Homoclinic orbits

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

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)). 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

Consider the space state model R deflned by Eq.(52), showing an equilibrium point such that the matrix of the linearized system at this point has a real negative eigenvalue A and a pair of complex eigenvalues a j/3, j = /—1) with positive real parts 0.. In this situation, the equilibrium point has onedimensional stable manifold and two-dimensional unstable manifold. If the condition A < a is verified, it is possible that an homoclinic orbit appears, which tends to the equilibrium point. This orbit is very singular, and then the Shilnikov theorem asserts that every neighborhood of the homoclinic orbit contains infinite number of unstable periodic orbits. 

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. 

According to the Shilnikov s theorem, the reactor presents a chaotic behavior. In order to test the presence of a strange attractor, it is necessary to raise the value of xe ax to introduce a perturbation in the vector field around the homoclinic orbit. Taking xemax = 5, the results of the simulation are shown in Figure 18, where the sensitive dependence on initial conditions has been corroborated. 

In regard dynamics and control scopes, the contributions address analysis of open and closed-loop systems, fault detection and the dynamical behavior of controlled processes. Concerning control design, the contributors have exploited fuzzy and neuro-fuzzy techniques for control design and fault detection. Moreover, robust approaches to dynamical output feedback from geometric control are also included. In addition, the contributors have also enclosed results concerning the dynamics of controlled processes, such as the study of homoclinic orbits in controlled CSTR and the experimental evidence of how feedback interconnection in a recycling bioreactor can induce unpredictable (possibly chaotic) oscillations. 

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

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.)...

We also have the hint of a new type of degeneracy associated with systems possessing multiple stationary states. It is possible for both the trace and the determinant of the Jacobian matrix to become zero simultaneously this gives the system two eigenvalues which are both equal to zero. These double-zero eigenvalue situations are important because they represent conditions at which a Hopf bifurcation point with an associated homoclinic orbit first appears. In the present case, tr(J) = det(J) = 0 only when k2 = Vg, but then the isola has shrunk to a point. 

Kaas-Petersen, C. and Scott, S. K. (1988). Homoclinic orbits in a simple model of an autocatalytic reaction. Physica, D 32, 461-70. 

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...

The double-zero eigenvalue points, such as M, represent the coalescence of Hopf bifurcation and stationary-state turning points. As mentioned above, they thus represent the points at which the Hopf bifurcation loci begin and end. They also have other significance. Such points correspond to the beginning or end of loci of homoclinic orbits. For the present model, with the given choices of k1 and k2, there are two curves of homoclinic orbit points, one connecting M to N, the other K to L, as shown schematically in Fig. 12.7. 

Fig. 12.7. Curves of homoclinic orbits.for Takoudis-Schmidt-Aris model with k, = 10 3 and k2 = 2 x lO 3 see text for details. (Adapted and reprinted with permission from McKarnin, M. A. et al. (1988). Proc. R. Soc., A4I5, 363-87.)...

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. 

Chow, S. N., Hale, J. K. Mallet-Paret, J. 1980 An example of bifurcation to homoclinic orbits. J. diffl Equal. 37, 351-373. 

Bifurcation diagram for equations (7.198) and (7.199) with one Hopf bifurcation point, two periodic limit points and one homoclinical orbit (infinite period bifurcation point)... 

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. 

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. 

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... 

The neutrally stable equilibria correspond to the particle at rest at the bottom of one of the wells, and the small closed orbits represent small oscillations about these equilibria. The large orbits represent more energetic oscillations that repeatedly take the particle back and forth over the hump. Do you see what the saddle point and the homoclinic orbits mean physically ... 

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