Fixed points of a map

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

There are many special cases of orbits. The most important orbit is the fixed point. A fixed point of a mapping function / is a point that satisfies... 

Thus the Poincare map converts problems about closed orbits (which are difficult) into problems about fixed points of a mapping (which are easier in principle, though not always in practice). The snag is that it s typically impossible to find a formula for P. For the sake of illustration, we begin with two examples for which P can be computed explicitly. 

A fixed point of a map is linearly stable if and only if all eigenvalues of the Jacobian satisfy A <1. Determine the stability of the fixed points of the Henon map, as a function of a and b. Show that one fixed point is always unstable, while the other is stable for a slightly larger than Show that this fixed point loses stability in a flip bifurcation (A = -1) at a, = (1 - b. ... 

It turns out that, in the CML, the local temporal period-doubling yields spatial domain structures consisting of phase coherent sites. By domains, we mean physical regions of the lattice in which the sites are correlated both spatially and temporally. This correlation may consist either of an exact translation symmetry in which the values of all sites are equal or possibly some combined period-2 space and time symmetry. These coherent domains are separated by domain walls, or kinks, that are produced at sites whose initial amplitudes are close to unstable fixed points of = a, for some period-rr. Generally speaking, as the period of the local map... 

For a later purpose (Chapter 7), we shall explain the perfectness of the Morse function given by the moment map of a torus action on a general symplectic manifold. However, when the fixed points of a torus action are all isolated, such as the case of the... 

The above scenario is accounted for by the normal form (4.9) truncated at fourth order in q with k = v = a = p = 0 and x < 0, taking p as the bifurcation parameter, which increases with energy (p thus plays a similar role as the total energy in the actual Hamiltonian dynamics). The antipitchfork bifurcation occurs at pa = 0. The fixed points of the mapping (4.8) are given by p = 0 and dv/dq = 0. Since the potential is quartic, there are either one or three fixed points that correspond to the shortest periodic orbits 0, 1, and 2 of the flow. 

The mapping function fr has two fixed A fixed point of a function is... 

To see why, we start by analyzing the fixed points of the map f. These are points z such that /(z ) = z, in which case z = z + = z +2 =. Figure 9.4.3 shows that there is one fixed point, where the 45° diagonal intersects the graph. It represents a closed orbit that looks like that shown in Figure 9.4.4. 

Solution If you tried this on your calculator, you found that x —> 0.739..., no matter where you started. What is this bizarre number It s the unique solution of the transcendental equation x = cosx, and it corresponds to a fixed point of the map. Figure 10.1.3 shows that a typical orbit spirals into the fixed point X = 0.739.. . as —> oo. ... 

In order to calculate the ratio 5 between the semi-minor and the semi-major axes of the ellipses surrounding the fixed points of a periodic orbit we consider a two dimensional area preserving mapping and a point [x, y) of a periodic orbit of frequency P/Q. Let us write the Jacobian of the Qth iteration of the map as follows ... 

The dynamic simulation approximates fixed points of the map F by iterating the map itself. If we define the map G = F-I, then fixed points of F become zeros of G. Newton s method generates approximations of a zero of G using the iteration scheme... 

POMEAU and tiANNEVILLE [51] have shown that intermittency can be understood in terms of a tangent bifurcation of a one-dimensional map at tangency a stable fixed point of the map disappears (or appears, depending on the direction in which the bifurcation parameter is changed,)... 

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. 

The answers to these questions are settled by the theory of bifurcations. In this chapter, we consider only local bifurcations, i.e. those which occur near critical equilibrium states, and near fixed points of a Poincare map. We restrict our study to the simplest but key bifurcations which have an immediate connection to the critical cases are discussed in the two last chapters. 

Since G p(x,e) is small when both x and e are small, the right-hand side of (11.3.4) is a monotonically increasing function of x. The fixed points of the map (11.3.4) are found from the condition G(x,e) = 0 their stability is determined by the sign of the derivative G (x,s) if this derivative is positive at a fixed point, the latter is imstable if the derivative is negative, the fixed point is stable. In other words, we have a complete analogy with the family of differential equations... 

In fact, no common upper bound exists on the number of the periodic orbits which can be generated from a fixed point of a smooth map through the given bifurcation. If the smoothness r of the map is finite, the absence of this upper estimate is obvious because it follows from the proof of the last theorem that to estimate the number of the periodic orbits within the resonant zone 1/ = M/N the map must be brought to the normal form containing terms up to order (AT — 1). In this case the smoothness of the map must not be less than (iV — 1). Hence, we can estimate only a finite number of resonant zones if the smoothness is finite. 

If (2/1 j 2/2) is a solution of this system, then (2/2,2/1) is a solution as well. There is also the solution 2/1 = 2/2 = 2/o where yo is the imique fixed point of the map (13.3.8), which always exists for p > 0. Therefore, to prove that there are no saddle-node orbits of period two, it suffices to check that system (13.3.8) has no more than three solutions, including multiplicity. This verification will be performed in Sec. 13.6 for a more general system (see (13.6.26)), corresponding to the bifurcation of a homoclinic loop of a multi-dimensional saddle with... 

We have shown the existence of infinitely many saddle fixed points of the map T which correspond to saddle periodic orbits (with two-dimensional unstable and m-dimensional stable manifolds) of the system. Those with even fc s have a negative unstable multiplier, and their unstable manifolds are non-orientable. The periodic orbits corresponding to odd fc s have a positive unstable multiplier, and hence, orientable unstable manifolds. 

C.3. 21. Prove that if the origin is a structurally stable equilibrium state of the system (C.3.3), then the corresponding fixed point of the map (C.3.2) is structurally stable as well. Furthermore, show that the topological types of the equilibrium state of (C.3.3) and the fixed point of (C.3.2) are the same. ... 

Theorem 7 ([tch82], [goles84], [golesQO]) All finite initiai states evolve, under the map majority) to a fixed point in a finite number of steps. 

Let us begin by considering the stability of homogeneous solutions and/or initial-conditions i.e. by considering the stability of a simple-diffusive CML when cri(O) = a for all sites i , where a is a fixed point of the local logistic map F(cr) = acr(l—cr). Following Waller and Kapral [kapral84], we first recast equations 8.23 and 8.24... 

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

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