Smooth dynamical system

G. Benettin, L. Galgani, and J-M. Strelcyn. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems a method for computing all of them. Meccanica, 15 9-20, 1980. 

In the two-dimensional case (two variables) "almost any C1-smooth dynamic system is rough (i.e. at small bifurcations its phase pattern deforms only slightly without qualitative variations). For rough two-dimensional systems, the co-limit set of every motion is either a fixed point or a limit cycle. The stability of these points and cycles can be checked even by a linear approximation. Mutual relationships between six different types of slow relaxations for rough two-dimensional systems are sharply simplified. 

The relationships, rather similar in sense, for smooth dynamic systems were introduced in ref. 34 (p. 220 etc.) for studying the random perturbations via a method of action functionals. Close concepts can also be found in ref. 39. 

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. 

Consider a one-parameter family of (r > 2) smooth dynamical systems in (n > 1). Suppose that when the parameter vanishes the system possesses a non-rough equilibrium state at the origin with one characteristic exponent equal to zero and the other n exponents lying to the left of the imaginary axis. We suppose also that the equilibrium state is a simple saddle-node, namely the first Lyapimov value I2 is not zero (see Sec. 11.2). Without loss of generality we assume /2 > 0. 

Anosov, D. V., Bronshtejn, I. U., Aranson, S. Kh. and Grines, V. Z. [1988] Smooth Dynamical Systems Dynamical Systems /, Encyclopedia of Mathematics Science I (Springer-Verlag New York), 149-233. 

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

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],... 

To prepare dynamic cyanohydrin systems under mild conditions, a range of aldehyde compounds and cyanide sources was evaluated. As a result, benzaldehydes 23A-E were selected due to their diverse substitution patterns and their inability to generate any side reactions. Even though there are many cyanide sources, acetone cyanohydrin 24 was chosen as cyanide source in presence of triethylamine base, resulting in smooth cyanide release. Dynamic cyanohydrin systems (CDS-3) were thus generated from one equivalent of each benzaldehyde 23A-E and acetone cyanohydrin 24 in chloroform- at room temperature (Scheme 10). One equivalent of triethylamine was added to accelerate the reversible cyanohydrin reactions and this amount was satisfactory to force the dynamic system to reach equilibrium even at low temperature. 

Abstract We consider the time evolution of a dynamic quantum system coupled to a repeatedly measured ancilla. Given the time lapse At between two subsequent measurements, the combined system may be described using a difference master equation whereas, in the Zeno-limit At — 0, the evolution of the dynamic system is unitary and defined by the state of the ancilla. For an arbitrary At, we also formulate a master equation that interpolates smoothly the exact evolution given by the difference equation. 

In recent years a great many studies have reported on the dynamic systems where a drop of liquid is placed on a smooth solid surface. ° The system liquid drop-solid is a very important system in everyday life, for example, rain drops on tree leaves or other surfaces. It is also significant in all kinds of systems where a spray of fluid is involved, such as in sprays or combustion engines. The dynamics of liquid drop evaporation rate is of much interest in many phenomena. The liquid-solid interface can be considered as follows. Real solid surfaces are, of course, made up of molecules not essentially different in their nature from the molecules of the fluid. The interaction between a molecule of the fluid and a molecule of the boundary wall can be regarded as follows. The molecules in the solid state are not as mobile as those of the fluid. It is therefore permissible for most purposes to regard the molecules in the solid state as stationary. However, complexity arises in those liquid-solid systems where a layer of fluid might be adsorbed on the solid surface, such as in the case of water-glass. 

As mentioned previously, for the N-body system, energy and total momentum are constants of motion—they do not change even as the bodies of the system move along their natural paths (defined by the equations of motion). Another term for constant of motion is first integral. In general, if we have a dynamical system z = /(z), first integral is a smooth function /(z) which is constant along solutions, for all values of the initial condition. Letz(f) f M be a solution, then... 

The generation of self-sustained oseillations is a particular case of bifurcation. The term bifurcation is often used in connection with the mathematical study of dynamical systems. It denotes a sudden qualitative ehange in the behavior of a system upon the smooth variation of a parameter, the so-eaUed bifureation parameter, and is applied to the point of the fundamental reeonstmetion of the phase portrait where the bifurcation parameter attains its critical value. The simplest examples of bifurcation are the appearance of a new rest point in the phase space, the loss of the rest-point stability, and the appearance of a new limit cycle. Bifurcation relates to physicochemical phenomena such as ignition and extinction, that is, a jump-like transition from one steady state to another one, the appearance of oscillations, or a chaotic regime, and so on. 

Further examination of the trade-offs to achieve a smooth transition to a globally sustainable nuclear fuel cycle would require additional dynamic system studies [11, 13]. 

From the perspective of a multi-decade transition to an integrated symbiotic fuel cycle, small reactors with a high conversion ratio CR > 1 offer an advantage over standard LWRs in securing a higher overall nuclear park conversion ratio. Further examination of the trade-offs to achieve a smooth transition to a globally sustainable nuclear fuel cycle requires additional dynamic system studies. 

Like it was done in the original definition of roughness, it is natural to impose some assumptions regarding the boundary dO of the region G namely, that dG must be a smooth closed curve without contact with the vector field (i.e. not tangent to it). Notice that in the case of dynamical systems on compact smooth surfaces, the domain G is just taken to coincide with the whole surface, so no conditions on the boundary appear. 

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