Phase space bifurcations

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... 

Leaving the details, the equation describing the motion of one particle in two electrostatic waves allows perturbation methods to be applied in its study. There are three main types of behavior in the phase space - a limit cycle, formation of a non-trivial bounded attracting set and escape to infinity of the solutions. One of the goals is to determine the basins of attraction and to present a relevant bifurcation diagram for the transitions between different types of motion. 

Some typical oscillatory records are shown in Fig. 4.6. For conditions close to the Hopf bifurcation points the excursions are almost sinusoidal, but this simple shape becomes distorted as the oscillations grow. For all cases shown in Fig. 4.6, the oscillations will last indefinitely as we have ignored the effects of reactant consumption by holding /i constant. We can use these computations to construct the full envelope of the limit cycle in /r-a-0 phase space, which will have a similar form to that shown in Fig. 2.7 for the previous autocatalytic model. As in that chapter, we can think of the time-dependent... 

This study [14] has shown that a period-doubling bifurcation associated with the Fermi resonance occurs in this subsystem at the energy E = 3061.3 cm-1 (with Egp = 0). Below the Fermi bifurcation, there exist edge periodic orbits of normal type, which are labeled by ( i, 22 -)normai- At the Fermi bifurcation, a new periodic orbit of type (2,1°, ->Fermi appears by period doubling around a period of 2T = 100 fs. This orbit is surrounded by an elliptic island that forms a region of local modes in phase space. Therefore, another family of edge periodic orbits of local type are bom after the Fermi bifurcation that may be labeled by the integers (n n , -)iocai- They are distinct... 

There are also two local bifurcations. The first one takes place for r 13.926..., when a homoclinic tangency of separatrixes of the origin O occurs (it is not shown in Fig. 20) and a hyperbolic set appears, which consists of a infinite number of saddle cycles. Beside the hyperbolic set, there are two saddle cycles, L and L2, around the stable states, Pi and P2. The separatrices of the origin O reach the saddle cycles Li and L2, and the attractors of the system are the states Pi and P2. The second local bifurcation is observed for r 24.06. The separatrices do not any longer reach to the saddle cycles L and L2. As a result, in the phase space of the system a stable quasihyperbolic state appears— the Lorenz attractor. The chaotic Lorenz attractor includes separatrices, the saddle point O and a hyperbolic set, which appears as a result of homoclinic tangency of the separatrices. The presence of the saddle point in the chaotic... 

For a given value of a, the brute force bifurcation diagram displays all the values of the relative arteriolar radius r that the model displays when the steady state trajectory intersects a specified hyperplane (the Poincare section) in phase space. Due to the coexistence of several stable solutions, the brute force diagram must be obtained by scanning a in both directions. 

In 1992 Wintgen and Friedrich reviewed the status of semiclassical methods in connection with the DKP. Traditional semiclassical methods are not applicable in the vicinity of bifurcations in phase space. An improved semiclassical theory, applicable at bifurcation points, is discussed and experimentally tested by Courtney et al. (1995). 

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