Topological fixing points

Data on the fractal forms of macromolecules, the existence of which is predetermined by thermodynamic nonequilibrium and by the presence of deterministic order, are considered. The limitations of the concept of polymer fractal (macromolecular coil), of the Vilgis concept and of the possibility of modelling in terms of the percolation theory and diffusion-limited irreversible aggregation are discussed. It is noted that not only macromolecular coils but also the segments of macromolecules between topological fixing points (crosslinks, entanglements) are stochastic fractals this is confirmed by the model of structure formation in a network polymer. 

The Dimension of the Sections of a Macromolecule Between Topological Fixing Points... 

Stability studies of the steady-state solutions of the TRAM are in general very complex and beyond the scope of this Report. The more successful approaches have involved the use of Liapunov functions, " of collocation methods, or of topological fixed-point methods. The genoation of r ons of stability inevitably involves a considerable amount of computational effort. For a full discussion of these methods the reader is again referred to Perlmutter s book. ... 

Hyperbolic Fixed Points, Topological Equivalence, and Structural Stability... 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

Hyperbolic fixed points also illustrate the important general notion of structural stability. A phase portrait is structurally stable if its topology cannot be changed by an arbitrarily small perturbation to the vector field. For instance, the phase portrait of a saddle point is structurally stable, but that of a center is not an arbitrarily small amount of damping converts the center to a spiral. 

In this broader context, what exactly do we mean by a bifurcation The usual definition involves the concept of topological equivalence (Section 6.3) if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred. Examples include changes in the number or stability of fixed points, closed orbits, or saddle connections as a parameter is varied. 

The Newtonian gravitational force is the dominant force in the N-Body systems in the universe, as for example in a planetary system, a planet with its satellites, or a multiple stellar system. The long term evolution of the system depends on the topology of its phase space and on the existence of ordered or chaotic regions. The topology of the phase space is determined by the position and the stability character of the periodic orbits of the system (the fixed points of the Poincare map on a surface of section). Islands of stable motion exist around the stable periodic orbits, chaotic motion appears at unstable periodic orbits. This makes clear the importance of the periodic orbits in the study of the dynamics of such systems. 

The fixed points on the phase space diagrams or phase spheres in Fig. 9.13 are labeled A, B, Ca, and C. Each corresponds to a periodic orbit that is said to organize the surrounding region of phase space that is filled with topologically similar quasiperiodic trajectories. 

Each trajectory is launched at chosen initial values of Jb and ipb and at fra = 0. Since any point on the 3-dimensional energy shell may be specified by three linearly independent coordinates, selection of initial values J , ip%, and ip°, implies a definite value of J°. Thus trajectories are launched at various [Jfi, ip%, ip° = 0, J°(J , ipl, ip°,) ] initial values until all of the qualitatively distinct regions of phase space are represented on the surface of section by either a family of closed curves (quasiperiodic trajectories) that surround a fixed point (a periodic trajectory that defines the qualitative topological nature of the neighboring quasiperiodic trajectories) or an apparently random group of points (chaos). Often, color is used to distinguish points on the surface of section that belong to different trajectories. 

The fractal dimension D of a chain fragment between the points of topological fixing (entanglements, clusters, crosslinks) is an important structural parameter, which controls the molecular mobility and deformability of polymers. Crucial factors accounting for the use of the dimension D are clearly defined limits of variation (1super-molecular structure of the polymer. It should be emphasised that all fractal relations contain at least two variables. 

Topological Methods. The topological properties of the stirred-reactor equations (37) can be used to predict the occurrence of multiple states and to determine their stability. More sophisticated tediniques can be enqiloyed in determining the oscillatory nature and limit-cycle bdiaviour of such systems. The introduction of topological methods in the study of chemical reactors was made by Oavalas in 196S fixed-point methods were introduced in the study of thermodynamically... 

Analytical studies of the TRAM have profited greatly from topological studies of the structure of the reactor equations. Luss and Amundson obtained a sufBcient condition for uniqueness of the steady state in adiabatic reactors by the use of fixed-point methods in deriving a linearized eigenvalue problem. Gavalas and Luss produced equivalent criteria for adiabatic TRAM and McGowin and Perlmutter and Han and Agrawal extended these results to non-adiabatic reactors. 

In Section 1.4 the topological concept of rotation was used to prove the existence of equilibrium states. When the reaction kinetics are not restricted by Postulate 1.5.1, each invariant manifold may include more than one equilibrium states and it is interesting to obtain information about the number and stability of these states. In the present section we shall use one more topological concept, the index of a fixed point, to show that the equilibrium states are odd in number, 2m+1, among which m at least are unstable. As in the preceding sections, the discussion concerns isolated systems, but extension to other closed systems should not present difficulties. 

This result is due to Palis, who had fotmd that two-dimensional diffeomor-phisms with a heteroclinic orbit at whose points an unstable manifold of one saddle fixed point has a quadratic tangency with a stable manifold of another saddle fixed point can be topologically conjugated locally only if the values of some continuous invariants coincide. These continuous invariants are called moduli. Some other non-rough examples where moduli of topological conju-gacy arise are presented in Sec. 8.3. 

Recall that a fixed point 0 x = xq) is called structurally stable if none of its characteristic multipliers, i.e. the roots of the characteristic equation (7.5.2), lies on the unit circle, A topological type (m,p) is assigned to it, where m is the number of roots inside the unit circle and p is that outside of the unit circle. If m = n (m = 0), the fixed point is stable (completely unstable). The fixed point is of saddle type when m 0,n. The set of all points whose trajectories converge to xq when iterated positively (negatively) is called the stable (unstable) manifold of the fixed point and denoted by Wq Wq). In the case where m = n, the attraction basin of O is Wq. If the fixed point is a saddle, the manifolds Wq and Wq are C -smooth embeddings of and MP in respectively. 

Fig. 10.5.4. Topology of the stable and unstable manifold of a resonant tt/3 fixed point in...

C.3. 21. Prove that if the origin is a structurally stable equilibrium state of the system (C.3.3), then the corresponding fixed point of the map (C.3.2) is structurally stable as well. Furthermore, show that the topological types of the equilibrium state of (C.3.3) and the fixed point of (C.3.2) are the same. ... 

Therefore, for all small fi the roots a will be close to those of (C.3.10). Thus, the fixed point will be structurally stable. Moreover, it has the same topological type as the equilibrium state of the averaged system. ... 

Theory of Bifurcations.—In the preceding sections we have reviewed a few more important points of the existing topological methods in the theory of oscillation, assuming that the topological configuration or the phase portrait remains fixed. 

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