We begin with the definition of stability and instability. Let ℜ be a topological space whose points we denote by þ, and let a be a certain point in ℜ. By a neighborhood here we will always mean a neighborhood of a in ℜ. Let þ1 = Sþ be a topological mapping of a neighborhood U1 onto a neighborhood B1 whereby a = Sa is mapped onto itself. The inverse mapping p-1 = S-1þ then carries B1 onto U1, and in general þn = Snþ (n = 0, ± 1, ± 2,…) is a topological mapping of a neighborhood Un onto a neighborhood Bn, having a as a fixed-point. For each point þ = þ0 in the intersection U1∩B1=M we construct the successive images þk+1 =Sþk (k = 0,1,…), as long as þk lies in U1 and similarly þ-k-1 =S-1þ-k as long as þ-k lies in B1. If the process terminates with a largest k+ 1 = n, then þ0,…, þn-1 all still lie in U1, but þn no longer does; similarly for the negative indices. In this way, to each þ in M there is associated a sequence of image points þk (k =…, — 1,0,1,…), which is finite, infinite on one side, or infinite on both sides.
KeywordsNormal Form Hamiltonian System Equilibrium Solution Formal Power Series Invariant Curve
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