Geometric properties of the scattering map of a normally hyperbolic invariant manifold

Quantile connectedness in the cryptocurrency market

In this paper we develop an alternative topological data analysis (TDA) approach for studying graph representations of time series of dynamical systems. Specifically, we show how persistent homology, a tool from TDA, can be used to yield a compressed, multi-scale representation of the graph that can distinguish between dynamic states such as periodic and chaotic behavior. We show the approach for two graph constructions obtained from the time series. In the first approach the time series is embedded into a point cloud which is then used to construct an undirected k-nearest-neighbor graph. The second construct relies on the recently developed ordinal partition framework. In either case, a pairwise distance matrix is then calculated using the shortest path between the graph's nodes, and this matrix is utilized to define a filtration of a simplicial complex that enables tracking the changes in homology classes over the course of the filtration. These changes are summarized in a persistence diagram's two-dimensional summary of changes in the topological features. We then extract existing as well as new geometric and entropy point summaries from the persistence diagram and compare to other commonly used network characteristics. Our results show that persistence-based point summaries yield a clearer distinction of the dynamic behavior and are more robust to noise than existing graph-based scores, especially when combined with ordinal graphs.

Persistent homology of complex networks for dynamic state detection

We present an efficient numerical method for computing Fourier--Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold The method requires only that some mild nonresonance conditions hold The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier--Taylor expansion A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional We also discuss computations of cycle-to-cycle connecting orbits 

/pdf/parameterization-of-invariant-manifolds-for-periodic-orbits-3wwaurt33u.pdf

Parameterization of invariant manifolds for periodic orbits (I) : efficient numerics via the Floquet normal form

This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

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Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps

We present a method of proving existence of center manifolds within specified domains. The method is based on a combination of topological tools, normal forms, and rigorous computer-assisted com- putations. We apply our method to obtain a proof of a center manifold in an explicit region around the equilibrium point L1 in the Earth-Sun planar restricted circular three-body problem. 1. Introduction. In this paper we give a method for proving existence of center manifolds for systems with an integral of motion. The aim of the paper is not to give yet another proof of the center manifold theorem, but to provide a practical tool which can be applied to nontrivial systems. There are a number of advantages to the method. First, the method is not perturbative. We thus do not need to start with an invariant manifold and then perturb it. All that is required is a good numerical approximation of the position of a center manifold. The conditions required in order to ensure existence of the manifold in the vicinity of the numerical approximation are such that it is possible to verify them using (rigorous, interval-based) computer-assisted computations. This is another advantage, since it allows for application to problems which cannot be treated analytically. The method gives explicit bounds on the position and on the size of the manifold. Moreover, under appropriate assumptions we can also prove that the manifold is unique. Our proof of existence of the center manifold is performed using purely topological ar- guments. This means that it can be applied to treat nonanalytic invariant manifolds. The main disadvantage of using topological tools, though, is that the proof ensures only Lipschitz continuity of the manifold even for manifolds with higher order regularity.

/pdf/existence-of-a-center-manifold-in-a-practical-domain-around-10qrwgzn2b.pdf

Existence of a Center Manifold in a Practical Domain around L1 in the Restricted Three-Body Problem ∗

We analyze four major cryptocurrencies (Bitcoin, Ethereum, Litecoin, and Ripple) before the digital asset market crash at the beginning of 2018. We also analyze Bitcoin before some of the mini-crashes that occurred during the period 2016–2018. All relevant time series exhibited a highly erratic behavior. We introduce a methodology that combines topological data analysis with a machine learning technique – k -means clustering – in order to characterize the emerging chaotic regime in a complex system approaching a critical transition. We first test our methodology on the complex system dynamics of a Lorenz-type attractor. Then we apply it to the four major cryptocurrencies. We find early warning signals for critical transitions, i.e., crashes, in the cryptocurrency markets.

