Homoclinic cycles

In this case the limit of a periodic trajectory as e —0 is a homoclinic cycle r composed of a simple saddle-node equilibrium state and its... 

If p, u > 0 —A > 0 and —A/p > 1 the equilibrium point is unstable, and a Shilnikov orbit may appear. For the reactor, with a value of X50 > 1 and X6max x6max)M (see Figure 15), by simulation it is possible to verify the presence of a homoclinic orbit to the equilibrium point. Figure 17 shows the homoclinic orbit for the model and R, when the steady state has been reached. Note that the Shilnikov orbit appear when the coolant flow rate is constrained. If there is no limitation of the coolant flow rate, a limit cycle is obtained both in models R and R, by simulation. 

The point at which the homoclinic orbit is formed must be calculated numerically, but once it has been located we can show that the limit cycle is still unstable as it approaches the loop formation. We do this by evaluating the trace of the Jacobian matrix for the saddle point solution a2, P2 corresponding to Tres if tr(J) is positive, the limit cycle is unstable (as we always find for this special case of the present model) if tr( J) is negative for the saddle... 

Fig. 8.6. Typical arrangement of local stabilities and development of unstable limit cycle, from a subcritical Hopf bifurcation, appropriate to cubic autocatalysis with decay and no autocatalyst inflow and with 9/256 < k2 < 1/16. The unstable limit cycle grows as t, decreases below t m, and terminates by means of the formation of a homoclinic orbit at rf . Stable stationary states, including the zero conversion branch 1 — a, = 0, are indicated by solid curves, unstable states and limit cycles by broken curves.

Fig. 8.7. Supercritical Hopf bifurcation for cubic autocatalysis with decay and /) = 0, appropriate for small dimensionless decay rate constant k2 < 9/256. A stable limit cycle emerges and grows as the residence time is increased above t s. At higher residence times, this disappears at rj , by merging with an unstable limit cycle born from a homoclinic orbit at t. (With non-zero autocatalyst inflow, (i0 > 0, the stable limit cycle itself may form a homoclinic orbit at long tres.)...

In some special cases, the two separatrices join up to give a loop which corresponds, in fact, to the formation of a homoclinic orbit. A limit cycle may... 

Fig. 8.8. Phase plane representations of the birth (or death) of limit cycles through homoclinic orbit formation. In the sequence (a)-fb)-(c) the system has two stable stationary states (solid circles) and a saddle point. As some parameter is varied, the separatrices of the saddle join together to form a closed loop or homoclinic orbit (b) this loop develops as the parameter is varied further to shed an unstable limit cycle surrounding one of the stationary states. The sequence (d)-(e)-(f) shows the corresponding formation of a stable limit cycle which surrounds an unstable stationary state. (In each sequence, the limit cycle may ultimately shrink on to the stationary state it surrounds—at a Hopf bifurcation point.)...

Fig. 12.4. Stationary-state solutions and limit cycles for surface reaction model in presence of catalyst poison K3 = 9, k2 = 1, k3 = 0.018. There is a Hopf bifurcation on the lowest branch p = 0.0237. The resulting stable limit cycle grows as the dimensionless partial pressure increases and forms a homoclinic orbit when p = 0.0247 (see inset). The saddle-node bifurcation point is at...

FIGURE 2 The birth and growth of limit cycle oscillations in the I - a, jS, Tr space for a system with non-zero e and k displaying a mushroom stationary-state pattern. Oscillatory behaviour originates from a supercritical Hopf bifurcation along the upper branch and terminates via homoclinic orbit formation. 

