Question

1. Recall that center manifolds are not unique. Also, locally, center manifolds are given as the graph of a function, whose Taylor series coefficients we can compute. Explain why these two facts are not contradictory using a simple example.

Answer 1

Consider the Skew product Syslem x= x y=-4 Here the linearization of the equilibrium at the origins has eigen values 7=o, and A=-1, so the slable space ES is the y-aris and the Center Space Ec is the è a- arcs .gt is clear that the local slable manifold is the y. axes since this is tangent to ES. and every point on the Y.anes femits to the origin. We are templed to say that we is the x-axis and this is Cerlainly an acceptable Center manifold et a clean an in variant set and is tangent to EC Of we solve the equation for the phase Cane by dividing the y equation by the al equation, we oblain dy = -4 yea)= Cet 07 TYY ) 2 an ñ a la loro When aco, each of these Curves is asymptotic a to the origin riain and as and its tangent to the n-aris So we could define a - Centar manifold by tollane phare potrait of the the Systema

cox aco 2 w oc conu)= {(x,y): y=cexco x70 D3 for any value of c. There is a one parameter family of possible Center manifold. ('observe fig. ) This example shows that the center manifold is not unique The example has another pathology: though the local slable manifold is the y anej thê global stable set - namely, the set of points - that are asymptotic to the origin is the left hall plane ws(o)={(1,0): 870) since this Set us asymptotic to the origin as Exo . We conclude that the Centro manifold was not unique nevertheless, every choice of c in ② gives a Curve with the same power series Claylor senes) expression, namely yex) = 0+0x tox2t... Consequently as far as the power Senes is concerned, there is a unique Centre manifold, the x-aaw. Indeed , whenever fisco. there is unique power senis expression for a Center manifold.