Oscillation nonlinear internal

Interaction with External Fields. The models considered exhibit cooperative behaviour through nonlinear internal oscillations (models 1, 2, 4) or through nonlinear resonances (model 3). This makes plausible the existence of effects, when the system is driven by weak external fields of appropriate frequency. 

Kuramoto, Y. (1975) Self-entrainment of a population of coupled nonlinear oscillators. In International Symposium on Mathematical Problems in Theoretical Physics, ed. by H. Araki, Lecture Notes Phys., Vol. 39 (Springer, New York) p. 420 Kuramoto, Y., Tsuzuki, T. (1976) Persistent propagation of concentration waves in dissipative media far from thermal equilibrium. Prog. Theor. Phys. 55, 356 Kuramoto, Y., Yamada, T. (1976a) Turbulent state in chemical reactions. Prog. Theor. Phys. 55, 679... 

Summary. In this chapter the control problem of output tracking with disturbance rejection of chemical reactors operating under forced oscillations subjected to load disturbances and parameter uncertainty is addressed. An error feedback nonlinear control law which relies on the existence of an internal model of the exosystem that generates all the possible steady state inputs for all the admissible values of the system parameters is proposed, to guarantee that the output tracking error is maintained within predefined bounds and ensures at the same time the stability of the closed-loop system. Key theoretical concepts and results are first reviewed with particular emphasis on the development of continuous and discrete control structures for the proposed robust regulator. The role of disturbances and model uncertainty is also discussed. Several numerical examples are presented to illustrate the results. 

In this expression, s is the number of internal harmonic oscillators, which is 3N — 5 for linear molecules and 3N—6 for nonlinear molecules, and N is the number of atoms in the species. These are the precise conditions under which the high-pressure rate parameters discussed in the previous sections are applicable. 

Externally driven limit cycles can exhibit a great variety of behaviour. Quite generally, one gets a nonlinear superposition of an internal nonlinear oscillation with an external oscillatory perturbation. Details of the resulting behaviour (including sub- and superharmonic oscillations, entrainment, quenching...) depend on both, the frequency and intensity of the applied fields and on the internal nonlinear kinetics of the considered system. 

In the resonance region the system oscillates with the external frequency X and with an increased amplitude (entrainment region). Far away from resonance, the internal free oscillations are present. This behaviour is completely absent in systems, where no self-sustained oscillations can exist. A typical example of such a system is a nonlinear conservative system. The resonance diagram has been drawn in Figure 2 for both, the small and the large ampli tude oscillation. 

F cos At. Again one has to investigate the competition between an internal nonlinear oscillation and an external oscillatory field. 

If, by internal means, the system is near c2> only a small amount of energy is necessary to drive the system into the highly excited state. Furthermore, we have been able to show that oscillations on the hysteresis are possible for A detailed inspection of the transport equations (generalizeS nonlinear Peierls-Boltzmann equations for phonons) shows that nonlinear kinetics, dissipation and energy supply via transport are indispensable for such a behaviour to occur. 

The above scenario is typical of nonlinear dynamical systems when the amplitude of the internally generated oscillations becomes sufficiently large. In the bifurcation diagram of Fig. 12.5 this occurs when the slope of the feedback characteristics exceeds a critical value. However, similar scenarios can be produced through variation of other parameters such as, for instance, the damping of the arteriolar oscillator. 

The Hamiltonian H(p,x,n, J ) is a model of diatomic molecules. h(p,x) represents the translational degrees of freedom and / ,(7t, Sj represents the internal vibrations of the molecules. If all the molecules are identical, we can assume that all frequencies are set to be equal. The internal part h n, E,) takes the form of uncoupled harmonic oscillators, so it looks specific. But this is not the case because all the nonlinear terms can be absorbed into the coupling term f P,X, 7I, ). 

A. J. Szeri and L. G. Leal, The onset of chaotic oscillations and rapid growth of a gas bubble at subcritical conditions in an incompressible liquid, Phys. Fluids A3, 551-5 (1991) Z. C. Feng and L. G. Leal, Bifurcation and chaos in shape and volume oscillations of a periodically driven bubble with two-to-one internal resonance, J. Fluid Mech. 266, 209-42 (1994) L. G. Leal and Z. C. Feng, Nonlinear bubble dynamics, Annu. Rev. Fluid Mech. 29, 201-42 (1997). 

