Bifurcation of the fixed point

The bifurcations of the fixed point are found from the system (compare with (13.6.22)) ... 

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. 

In the region Di there are two fixed points, one of which is a saddle, and the other one is stable for (o,6) Df, and repelling when (a, 6) G Di. Transition from Di to D2 is accompanied with the period-doubling bifurcations of the fixed point, correspondingly, stable on the route Df and repelling on... 

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. 

Moreover, the breakdown of normal hyperbolicity leads to the bifurcation from the fixed point to the limit cycle. Suppose that under a smooth variation of parameters we change the flow from the one in Fig. 30 to the one in Fig. 31. Then, in order for the fixed point PI in Fig. 30 to shift to P2 in Fig. 31, it should go through the point where normal hyperbolicity breaks down. 

Equations similar to = -x+ /Jtanhx arise in statistical mechanical models of magnets and neural networks (see Exercise 3.6.7 and Palmer 1989). Show that this equation undergoes a supercritical pitchfork bifurcation as P is varied. Then give a numerically accurate plot of the fixed points for each p. 

We now see that a supercritical pitchfork bifurcation occurs at 7 = 1. It s left to you to check the stability of the fixed points, using linear stability analysis or graphical methods (Exercise 3.5.2). 

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. ... 

Figure 26. Skeleton bifurcation diagram in the t/-p parameter plane for the model equation (16). Shown are Hopf and saddle-node bifurcations (SUN = saddle-unstable-node bifurcation) as well as the border of the focus-node transition (dashed line) mixed-mode wave forms exist close to the dark region (which marks the region where a fixed point is a ShQ nikov saddle focus). The phase portraits sketch the Unear stability of the fixed point(s). (Reprinted with permission from M. T. M. Koper and P. Gaspard, J. Chem. Phys. 96, 7797, 1992. Copyright 1992, American Institute of Physics.)...

In this chapter, we describe an algorithm for predicting feasible splits for continuous single-feed RD that is not limited by the number of reactions or components. The method described here uses minimal information to determine the feasibility of reactive columns phase equilibrium between the components in the mixture, a reaction rate model, and feed state specification. This is based on a bifurcation analysis of the fixed points for a co-current flash cascade model. Unstable nodes ( light species ) and stable nodes ( heavy species ) in the flash cascade model are candidate distillate and bottom products, respectively, from a RD column. Therefore, we focus our attention on those splits that are equivalent to the direct and indirect sharp splits in non-RD. One of the products in these sharp splits will be a pure component, an azeotrope, or a kinetic pinch point the other product will be in material balance with the first. 

The two-dimensional (c,T) phase space for each of these systems can be partitioned into basins of attraction of the distinct attractors. Naturally, the basin structure is more complicated than that of the one-dimensional model studied in Sec. 3, but it is still quite simple. In the one-dimensional model the two stable fixed points were separated by an unstable fixed point, which formed a boundary separating the basins of attraction of the fixed points. In the GK model for the specified parameter setting the situation differs in that one of the stable fixed points has undergone a Hopf bifurcation to yield a stable limit cycle. The basin boundary in the two-dimensional space now consists of the unstable fixed point along with its stable manifolds these are shown in Fig. 1 and labelled Bi and B2. 

This equation is analogous to the equation yielding the coordinates of the fixed point which emerge from the saddle-node bifurcation (see (11.2.12)). So, one may verify that the equation has no real roots when /iZi > 0, but it has two roots of opposite signs when pli <0 namely ... 

The plane /io = 0 corresponds to the loss of stability of the fixed point at the origin. No other bifurcations occur with the fixed point at /io 0. To study the trajectories of period-two we need to examine the second iteration of the map... 

Thus, the bifurcation of a fixed point with one multiplier equal to —1, and with k — 1) zero Lyapunov values, cannot produce more than k orbits of period two. Moreover, it is easy to specify the precise parameter values for which Eq. (11.4.21) has a prescribed number of positive roots, within the range from 0 to k. This implies that in the parameter space of the map (11.4.16), there are regions where the family has any prescribed number (from 0 to k) of period-two orbits. 

If Li > 0, then when /i > 0, the fixed point is a saddle-focus of the above type, but its unstable manifold is the whole plane y = 0. Upon entering the region M < 0, the fixed point becomes stable. Meanwhile a saddle invariant curve C bifurcates from the fixed point its unstable manifold is (m -h 1)-dimensional and consists of the layers x — constant, restored at the points of the invariant curve. The stable manifold separates the attraction basin of the point O all trajectories from the inner region tend to O, and all those from outside of Wq leave a neighborhood of the origin. 

In the simplest case (say, when fo (p) = sin27ry with C < 1), the bifurcations proceed as follows (see Fig. 12.3.4) on some interval of p one of the fixed points (( o) is stable and another ( i) is unstable. Then at some value of p, the fixed point (po loses its stability and a stable orbit of period two is born. After that, in some interval of p, there exist two unstable fixed points and a stable orbit of period two then the stable orbit collapses into the fixed point (pi which becomes stable now, and then the bifurcation process is repeated the stable fixed point again loses its stability via period-doubling, and so on, infinitely many times without ever ending. 

This map is a monotonically increasing (since >1 > 0 for systems on the plane) one-dimensional map. The only possible bifurcation occurring in such maps is the bifurcation of a fixed point with a multiplier equal to -hi. The y-coordinate of such a point must satisfy... 

I C.6. 74. Find analytically the equations of the basic bifurcation curves of the fixed points and period-2 cycles of these maps. 

Figure C3.6.5 The first two periodic orbits in the main subhannonic sequence are shown projected onto the (c, C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cycle or period-1 orbit at k 2 = 0.11. (b) The first subhannonic or period-2 orbit at k 2 = 0.095.

Fig. 4.4 A schematic representation of the pitchfork bifurcation a stable fixed point bifurcates into a period-2 limit cycle plus an unstable fixed point.

Landau proposed in 1944 that turbulence arises essentially through the emergence of an ever increasing number of quasi-periodic motions resulting from successive bifurcations of the fluid system [landau44]. For small TZ, the fluid motion is, as we have seen, laminar, corresponding to a stable fixed point in phase space. As Ti is... 

To investigate the effect of composition at high pressures, two-parameter bifurcation diagrams are constructed. An example is shown in Fig. 26.2. Cuts at fixed compositions are shown in Fig. 26.1. A nonextinction regime is found on each side of the stoichiometric point, within which the flame cannot be... 

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. 

Among all of the points on the period 1 Hopf curve, some will have complex Floquet multipliers A with a phase angle 6 of mln)2u with n = 3 or 4 (i.e. third or fourth roots of unity) and are called hard reasonances. Because these points are fixed points for Fn that have multipliers equal to A" = 1, it is not surprising to find that subharmonic fixed points of period n are involved in addition to the bifurcating period 1 fixed point. 

To see what is happening when 9 passes through the bifurcation value 9 = 3, we examine the stability at the second iteration. The second iteration can be thought of as a first iteration in a model where the iterative time step is 2. The fixed points are solutions of... 

Two further bifurcations occur at r = r2 where both x and X4 become unstable. The third iterate of fr has to be consulted for an explanation of this behaviour. Since the third iterate is a polynomial of eighth degree, it has eight fixed points, four of which are identical to the fixed points of This leaves four fixed points which are visited by the iterates of Xn explaining the four branches of Xoo r) in the interval r2 [Pg.16]

---