Periodically forced self-oscillating systems

We emphasize the link with the name of the already formed class of kick-excited self-adaptive systems and phenomena the external force is linked, through the function e(x), with the motion coordinate in an adaptive mode, and at the same time it exerts action in the form of short impulses much shorter than the oscillation period of the system. 

A comparative study was done by Kevrekidis and published as I. G. Kevrekidis, L. D. Schmidt, and R. Aris. Some common features of periodically forced reacting systems. Chem. Eng. Sci. 41,1263-1276 (1986). See also two papers by the same authors Resonance in periodically forced processes Chem. Eng. Sci. 41, 905-911 (1986) The stirred tank forced. Chem. Eng. Sci. 41,1549-1560 (1986). A full study of the Schmidt-Takoudis vacant site mechanism is to be found in M. A. McKamin, L. D. Schmidt, and R. Aris. Autonomous bifurcations of a simple bimolecular surface-reaction model. Proc. R. Soc. Lond. A 415,363-387 (1988) Forced oscillations of a self-oscillating bimolecular surface reaction model. Proc. R. Soc. Lond. A 415,363-388 (1988). 

We begin our discussion of synchronization phenomena by considering the simplest case, an entrainment of a self-sustained oscillator by an external periodic force. Before we describe this effect in mathematical terms, we illustrate it by an example. We will again speak about clocks, but this time about biological clocks that regulate daily and seasonal rhythms of living systems - from bacteria to humans. 

In physics such oscillatory objects are denoted as self-sustained oscillators. Mathematically, such an oscillator is described by an autonomous (i.e., without an explicit time dependence) nonlinear dynamical system. It differs both from linear oscillators (which, if a damping is present, can oscillate only due to external forcing) and from nonlinear energy conserving systems, whose dynamics essentially depends on initial state. Dynamics of oscillators is typically described in the phase (state) space. Periodic oscillations, like those of the clock, correspond to a closed attractive curve in the phase space, called the limit cycle. The limit cycle is a simple attractor, in contrast to a strange (chaotic) attractor. The latter is a geometrical image of chaotic self-sustained oscillations. 

An important area of dynamics has to do with problems where bodies oscillate in response to external disturbances. The exciting disturbances may be due to imbalance of rotating machinery a sudden shock, as in an earthquake or a periodic series of forces, as in an internal combustion engine. Self-excited vibrations exist when a system, once disturbed, vibrates indefinitely. This occurs when the energy stored exceeds that which is released during a cycle of vibration. In such cases, the amplitude of vibration continues to increase imtil the increase in energy lost per cycle in... 

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