A homoclinic

What cannot be obtained through local bifurcation analysis however, is that both sides of the one-dimensional unstable manifold of a saddle-type unstable bimodal standing wave connect with the 7C-shift of the standing wave vice versa. This explains the pulsating wave it winds around a homoclinic loop consisting of the bimodal unstable standing waves and their one-dimensional unstable manifolds that connect them with each other. It is remarkable that this connection is a persistent homoclinic loop i.e. it exists for an entire interval in parameter space (131. It is possible to show that such a loop exists, based on the... 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

The system (7) with e = 0 is referred as unperturbed system. About it we shall assume that it possesses a hyperbolic fixed point xQyh connected to itself by a homoclinic orbit Xh(t) = x (t), x (t)). 

In this situation, a periodic variation of coolant flow rate into the reactor jacket, depending on the values of the amplitude and frequency, may drive to reactor to chaotic dynamics. With PI control, and taking into account that the reaction is carried out without excess of inert (see [1]), it will be shown that it the existence of a homoclinic Shilnikov orbit is possible. This orbit appears as a result of saturation of the control valve, and is responsible for the chaotic dynamics. The chaotic d3mamics is investigated by means of the eigenvalues of the linearized system, bifurcation diagram, divergence of nearby trajectories, Fourier power spectra, and Lyapunov s exponents. 

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 non-linear dynamics of the reactor with two PI controllers that manipulates the outlet stream flow rate and the coolant flow rate are also presented. The more interesting result, from the non-linear d mamic point of view, is the possibility to obtain chaotic behavior without any external periodic forcing. The results for the CSTR show that the non-linearities and the control valve saturation, which manipulates the coolant flow rate, are the cause of this abnormal behavior. By simulation, a homoclinic of Shilnikov t3rpe has been found at the equilibrium point. In this case, chaotic behavior appears at and around the parameter values from which the previously cited orbit is generated. 

The critical nucleus, which can be found explicitly, is described by a homoclinic trajectory of the Euler-Lagrange equation ew = g w) (see, for instance Bates and Fife, 1993). The fact that this perturbation plays a role of a threshold is clear from Fig.9 which demonstrates extreme sensitivity of the problem to slight variations around the critical nucleus representing particular initial data (see Ngan and Truskinovsky (1996b) for details). 

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. 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...

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).

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. 

Also interesting is the dynamical behavior associated with the fixed point at infinity, that is, q,p) = (oo,0). Here we introduce the concept of homoclinic orbit, which is a trajectory that goes to an unstable fixed point in the past as well as in the future. A homoclinic orbit thus passes the intersection between the unstable and stable manifolds of a particular fixed point. Indeed, as shown in Fig. 6, these manifolds generate a so-called homoclinic web. In particular. Fig. 6a displays a Smale horseshoe giving a two-symbol subdynamics, indicating that the fixed point (oo,0) is not a saddle. Nevertheless, it is stUl unstable with distinct stable and unstable manifolds, with its dynamics much slower than that for a saddle. Figure 6b shows an example of a numerical plot of the stable and unstable manifolds. 

Here, we limit our argument to a system with a homoclinic connection—that is, a separatrix connecting a saddle with itself. The following argument can be straightforwardly extended to a system with a heteroclinic connection— that is, a separatrix connecting different saddles. 

Recently Tambe et al. (284) extended this model and included two different types of adsorption sites for A and B, while permitting the conversion of sites from one type to the other. The authors used the same coverage dependency and the same parameters as Pikios and Luss (283). Introducing the possibility of adsorption on different sites generated a qualitatively new dynamic behavior for the system characterized by a finite amplitude/ infinite period bifurcation that yielded a homoclinic orbit. This new feature was observed when the equilibration between the two types of sites was slow compared to the other reactions. However, if equilibration is fast and the equilibrium constant is assumed to be one, this model is equivalent to the one discussed by Pikos and Luss (283). 

An important event in the phase plane occurs if the inset to a saddle manages to join up with an outset from the same saddle point. This then gives rise to a closed loop with the saddle point lying on it as a corner . Such a loop is known as a homoclinic orbit as it forms a path connecting... 

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.

Thus solutions of the system are typically periodic, except for the equilibrium solutions and two very special trajectories these are the trajectories that appear to start and end at the origin. More precisely, these trajectories approach the origin as t . Trajectories that start and end at the same fixed point are called homoclinic orbits. They are common in conservative systems, but are rare otherwise. Notice that a homoclinic orbit does not conespond to a periodic... 

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). 

Homoclinic bifurcation) Using numerical integration, find the value of at which the system. i = //x + y - x, y = -x + fiy + 2x undergoes a homoclinic bifurcation. Sketch the phase portrait just above and below the bifurcation. 

Scaling near a homoclinic bifurcation) To find how the period of a closed orbit scales as a homoclinic bifurcation is approached, we estimate the time it takes for a trajectory to pass by a saddle point (this time is much longer than all others in the problem). Suppose the system is given locally by x x, y = -2, y. Let a trajectory pass through the point (/i,l), where f.i 1 is the distance from the stable manifold. How long does it take until the trajectory has escaped from the saddle, say out to x t 1 (See Gaspard (1990) for a detailed discussion.)... 

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. 

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