Stability dynamic system

The characteristic time of surfactant adsorption at a fluid interface is an important parameter for suriaetant-stabilized dynamic systems sueh as emulsions. Sutherland (22) derived an expression deseribing flie relaxation of a small dilatation of an initially equilibrium adsorption monolayer... 

To obtain the unconditional stability of the midpoint method in local coordinates, one would have to consider the decoupling transformation from cartesian to local coordinates for the discrete variables etc. But this transformation, which for the continuous variables is not constant, necessarily is in error which depends on k, not e. The stability properties of the discrete dynamical systems obtained by the midpoint discretization in the different sets of coordinatc.s may therefore be significantly different when it 3> e [3]. 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

The motion of particles of the film and substrate were calculated by standard molecular dynamics techniques. In the simulations discussed here, our purpose is to calculate equilibrium or metastable configurations of the system at zero Kelvin. For this purpose, we have applied random and dissipative forces to the particles. Finite random forces provide the thermal motion which allows the system to explore different configurations, and the dissipation serves to stabilize the system at a fixed temperature. The potential energy minima are populated by reducing the random forces to zero, thus permitting the dissipation to absorb the kinetic energy. 

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by a nonlinear (inhomogeneous) external periodic excitation (as regards the coordinates of the excited system) and is remarkable for the occurrence of the following objective regularities the phenomenon of discrete oscillation excitation in macro-dynamical systems having multiple branch attractors and strong self-adaptive stability. 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

In 2004, Rayner and coworkers reported a dynamic system for stabilizing nucleic acid duplexes by covalently appending small molecules [34]. These experiments started with a system in which 2-amino-2 -deoxyuridine (U-NH ) was site-specifically incorporated into nucleic acid strands via chemical synthesis. In the first example, U-NH was incorporated at the 3 end of the self-complementary U(-NH2)GCGCA DNA. This reactive amine-functionalized uridine was then allowed to undergo imine formation with a series of aldehydes (Ra-Rc), and aldehyde appendages that stabilize the DNA preferentially formed in the dynamic system. Upon equilibration and analysis, it was found that the double-stranded DNA modified with nalidixic aldehyde Rc at both U-NH positions was amplified 34% at the expense of Ra and Rb (Fig. 3.16). The Rc-appended DNA stabilizing modification corresponded to a 33% increase in (melting temperature). Furthermore, imine reduction of the stabilized DNA complex with NaCNBH, resulted in a 57% increase in T. ... 

The results of these three dynamic systems highlight the ability of DCC to identify the sites and appendages that most efficiently stabilize nucleic acid interactions. 

It is interesting that in the soluble case studied in Section VI a close connection appears between initial value problems of dynamics and boundary value problems for dissipative structures. In discussion of initial value problems the concept of stability plays an important role. Only for simple dynamical systems such as separable systems do we find, in general, stability in the sense that trajectories originating from neighboring points remain close for all times. It would be very interesting to investigate along similar lines the stability of dissipative structures, and... 

Such was the state of the art when Amundson and Bilous s paper was published in the first volume of the newly founded A.I.CH.E. Journal (Bilous and Amundson, 1955). This for the first time treated the reactor as a dynamical system and, using Lyapounov s method of linearization, gave a pair of algebraic conditions for local stability. One of these corresponded to the slope condition of previous analyses, and there was a brief flurry of attempts to invest the other with a similarly physical explanation. For the global picture they introduced the phase plane (another feature of the theory of dynamical systems) and, with consummate skill, Bilous conjured the now classic figures from a Reeves electronic analogue computer. Even in this early paper, they had touched upon the consecutive reaction scheme A - B - C and had shown that up to five steady states might be expected under some conditions. 

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 scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics variational formulations computational mechanics statics, kinematics and dynamics of rigid and elastic bodies vibrations of solids and structures dynamical systems and chaos the theories of elasticity, plasticity and viscoelasticity composite materials rods, beams, shells and membranes structural control and stability soils, rocks and geomechanics fracture tribology experimental mechanics biomechanics and machine design. 

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