Maciej J. Capiński

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

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.

Transition Tori in the Planar Restricted Elliptic Three Body Problem

We consider the elliptic three-body problem as a perturbation of the circular problem. We show that for sufficiently small eccentricities of the elliptic problem, and for energies sufficiently close to the energy of the libration point L2, a Cantor set of Lyapunov orbits survives the perturbation. The orbits are perturbed to quasi-periodic invariant tori. We show that for a certain family of masses of the primaries, for such tori we have transversal intersections of stable and unstable manifolds, which lead to chaotic dynamics involving diffusion over a short range of energy levels. Some parts of our argument are nonrigorous, but are strongly backed by numerical computations.

Computer Assisted Existence Proofs of Lyapunov Orbits at

We present a computer assisted proof of existence of a family of Lyapunov orbits which stretches from

L_2

L_2

and Transversal Intersections of Invariant Manifolds in the Jupiter--Sun PCR3BP

(the collinear libration point between the primaries) up to half the distance to the smaller primary in the Jupiter--Sun planar circular restricted three body problem. We then focus on a small family of Lyapunov orbits with energies close to comet Oterma and show that their associated invariant manifolds intersect transversally. Our computer assisted proof provides explicit bounds on the location and on the angle of intersection.

Hedging Conditional Value at Risk with options

We present a method of hedging Conditional Value at Risk of a position in stock using put options. The result leads to a linear programming problem that can be solved to optimise risk hedging.

Validated Computation of Heteroclinic Sets

In this work we develop a method for computing mathematically rigorous enclosures of some one dimensional manifolds of heteroclinic orbits for nonlinear maps. Our method exploits a rigorous curve following argument build on high order Taylor approximation of the local stable/unstable manifolds. The curve following argument is a uniform interval Newton method applied on short line segments. The definition of the heteroclinic sets involve compositions of the map and we use a Lohner-type representation to overcome the accumulation of roundoff errors. Our argument requires precise control over the local unstable and stable manifolds so that we must first obtain validated a-posteriori error bounds on the truncation errors associated with the manifold approximations. We illustrate the utility of our method by proving some computer assisted theorems about heteroclinic invariant sets for a volume preserving map of

Geometric proof of strong stable/unstable manifolds with application to the restricted three body problem

\mathbb{R}^3

Isolating segments for Carathéodory systems and existence of periodic solutions

.

Portfolio Theory and Risk Management: Risk and return

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.