By Frederick S. And Frederick H. Bailey Woods

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It uses the fact that periodic orbits correspond to fixed points of Poincare maps. We chose the plane U = 0 as one sided Poincare section. (The 'side' of the section is here defined by U becoming positive) Let us recall the main steps in locating periodic orbits by using the Poincare map method : we apply the Newton-Raphson algorithm to the application H(X) = P(X) − X where P(X) is the Poincare map associated to our system which can be written as : X(0) = X0 where ε denotes the set of the control parameters.

1. u verifies initial conditions. Various perturbative methods are presented now. Regular perturbation Solving method can be described as follows: Algorithm: 1. Differential equation is written as: 1. The solution u0 of the problem when ε is zero is known. 2. General solution is seeked as: 1. Function N(u) is developed around u0 using Taylor type formula: 1. A hierarchy of linear equations to solve is obtained: This method is simple but singular problem my arise for which solution is not valid uniformly in t.

When stars of very large mass explode in a supernova, they leave behind a core which is so massive (greater than about 3 solar masses) that it cannot be stabilized against gravitational collapse by an known means, not even neutron degeneracy. Such a core is detined to collapse indefinitely until it forms a black hole, and object so dense that nothing can escape its gravitational pull, ot even light. Let us now start the listing of matter forms observed at a super nuclear scale (sacle larger than the nuclear scale).

### A Course in Mathematics Volume II by Frederick S. And Frederick H. Bailey Woods

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