FIGURE I Steady-state bifurcation diagrams for variations in the reactant partial pressures, (a) Partial two-parameter bifurcation diagram representing the projection of turning points, Hopf bifurcation points, and apparent triple points. (b)-(Q One parameter sections of the steady-state ffe surface. The vertical axes are the steady-state (k and range from 0 to I. The horizontal axes correspond to the appropriate axis of the two parameter diagram (a). Steady-states are stable or unstable for solid or dashed curves respectively and periodic branches are denoted by pairs of chained curves which represent the minimum and maximum values of ffe on the limit cycle. The periodic branches all terminate in Hopf bifurcations or, when a saddle is present, homoclinic (infinite period) bifurcations. (b)-(e) a, = 0.017, 0.019, 0.021, 0.025 (f)-(i) oti = 0.031, 0.028, 0.024, 0.022. 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

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... 

Figure 12 (A-2) shows one Hopf bifurcation point, one periodic limit point and the stable limit cycle terminates at a homoclinical orbit (infinite period bifurcation).

This limit cycle represents a trajectory that starts at the static saddle point and ends after one period at the same saddle point. This trajectory is called the homoclinical orbit and will occur at some critical value jiuc- It has an infinite period and therefore this bifurcation point is called infinite period bifurcation . For p < hc the limit cycle disappears. This is the second most important type of dynamic bifurcation after Hopf bifurcation. 

Limit cycles (periodic solutions) emerging from the Hopf bifurcation point and terminating at another Hopf bifurcation point or at a homoclinical orbit (infinite period bifurcation point) represent the highest degree of complexity in almost all two- dimensional autonomous systems. 

If the x-motion is whole, not P+ -stable [x ca(x)] and a(x) (°) co(x) 4>> then the x-trajectory will be called a loop. An example of a loop on the plane is a loop of the sepatrix that is a trajectory going from a singular point and back to it. Another example is a homoclinic trajectory for which the same saddle limit cycle is both a a-limit and an co-limit set. 

The bursting dynamics ends in a different type of process, referred to as a global (or homoclinic) bifurcation. In the interval of coexisting stable solutions, the stable manifold of (or the inset to) the saddle point defines the boundary of the basins of attraction for the stable node and limit cycle solutions. (The basin of attraction for a stable solution represents the set of initial conditions from which trajectories asymptotically approach the solution. The stable manifold to the saddle point is the set of points from which the trajectories go to the saddle point). When the limit cycle for increasing values of S hits its basin of attraction, it ceases to exist, and... 

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

Fig. 5.13. Formation of a stable limit cycle about an unstable steady-state through a homoclinic orbit as the inset and outset of a saddle point merge.

In this scenario, part of a limit cycle moves closer and closer to a saddle point. At the bifurcation the cycle touches the saddle point and becomes a homoclinic or-... 

Our goal now is to visualize the corresponding bifurcation in phase space. In Exercise 8.5.2, you re asked to show (by numerical computation of the phase portrait) that if a is sufficiently small, the stable limit cycle is destroyed in a homoclinic bifurcation (Section 8.4). The following schematic drawings summarize the results you should get. 

As I decreases, the stable limit cycle moves down and squeezes U closer to the stable manifold of the saddle. When / = /<., the limit cycle merges with t/ in a homoclinic bifurcation. Now f/ is a homoclinic orbit—it joins the saddle to itself (Figure 8.5.8). 

Consider the driven pendulum + atj) + sintl) = I. By numerical computation of the phase portrait, verify that if a is fixed and sufficiently small, the system s stable limit cycle is destroyed in a homoclinic bifurcation as / decreases. Show that if a is toolarge, the bifurcation is an infinite-period bifurcation instead. 

Now for the new results. As we decrease r from, the unstable limit cycles expand and pass precariously close to the saddle point at the origin. At r 13.926 the cycles touch the saddle point and become homoclinic orbits hence we have a homoclinic bifurcation. (See Section 8.4 for the much simpler homoclinic bifurcations that occur in two-dimensional systems.) Below r= 13.926 there are no limit cycles. Viewed in the other direction, we could say that a pair of unstable limit cycles are created as r increases through r = 13.926. 

Quasiperiodicity is significant because it is a new type of long-term behavior. Unlike the earlier entries (fixed point, closed orbit, homoclinic and heteroclinic orbits and cycles), quasiperiodicity occurs only on the torus. 

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