## A12 Second Order Differential Equations

Second-order differential equations (by using explicit representation) can be written y'' f (x,y,y'), where f is a function defined on G c R3 having real values. A function F Ix R defined by x y F (x) is a solution of the differential equation if it is twice differentiable and if for any x e Ix, (x,F(x),F'(x)) e G and F''(x) f (x, F(x), F'(x)). A solution denoted FA is said solution of the problem of initial values (x0,y0,y0) e G, if (x0,FA(x0),F'A(x0)) (x0,y0,y0). Resolve a second-order...

## Lyapunov Exponent Calculation

The Lyapunov exponent A(x0) measures the gap of trajectories. Let x0,x0 + e be two close points, then we write the Lyapunov exponent by eenX(x fn (x0 + e)-f (x0) . For the limits of e and n, we have It is pointed out that xi f (x0) and fn(x0) f (fn 1 (x0)), consequently dfn(x0) dx f (xn-1) f(xn-2) f (x1) f (xn-1). 0) is written

## A11 Relations and Diffeomorphisms

Before giving the definitions of maps until the diffeomorphism, we recall an extremely useful elementary notion of the set theory which is the partition. Definition A.1 (Partition). If U Ai F and if all the Aj are assumed 0 and pairwise disjoint (i.e. mutually disjoint), the set Aj is called a partition of F. The main partitions in mathematics are obtained in the form of equivalence classes1 according to an equivalence relation.2 Thus, the vectors or the rational numbers and many others...

## A42 Classification of Closed Surfaces

The homeomorphic relation between closed surfaces of Rn (n > 3) is an equivalence relation, and each class admits a polyhedral representative. The problem of this classification is completely resolved. If we represent a class by a polyhedron, it is interesting to choose a polyhedron whose number of triangular faces is minimum. Then in R3, the skeleton of dimension 2 of the tetrahedron is convenient to represent the class of S2 (S2 corresponds to a sphere of R3, subspace of (R3, R3), i.e. in...

## Shilnikov Phenomenon Homoclinic Orbit in R3

Let us consider a three-dimensional phase flow of the following autonomous equation x f (x), x e R3.63 Then suppose that x 0 is an equilibrium point with a complex pair of conjugate (unstable) eigenvalues X, X and real positive eigenvalue Y > 0. It is supposed that ReX < 0, ReX < Y and there exists one homoclinic orbit Fig. 1.89 Domain A (left). Horseshoe map g(A) (right) Fig. 1.89 Domain A (left). Horseshoe map g(A) (right) 63 See for an introduction Wiggins (1988), and also Glendinning...

## Ramsey and Zhang Approach of Stock Market Crises by Matching Pursuit with Time Frequency Atom Dictionaries High

It would be interesting to test this method on a stock market index offering a continuity throughout the twentieth century and compare this series and its decomposition with the historical events which impacted this financial index and the economy. The French index does not allow probably this experiment, indeed, it would be necessary to study the continuity between the French index before and after 1988, since in 1988 a new index has been created (i.e. Cac40). The Ramsey and Zhang work (using...

## A62 Euclidean and Non Euclidean Metrics A621 Rectangle Axiom

The introduction of new axioms are needed to classify metric planes. Rectangle axiom (R) There exist two distinct lines, which have two distinct perpendiculars in common. Axiom -R This axiom means that any pair of distinct lines has at most one common perpendicular. A quadrilateral also said quadrangle (tetragon) is a four-sided polygon. In a quadrilateral with three right angles the fourth angle is also a right angle. Definition A.67 (Euclidean or non-Euclidean metric plane). A metric plane...

## Efficiency and Random Walk

This chapter, which corresponds to the second subdivision of the part, attempts to characterize stock markets. The stock markets are known as efficient and are representative of the perfect competition. They are also known as advanced indicators of the global economic activity. The evolution of stock market indexes show however trajectories whose amplitudes are often considerably higher than those of gross domestic product. The concept and the models of rational expectations arbitrage and stock...

## The Atomic Decompositions of Signals

6.1 A Hybrid Transformation Evolution of the Matching Pursuit Towards the Mallat and Zhang Version It is one of the applications of waveforms theory, which corresponds to the transformation by the Pursuit algorithm with adaptive window. This technique will be applied in this chapter to a stock index, i.e. the French stock index Cac40. We know that this transformation decomposes the signal in a time-frequency plane, the analyzing function is usually Gaussian of a variable width. The variable...

## Reminders Statistics and Probability 311 Random Experiment and Measurement

A random experiment is represented by a triplet Q, a, P where the following conditions are verified 1 Q is the set of possible results of the experiment, 2 a is a a-algebra, i.e. a set of parts of Q called events , containing the parts 0 and Q, stable by complementation and by denumerable union, 3 P is a set function,1 i.e. a map from a to R , which satisfies P Q 1 and the condition of a-additivity if An is a sequence of disjoint events and if InAn indicates...

## Info

It is possible to write that P implies -R and C -C and H implies -R -H implies -C. A.6.3 Affine and Projective Planes A.6.3.1 Affine Planes Definition A.75 Affine plane of incidence . An affine plane of incidence is any set of points and lines which verifies A1 For any pair of distinct points A and B, there exists one and only one line l incident to A and B. A2 For any line l there exists at least one point A which is not incident to l. A3 Given a line l and a point D which is non incident to...

## A8 Distribution Theory

Distribution theory, created in the 1950s by Laurent Schwartz, made possible to make rigorous certain heuristic process i.e. symbolic calculation ofHeaviside, delta of Dirac , to clarify the notion of weak solution of a partial differential equation PDE , and lastly to provide a general framework to Fourier transform. Distribution theory is a vast generalization of the function notion of several variables The fundamental idea is that of duality Distributions are, by definition, linear forms on...

## Smale Birkhoff Homoclinic Theorem

Closely related to the Smale horseshoe map topology43 and to the hyperbolicity concept, it is interesting to present the following fundamental theorem. Theorem 1.11 Smale-Birkhoff homoclinic theorem . Let f be a diffeomor-phism C1 and suppose p is a hyperbolic fixed point. A homoclinic point is a point q p which is in the stable and unstable manifold. If the stable and unstable manifold intersect transversally44 at q, then q is called transverse. This implies that there is a homoclinic orbit y...

## Nonlinear Theory

In the general introduction we observed that the irruption of the nonlinear led to a profound transformation of a great number of scientific fields. The behaviors resulting from the nonlinear make it possible to better understand the natural phenomena considered as complex. The nonlinear introduced a set of concepts and tools, i.e. analysis and investigation instruments of dynamics generated by the nonlinear. We try to gather these investigation tools, knowing that today it is possible to say...