If a semiconductor element with negative differential conductance is operated in a reactive circuit, oscillatory instabilities may be induced by these reactive components, even if the relaxation time of the semiconductor is much smaller than that of the external circuit so that the semiconductor can be described by its stationary I U) characteristic and simply acts as a nonlinear resistor. Self-sustained semiconductor oscillations, where the semiconductor itself introduces an internal unstable temporal degree of freedom, must be distinguished from those circuit-induced oscillations. The self-sustained oscillations under time-independent external bias will be discussed in the following. Examples for internal degrees of freedom are the charge carrier density, or the electron temperature, or a junction capacitance within the device. Eq.(5.3) is then supplemented by a dynamic equation for this internal variable. It should be noted that the same class of models is also applicable to describe neural dynamics in the framework of the Hodgkin-Huxley equations [16]. 

Y. Kuramoto. Self-entrainment of a population of coupled nonlinear oscillators. In H. Araki, editor. International Symposium on Mathematical Problems in Theoretical Physics, page 420, New York, 1975. Springer Lecture Notes Phys., v. 39. 

F. Mashayek and N. Ashgriz, Nonlinear oscillation of liquid drops With internal circulation, Phys. Fluids 10(5), 1071-1082, May 1998. 

Tsiglifis, K., N. A. Pelekasis Nonlinear oscillations and collapse of elongated bubbles subject to weak viscous effects effect of internal overpressme, Phys. Fluids 19, 072106 (2007). 

The feedforward system imposes an external material balance as well as a an internal material balance on the process. The internal balance is maintained by liquid level control on the discharge of each effect. Analysis of a level loop indicates that a narrow proportional band (less than 10%) can achieve stable control. However, because of the resonant nature of the level loop which causes the process to oscillate at its natural frequency, a much lower controller gain must be used (proportional bands 50-100%). A valve positioner is recommended to overcome the nonlinear nature of valve hysteresis. 

With fixed-point arithmetic it is possible for filter calculations to overflow. The term overflow oscillation, sometimes also called adder overflow limit cycle, refers to a high-level oscillation that can exist in an otherwise stable filter due to the nonlinearity associated with the overflow of internal filter calculations. 

A limit cycle, sometimes referred to as a multiplier roundoff limit cycle, is a low-level oscillation that can exist in an otherwise stable filter as a result of the nonlinearity associated with rounding (or truncating) internal filter calculations (Parker and Hess, 1971). Limit cycles require recursion to exist and do not occur in nonrecursive FIR filters. 

These results are perhaps surprising because the side-reboiler for the diabatie design introduces strong cross-coupling effects causing considerable oscillations within the column. However the improved controlled response is due to the removal of the sharp nonlinearity in the temperature profile (Fig 7). Shifts in the internal temperature, used as a measured variable in the control scheme, are less drastic producing a more even elosed loop response. 

For the diatomic molecule the same results apply except that the X displacement coordinate is replaced by the internal coordinate r — rj. For polyatomic molecules, each nonlinear molecule acts as though it consisted of 3N - 6 separate harmonic oscillators in each of which the X displacement coordinate is replaced by the appropriate normal coordinate Q. 

When nonlinear, dissipative systems are driven far from equilibrium, they often exhibit multistability, whereby there exist two or many coexisting steady states and/or oscillating states. Through internal or external noise, such systems may be driven to undergo transitions between these states. Moreover, the noise itself may create new states and transitions between them [9]. A wide variety of systems chosen from the fields of chemistry [10], optics [11] and biology [12] display such a behavior. 

The school from Brussels have found that for the systems maintained far from thermodynamic equilibrium and when the kinetic laws are suitably nonlinear, the excess entropy production, S P becomes negative for t > t. In this case, the nonequilibrium steady state becomes unstable and sometimes, when the internal structure permits, the system executes limit cycle oscillations [1-3]. 

Using the Oldroyd determination of derivatives In the law (1.1) and keeping nonlinear terms In the balance equations of mass and momentum, one can get the system of dynamic equations for the medium with Internal oscillators. 

A gas phase species with N atoms has 3N degrees of freedom associated with it. For a nonlinear molecule, three of these are translations and three are rotations (two, if linear). The remaining 3N - 6 (3N - 5, if linear) movements are the internal or normal modes of vibration for the species. The single vibration of a diatomic species can be modeled quantum mechanically in terms of a one-dimensional harmonic oscillator. Figure 3.4.1.1, giving the energy levels... 

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