Rough systems

Cyclone separators Often used as roughing systems in process industries before other forms of emission control equipment, seldom satisfactory as stand-alone equipment. 

The Lotka model is an example of a rough system deviations of concentrations from their asymptotic values (2.1.24) occur independently on chosen parameters p, K, (3, i.e., small variations of these parameters cannot affect the way a system strives for the equilibrium state. 

Structurally stable systems are not dense this is the title of Smale s study [41] that has opened a new period in understanding dynamics. Structurally stable (rough) systems are those whose phase patterns undergo no qualitative changes at small perturbations (for accurate definitions with comprehensive motivation, see ref. 11). Smale constructed such a structurally unstable system that any system sufficiently close to it is also structurally unstable. This result has destroyed any hope of the possibility of classifying "almost all dynamic systems. Such hopes were associated with the advance in the classification of two-dimensional systems, among which the structurally stable ones are dense. 

The multi-dimensional extension of two-dimensional rough systems is the Morse-Smale systems discussed in Sec. 7.4. The list of limit sets of such a system includes equilibrium states and periodic orbits only furthermore, such systems may only have a finite number of them. Morse-Smale systems do not admit homoclinic trajectories. Homoclinic loops to equilibrium states may not exist here because they are non-rough — the intersection of the stable and unstable invariant manifolds of an equilibrium state along a homoclinic loop cannot be transverse. Rough Poincare homoclinic orbits (homoclinics to periodic orbits) may not exist either because they imply the existence of infinitely many periodic orbits. The Morse-Smale systems have properties similar to two-dimensional ones, and it was presumed (before and in the early sixties) that they are dense in the space of all smooth dynamical systems. The discovery of dynamical chaos destroyed this idealistic picture. 

However, rough systems (both types — with simple and complex dynamics) with dimension (of the phase space) greater than two are not dense in the space of dynamical systems. In fact, it turns out that a key role must have been given to non-rough attracting limit sets with unstable behaviors in their trajectories. 

Surprisingly, even non-rough systems of codimension one may have infinitely many moduli. Of course, since the models of nonlinear dynamics are explicitly defined dynamical systems with a finite set of parameters, this creates a new obstacle which the classical bifurcation theory has not nm into. Although the case of homoclinic loops of codimension one does not introduce any principal problem, nevertheless codimensions two and higher are much less trivial as, for example, in the case of a homoclinic or heteroclinic cycle including a saddle-focus where the structure of the bifurcation diagrams is directly determined by the specific values of the corresponding moduli. 

Rough systems on a plane. Andronov-Pontryagin theorem... 

It follows from the above theorem that a rough system on the plane may possess only rough equilibrium states (nodes, foci and saddles) and rough limit cycles. As for separatrices of saddles, they either tend asymptotically to a node, a focus, or a limit cycle in forward or backward time, or leave the region G after a finite interval of time. 

Obviously, this picture is preserved under small smooth perturbations. Therefore, the rough systems form an open subset of Bq-... 

Moreover, it follows from simple arguments based on the rotation of a vector field to be presented below that, if is a non-rough system, then given any > 0 there exists a rough system X which is -close to X. In other words, the rough systems form a dense set in Bq-... 

It follows immediately from the Andronov-Pontryagin theorem that a rough system may possess only a finite number of equilibrium states and periodic orbits in G. 

Equilibrium states, periodic orbits and separatrices of saddles are special trajectories. Together they determine a scheme — a complete topological invariant (see Chap. 1 for details). One may easily conclude that all systems (5-close to a given rough system have the same scheme.,... 

Let us now explain why there are no separatrices which connect saddles in rough systems. 

Rough systems are also dense in the space of systems on two-dimensional orientable compact surfaces for which the necessary and sufficient conditions of roughness are analogous to those in the Andronov-Pontryagin theorem. The theory of such systems was developed by Peixoto [107]. The key element in this theory proves the absence of unclosed Poisson-stable trajectories in rough systems (they may be eliminated by a rotation of the vector field). 

Compared to the definition of rough systems, the above definition has an advantage it follows immediately that structurally stable systems form in... 

For rough systems on a plane, the Andronov-Pontryagin theorem gives a = 1. The case where a = 2 takes place in systems which has a loop of separatrix F to a saddle O, the loop is the limit trajectory for nearby orbits (see Fig. 7.2.1) and is non-wandering. Here, Aii = F U O. On the second step of the above procedure, one obtains Ai2 = O, i.e. the center of the region G is minimized to the equilibrium state. 

Since each point on a P-trajectory is non-wandering, this result is also valid for points stable in the sense of Poisson. The closing lenuna implies the following meaningful corollary a rough system with a P-trajectory possesses infinitely many periodic orbits. 

The modern theory of bifurcations of dynamical system is directly linked to the notion of non-roughness, or structural instability of a system. The main motivation is that the analysis of a system will be rather incomplete if we restrict our consideration to only the regions of structural stability of the system. Indeed, by changing parameters we can move from one structurally stable system to another, a qualitatively different system, upon crossing some boundaries in the parameter space that correspond to non-rough systems. 

In the two-dimensional case, rough systems compose an open and dense set in the space of all systems on a plane. The non-rough systems fill the boundaries between different regions of structural stability in this space. This nice structure allows for a mathematical description for transformations of... 

The analysis undertaken by Andronov and Leontovich suggests that the first-degree non-rough systems must have one of the following non-rough trajectories. 

To conclude this section, let us elaborate further on the restrictions (D) and (E). In case (D) the surface corresponding to the double cycle is of codimension-one, and therefore, it divides a neighborhood of the non-rough system Xq into two regions and D. Assume that in the double limit cycle is decomposed into two limit cycles, and that it disappears in D. The situation in -D is simple — all systems there are structurally stable and, moreover, of the same type. As for D the situation is less trivial if (D) is violated, then it is obvious that besides structurally stable systems in there are structurally unstable ones whose non-roughness is due to the existence of a heteroclinic trajectory between two saddles, as shown in Fig. 8.1.6(a). Moreover, this picture takes place in any neighborhood of Xq- In other words, in the region, there exists a countable number of the associated bifurcation surfaces of codimension-one which accumulate to In such cases the surface is said to be unattainable from one side. 

The study of a bifurcation means to describe the change in the phase portrait of a non-rough system in transition to an arbitrarily close system (with respect to some C -metric the choice of r depends on the character of nonroughness, and hence must be specified in each concrete case). 

The main idea of this approach is the following to a non-rough system X o some CO dimension k can be assigned. In the case of a finite degeneracy, the codimension k is identified with k equality-like conditions and a finite number of conditions of inequality type. Hence, Xeo is considered as a point on some Banach submanifold of codimension k in the space of dynamical systems. In other words, we have k smooth functionals defined in a neighborhood of Xeq whose zero levels intersect at B. In general, the inequality-like conditions secure the smoothness of B. In the case of codimension one Sotomayor [144, 145] had proved the smoothness of these functionals, and the smoothness of... 

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