Persistent dynamical system

Recently, Wiggins et al. [15] provided a firm mathematical foundation of the robust persistence of the invariant of motion associated with the phase-space reaction coordinate in a sea of chaos. The central component in RIT that is, unstable periodic orbits, are naturally generalized in many DOFs systems in terms of so-called normally hyperbolic invariant manifold (NHIM). The fundamental theorem on NHIMs, denoted here by M, ensures [21,53] that NHIMs, if they exist, survive under arbitrary perturbation with the property that the stretching and contraction rates under the linearized dynamics transverse to jM dominate those tangent to M. Note that NHIM only requires that instability in either a forward or backward direction in time transverse to M is much stronger than those tangential directions of M, and hence the concept of NHIM can be applied to any class of continuous dynamical systems. In the case of the vicinity of saddles for Hamiltonian problems with many DOFs, the NHIM is expressed by a set of all (p, q) satisfying both q = p = Q and o(Jb) + En=i (Jb, b) = E, that is. 

System (2.4) is the one that will receive most of the analysis. Several of the results in the appendices will be used the theory of monotone systems and the persistence results will be particularly useful. It is generally not possible to analyze a four-dimensional system such as (2.4) because the dynamics can be very complicated indeed, they can be chaotic. One must work very hard, using the theory developed, to show rigorously that the dynamics are, in fact, very simple. From the standpoint of dynamical systems, this is extraordinary luck from the standpoint of the biology, it is expected. What is new, biologically, is that coexistence is possible and the competition uncomplicated. 

The assumption of the invariance of the boundary is stronger than needed, and the assumption that the space is locally compact can be removed at the expense of further assumptions on the dynamical system. The combination of dissipative and uniform persistence allows the use of fixed-point theorems. The following is sufficient for our applications in K". 

BW] G. Butler and P. Waltman (1986), Persistence in dynamical systems, Journal of Differential Equations 63 255-63. 

JST] W. Jager, H. Smith, and B. Tang (1991), Some aspects of competitive coexistence and persistence, in S. Busenberg and M. Martelli (eds.), Delay Differential Equations and Dynamical Systems. Berlin Springer, pp. 200-9. 

Hutson, V. Moron, W. (1985). Persistence in systems with diffusion. In Dynamics of macrosystems, eds J.-P. Aubin, D. Saari K. Sigmund, (Lecture Notes in Economy and Mathematical Systems), pp. 43-8. Springer Verlag, Berlin. 

One important feature of reaction-diffusion fields, not shared by fluid dynamical systems as another representative class of nonlinear fields, is worth mentioning. This is the fact that the total system can be viewed as an assembly of a large number of identical local systems which are coupled (i.e., diffusion-coupled) to each other. Here the local systems are defined as those obeying the diffusionless part of the equations. Take for instance a chemical solution of some oscillating reaction, the best known of which would be the Belousov-Zhabotinsky reaction (Tyson, 1976). Let a small element of the solution be isolated in some way from the bulk medium. Then, it is clear that in this small part a limit cycle oscillation persists. Thus, the total system may be imagined as forming a diffusion-coupled field of similar limit cycle oscillators. 

These patterns are an example of what are sometimes called dissipative structures, which arise in many complex systems. Dissipative structures are dynamical patterns that retain their organized state by persistently dissipating matter and energy into an otherwise thermodynamically open environment. 

Because in an autonomous system many of the invariant manifolds that are found in the linear approximation do not remain intact in the presence of nonlinearities, one should expect the same in the time-dependent case. In particular, the separation of the bath modes will not persist but will give way to irregular dynamics within the center manifold. At the same time, one can hope to separate the reactive mode from the bath modes and in this way to find the recrossing-free dividing surfaces and the separatrices that are of importance to TST. As was shown in Ref. 40, this separation can indeed be achieved through a generalization of the normal form procedure that was used earlier to treat autonomous systems [34]. 

To characterize dendrimers, analytical methods used in synthetic organic chemistry as well as in macromolecular chemistry can be applied. Mass spectrometry and NMR spectroscopy are especially useful tools to estimate purity and structural perfection. To get an idea of the size of dendrimers, direct visualization methods such as atomic force microscopy (AFM) and transmission electron microscopy (TEM), or indirect methods such as size exclusion chromatography (SEC) or viscosimetry, are valuable. Computer aided simulation also became a very useful tool not only for the simulation of the geometry of a distinct molecule, but also for the estimation of the dynamics in a dendritic system, especially concerning mobility, shape-persistence, and end-group disposition. 

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