Completely unstable fixed point

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

Recall that a fixed point 0 x = xq) is called structurally stable if none of its characteristic multipliers, i.e. the roots of the characteristic equation (7.5.2), lies on the unit circle, A topological type (m,p) is assigned to it, where m is the number of roots inside the unit circle and p is that outside of the unit circle. If m = n (m = 0), the fixed point is stable (completely unstable). The fixed point is of saddle type when m 0,n. The set of all points whose trajectories converge to xq when iterated positively (negatively) is called the stable (unstable) manifold of the fixed point and denoted by Wq Wq). In the case where m = n, the attraction basin of O is Wq. If the fixed point is a saddle, the manifolds Wq and Wq are C -smooth embeddings of and MP in respectively. 

If I > 0, then the function F is a Lyapunov function for the system obtained from (10.5.30) by inversion of time. Thus, the equilibrium state of system (10.5.29) and hence the fixed point of the map (10.5.27) is completely unstable here. 

In fact, resonant fixed points are not restricted to only saddles and stable (completely unstable) points. An example of the other structure is given by the map... 

The Jacobian of the Henon map is constant and equal to h. Therefore, when 6 > 0, the Henon map preserves orientation in the plane, whereas orientation is reversed when 6 < 0. Note also that if 6 < 1, the map contracts areas, so the product of the multipliers of any of its fixed or periodic points is less than 1 in absolute value. Hence, in this case the map cannot have completely unstable periodic orbit (only stable and saddle ones). On the contrary, when b > 1, no stable orbits can exist. When 6 = 1, the map becomes conservative. At b = 0, the Henon map degenerates into the above logistic map, and therefore one should expect some similar bifurcations of the fixed points when b is suflSciently small. 

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