Non-roughness system

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. 

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-... 

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... 

Closed trajectories around the whirl-type non-rough points cannot be mathematical models for sustained self-oscillations since there exists a wide range over which neither amplitude nor self-oscillation period depends on both initial conditions and system parameters. According to Andronov et al., the stable limit cycles are a mathematical model for self-oscillations. These are isolated closed-phase trajectories with inner and outer sides approached by spiral-shape phase trajectories. The literature lacks general approaches to finding limit cycles. 

The majority of the above examples are non-rough (structurally unstable) systems. The rough dynamic systems on the plane cannot demonstrate the properties shown by the above examples. If Tt is specified by a rough individual (without parameters) system on the plane, there cannot exist th, rj2 slow relaxations and rh 2,3 and tj3 slow relaxations can take place only simultaneously. This can be confirmed by the results given below and the data of some classical studies concerning smooth rough two-dimensional systems [20, 21],... 

At present, when working with high-purity materials, smooth solid surfaces and low P02 atmospheres, the thermodynamic contact angle in a particular system can be determined at best within about five degrees. Roughness must be very low, particularly in non-wetting systems in order to obtain such an accuracy. The control of P02 is critical for oxidisable liquids and solids, specially at relatively low temperatures, and dynamic vacuum is often preferable to a static neutral gas atmosphere. 

In some cases (30% as a rough cstimale), we have observed, from electron microscopy evidence, a polymerization of styrene miniemulsions during or after ultrasonification. These miniemulsions showed an extraordinary stability during storage when compared to non-polymerised systems. 

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. 

The fundamental question of what distinguishes systems with simple dynamics from systems with chaotic dynamics can only be answered if we can correspond certain types of trajectories to physically observable processes. We began the classification with the study of quasiperiodic trajectories (Chap. 4 in the first part of this book). Even though these trajectories are non-rough, they were shown to model adequately such phenomena as beats and modulations. 

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. 

Another typical codimension-one bifurcation (left untouched in this book) within the class of Morse-Smale systems includes the so-called saddle-saddle bifurcations, where a non-rough saddle equilibrium state with one zero characteristic exponent (the others lie in both left and right half-planes) coalesces with another saddle having a different topological type. If, in addition, the stable and unstable manifolds of the saddle-saddle point intersect each other transversely along some homoclinic orbits, then as the bifurcating point disappears, saddle periodic orbits are born from the homoclinic loops. If there is only one homoclinic loop, then only one periodic orbit is born from it, and respectively, this bifurcation does not lead the system out of the Morse-Smale class. However, if there are more than one homoclinic loops, a hyperbolic limit set with infinitely many saddle periodic orbits will appear after the saddle-saddle vanishes [135]. 

Similarly, if there were a separatrix loop to a saddle at // = 0, it would be split for some non-zero /i, as shown in Fig. 7.1.2. We see that an arbitrarily small smooth perturbation of the vector field will modify the phase portrait of a system with a homoclinic loop or a heteroclinic connection this obviously means that such a system is non-rough. 

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. 

We remark that an equilibrium state, or a periodic orbit, may be arbitrarily degenerate. It is therefore logical to begin our study with the simplest structurally unstable systems which Andronov and Leontovich called systems of first degree of non-roughness. 

In essence, systems of first degree of non-roughness are structurally stable in the set of structurally unstable systems. 

Altogether the above requirements comprise the list of necessary and sufficient conditions which a system with first-degree of non-roughness must satisfy. 

Therefore, to study the transition from D to Z>, it is sufficient to examine a one-parameter family of systems such that XyxQ G D. Furthermore, since all qualitative changes in the phase portrait must occur in a small neighborhood of some non-rough special trajectory, we can restrict our consideration to the given neighborhood. 

This is the reason why the classification of principal bifurcations in multidimensional systems is not stated in terms of the degree of non-roughness, but it rather focuses on bifurcation sets of codimension-one. 

All non-rough two-dimensional systems in a small neighborhood of a system with first-order of non-roughness are now known to form a surface of codimension-one. Moreover, due to Leontovich and Mayer, we know that all of them are identical in the sense that they have an identical topological... 

Consider some finite-parameter family of smooth systems Xg, where e = ( 1,..., 6p) assumes its values from some region V e W. If is non-rough, then q is said to be a bifurcation parameter value. The set of all such values in V is called a bifurcation set. It is obvious that once we know the bifurcation set, we can identify all regions of structural stability in the parameter space. Hence, the first step in the study of a model is identifying its bifurcation set. This emphasizes a special role of the theory of bifurcations among all tools of nonlinear dynamics. 

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