Topological recognition of critical transitions in time series of cryptocurrencies

For a given a normally hyperbolic invariant manifold, whose stable and unstable
manifolds intersect transversally, we consider several tools and techniques to detect trajectories with prescribed itineraries:
the scattering map, the transition map, the method of correctly aligned windows, and the shadowing lemma. We provide an user's guide on how to apply these tools and techniques to detect unstable orbits in a Hamiltonian system. This consists in the following steps: (i) computation of the scattering map and of the transition map for the Hamiltonian flow, (ii) reduction to the scattering map and to the transition map, respectively, for the return map to some surface of section, (iii) construction of sequences of windows within the surface of section, with the successive pairs of windows correctly aligned, alternately, under the transition map, and under some power of the inner map, (iv) detection of trajectories which follow closely those windows. We illustrate this strategy with two models: the large gap problem for nearly integrable Hamiltonian systems, and the the spatial circular restricted three-body problem.

Transition map and shadowing lemma for normally hyperbolic invariant manifolds

We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun--Jupiter--asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio $\mu=10^{-3}$ and a small eccentricity $e_0>0$. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept almost constant. The proof relies on the careful analysis of the circular problem, which has a hyperbolic structure, but for which diffusion is prevented by KAM tori. We verify certain non-degeneracy conditions numerically. Based on the work of Treschev, it is natural to conjecture that diffusion time for this problem is $\sim \frac{-\ln (\mu e_0)}{\mu^{3/2} e_0}$. We expect our instability mechanism to apply to realistic values of $e_0$ and we give heuristic arguments in its favor. If so, the applicability of Nekhoroshev theory to the three-body problem as well as the long time stability become questionable. It is well known that, in the Asteroid Belt, located between the orbits of Mars and Jupiter, the distribution of asteroids has the so-called Kirkwood gaps exactly at mean motion resonances of low order. Our mechanism gives a possible explanation of their existence. To relate the existence of Kirkwood gaps with Arnold diffusion, we state a conjecture on its existence for a typical $\eps$-perturbation of the product of a pendulum and a rotator. Namely, we predict that a positive conditional measure of initial conditions concentrated in the main resonance exhibits Arnold diffusion on time scales $\frac{- \ln \eps}{\eps^{2}}$.

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=18&iss=10&rank=3

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

Let $A_1$ and $A_2$ be two normally hyperbolic invariant manifolds for a flow, such that the stable manifold of $A_1$ intersects the unstable manifold of $A_2$ transversally along a manifold Γ. The scattering map from $A_2$ to $A_1$ is the map that, given an asymptotic orbit in the past, associates the corresponding asymptotic orbit in the future through a heteroclinic orbit. It was originally introduced to prove the existence of orbits of unbounded energy in a perturbed Hamiltonian problem using a geometric approach.
 &nbsp
We recently computed the scattering map in the planar restricted three body problem using non-perturbative techniques, and we showed that it is a (nontrivial) integrable twist map. &nbsp
In the present paper, we compute the scattering map in a problem with three degrees of freedom using also non-perturbative techniques. Specifically, we compute the scattering map between the normally hyperbolic invariant
manifolds $A_1$ and $A_2$ associated to the equilibrium points $L_1$ and $L_2$ in the spatial Hill's problem.
 &nbsp
In the planar problem, for each energy level (in a certain range) there is a unique Lyapunov periodic orbit around $L_{1,2}$. In the spatial problem, this periodic orbit is replaced by a three-dimensional invariant manifold practically full of invariant 2D tori. There are heteroclinic orbits between $A_1$ and $A_2$ connecting these invariant tori in rich combinations. Hence the scattering map in the spatial problem is more complicated, and it allows nontrivial transition chains.
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Scattering maps have application to e.g. mission design in Astrodynamics, and to the construction of diffusion orbits in the spatial Hill's problem.

Pablo Roldan

Papers

Existence of a Center Manifold in a Practical Domain around L1 in the Restricted Three-Body Problem ∗

Topological recognition of critical transitions in time series of cryptocurrencies

Transition map and shadowing lemma for normally hyperbolic invariant manifolds

Kirkwood gaps and diffusion along mean motion resonances in the restricted planar three-body problem

Computing the scattering map in the spatial Hill's problem