Saddle chaotic

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. 

H. Waalkens, A. Burbanks, and S. Wiggins, A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill s problem, J. Phys. A 37, L257 (2004). 

Figure 3. The complexity of nodal lines, nodal points and saddles for the transmission through chaotic (Sinai) (left) and regular billiard (right). 

Instead of nodal lines in closed systems we are interested in the statistics of NPs for open chaotic billiards since they form vortex centers and thereby shape the entire flow pattern (K.-F. Berggren et.al., 1999). Thus we will focus on nodal points and their spatial distributions and try to characterize chaos in terms of such distributions. The question we wish to ask is simply if one can find a distinct difference between the distributions for nominally regular and irregular billiards. The answer to this question is clearly positive as it is seen from fig. 3. As shown qualitatively NPs and saddles are both spaced less regularly in chaotic billiard in comparison to the integrable billiard. The mean density of NPs for a complex RGF (9) equals to k2/A-k (M.V. Berry et.al., 1986). This formula is satisfied with good accuracy in both chaotic and integrable billiards. 

Figure 4 Distributions for separations between the nearest distances nearest NPs, saddle points, NPs with the same (++) and opposite winding numbers (+-) in a chaotic Sinai billiard. The radial distribution of nearest distances for completely random points (26) is shown by the dashed curve in (a). The corresponding distribution for the Berry model function for a chaotic state (2) and random superposition of 16 eigen functions for a rectangular box with the same size and energy are shown by dots and thin curves, respectively.

Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.

Besides the two most well-known cases, the local bifurcations of the saddle-node and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297 301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302 304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 

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]. 

While the conditions 1,2 can be verified approximately by simulation, proving the condition 3 is very difficult. Note that in many studies of chaotic behavior of a CSTR, only the conditions 1,2 are verified, which does not imply chaotic d3mamics, from a rigorous point of view. Nevertheless, the fulfillment of conditions 1,2, can be enough to assure the long time chaotic behavior i.e. that the chaotic motion is not transitory. From the global bifurcations and catastrophe theory other chaotic behavior can be considered throughout the disappearance of a saddle-node fixed point [10], [19], [26]. 

Subsidiary elliptic islands of very small area continue to exist until a last homoclinic tangency occurs at Eht, above which all the trapped orbits of the invariant set are unstable of saddle type. The system is then fully chaotic. According to this scenario, the invariant set may contain quasiperi-odic motion for energies Ea < E < Eht, while the main elliptic island exists only for Ea < E < Ed < Eh,- The interval /, - Ea turns out to be small as compared with the energy interval above Eht, where full chaos has set in and the invariant set is a repeller. 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. 

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

In the presence of weak noise there is a finite probability of noise-induced transitions between the chaotic attractor and the stable limit cycle. In Fig. 14 the filled circles show the intersections of one of the real escape trajectories with the given Poincare section. The following intuitive escape scenario can be expected in the Hamiltonian formalism. Let us consider first the escape of the system from the basin of attraction of a stable limit cycle that is bounded by an saddle cycle. In general, escape occurs along a single optimal trajectory qovt(t) connecting the two limit cycles. 

Since the basin of attraction of the CA is bounded by the saddle cycle SI, the situation near SI remains qualitatively the same and the escape trajectory remains unique in this region. However, the situation is different near the chaotic attractor. In this region it is virtually impossible to analyze the Hamiltonian flux of the auxiliary system (37), and no predictions have been made about the character of the distribution of the optimal trajectories near the CA. The simplest scenario is that an optimal trajectory approaching (in reversed time) the boundary of a chaotic attractor is smeared into a cometary tail and is lost, merging with the boundary of the attractor. 

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... 

The saddle cycles L and L2 surround the stable states P and P2 and are located at the intersection of the unstable Wu and stable Ws manifolds. The unstable manifold goes to the stable state P from one side and to the chaotic attractor from the other side. The stable manifold Ws forms a tube in the vicinity of the stable state [183]. The saddle cycles L and L2 have the multipliers (1.0000,1.0280,0.0001), and therefore trajectories will go slowly away along the unstable manifold, and they will approach quickly along the stable manifold. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

Secondly, the chaotic state of a resonant system is an infinite collection of unstable periodic motions (see Figure 14.9). The basic idea of the controlling chaos is to switch a chaotic system by a small perturbation among many different periodic orbits. Ott, Grebogy and Yorke [10] developed a scheme in which a chaotic system can be forced to follow one particular unstable periodic orbit. The problem is to calculate the perturbation that will shift the system towards the desired periodic orbit. This process is similar to balancing a marble on a saddle. To keep the marble from rolling off, one needs to move the saddle quickly from side to side. And... 

The jS-cell model displays chaotic dynamics in the transition intervals between periodic spiking and bursting and between the main states of periodic bursting. A careful description of the bifurcation diagram involves a variety of different transitions, including Hopf and saddle-node bifurcations, period-doubling bifurcations, transitions to inter-mittency, and homoclinic bifurcations. 

X.-J.Wang Genesis of bursting oscillations in the Hindmarsh-Rose model and ho-moclinicity to a chaotic saddle. Physica D 1993, 62 263-274. 

One may wonder if this expansion is possible for chaotic orbits, since the distance between the original orbit and the one under the perturbation increases exponentially in time for chaotic evolution. We will present an intuitive answer to this question. For example, the distance between XQ t) and a (f) increases exponentially for f oo. Then, the perturbative analysis will break down for this time interval. On the other hand, their distance shrinks exponentially for t —cxD, since they go to the same saddle. Similar reasoning would also hold for the stable orbit a (f). Thus, for some time to, the perturbative analysis will be applicable to Xu for f S [—oo, fo] and to Xg for f [fo, co], respectively. 

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