By Philipp Hövel
This examine addresses hold up results in nonlinear structures, that are ubiquitous in numerous fields of physics, chemistry, biology, engineering, or even in social and financial structures. they could come up because of processing occasions or end result of the finite propagation pace of knowledge among the components of a posh method. Time hold up has complementary, counterintuitive and virtually contradictory features. at the one hand, hold up is ready to set off instabilities, bifurcations of periodic and extra complex orbits, multi-stability and chaotic movement. nonetheless, it may suppress instabilities, stabilize risky desk bound or periodic states and should keep watch over advanced chaotic dynamics. This thesis bargains with either elements, and offers novel basic effects at the controllability of nonlinear dynamics by way of time-delayed suggestions, in addition to purposes to lasers, hybrid-mechanical structures, and paired neural systems.
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Extra info for Control of Complex Nonlinear Systems with Delay
In: AIP conference proceedings, vol 548. American Institute of Physics, Melville, New York 105. Wieczorek S, Krauskopf B, Lenstra D (2002) Multipulse excitability in a semiconductor laser with optical injection. Phys Rev Lett 88:063901 106. Wieczorek S, Krauskopf B, Simpson T, Lenstra D (2005) The dynamical complexity of optically injected semiconductor lasers. Phys Rep 416:1 Chapter 3 Control of Steady States Was ist Chaos? 1 Stanisław Jerzy Lec, Sämtliche unfrisierte Gedanken Next to periodic orbits, steady states or equilibria2 play an important role in a large variety of optical, electronic, chemical, biological, and other nonlinear systems [1–6].
1 What is chaos? It is the order which was destroyed at the creation of the world. In the mathematical literature equilibria denote fixed points or steady states of a dynamics system, not to be confused with equilibrium in the physical literature. All systems considered here are far from thermodynamic equilibrium. 2 P. 1007/978-3-642-14110-2_3, Ó Springer-Verlag Berlin Heidelberg 2010 43 44 3 Control of Steady States Asymptotic properties of time-delayed feedback control will be the topic of Sect.
The eigenvalues of this system are K0 = k ± ix. , z(t) = x(t) + iy(t) = r(t)eih(t), yields equivalently to Eqs. 4b dzðtÞ ¼ ðk Æ ixÞzðtÞ dt ð3:1:5Þ or written in amplitude r(t) and phases h(t) drðtÞ ¼ krðtÞ dt ð3:1:6aÞ dhðtÞ ¼ Æx: dt ð3:1:6bÞ One can see from the last system of equations that k acts as a damping rate whereas x influences the phase. For x = 0 and positive k, the system describes an unstable focus. , 46 3 Control of Steady States Fig. , dependence of the amplitude r on the bifurcation parameter k.