one obtains any perturbation of the derivative of a diffeomorphism along a periodic orbit by a $C^1$-perturbation of the whole diffeomorphism on a small neighbourhood of the orbit. However, one does not control where the invariant manifolds of the orbit are, after perturbation. We show that if the perturbated derivative is obtained by an isotopy along which some strong stable/unstable manifolds of some dimensions exist, then the Franks perturbation can be done preserving the corresponding stable/unstable semi-local manifolds. This is a general per…
We develop the geometrical, analytical, and computational framework for reactive island theory for three degrees-of-freedom time-independent Hamiltonian systems. In this setting, the dynamics occurs in a 5-dimensional energy surface in phase space and is governed by four-dimensional stable and unstable manifolds of a three-dimensional normally hyperbolic invariant sphere. The stable and unstable manifolds have the geometrical structure of spherinders and we provide the means to investigate the ways in which these spherinders and their intersections determine the dynamical evolution of trajecto…
We develop a validated numerical procedure for continuation of local stable/unstable manifold patches attached to equilibrium solutions of ordinary differential equations. The procedure has two steps. First we compute an accurate high order Taylor expansion of the local invariant manifold. This expansion is valid in some neighborhood of the equilibrium. An important component of our method is that we obtain mathematically rigorous lower bounds on the size of this neighborhood, as well as validated a-posteriori error bounds for the polynomial approximation. In the second step we use a rigorous …
In the first part of this paper, we formulate a general setting in which to study the ergodic theory of differentiable $\mathbb{Z}^d$-actions preserving a Borel probability measure. This framework includes actions by $C^{1+\text{Hölder}}$ diffeomorphisms of compact manifolds. We construct intermediate and coarse unstable manifolds for the action and establish controls on their local geometry. In the second part we consider the relationship between entropy, Lyapunov exponents, and the geometry of conditional measures for rank-1 systems given by a number of generalizations of the Ledrappier--You…
A heterodimensional cycle consists of a pair of heteroclinic connections between two saddle periodic orbits with unstable manifolds of different dimensions. Recent theoretical work on chaotic dynamics beyond the uniformly hyperbolic setting has shown that heterodimensional cycles may occur robustly in diffeomorphisms of dimension at least three. We study a concrete example of a heterodimensional cycle in the continuous-time setting, specifically in a four-dimensional vector field model of intracellular calcium dynamics. By employing advanced numerical techniques, Zhang, Krauskopf and Kirk [Dis…
