Noisy Dynamical Systems with Time Delay: Some Basic Analytical Perturbation Schemes with Applications
Systems with time delay, a rather prominent branch in applied dynamical systems theory, constitute a special case of functional differential equations for which the general mathematical theory is fairly well developed and largely parallels the theory of ordinary differential equations. Hence analytic concepts like bifurcation theory, adiabatic elimination, global attractors, invariant manifolds and others can be used to study dynamical behaviour of systems with time delay if some care is applied to take special features of infinite dimensional phase spaces into account. Simple analytic perturbation schemes, frequently used to gain insight for ordinary differential equations, can be applied to time delay dynamics as well. However, such approaches seem to be used infrequently within the physics community, probably because of a lack of easily accessible expositions. Here we review some elementary and well established concepts for the analytical treatment of time delay dynamics, even when subjected to noise. We cover normal form reduction and adiabatic elimination, stochastic linearisation of time delay dynamics with noise, and some elements of weakly nonlinear- and bifurcation analysis. These tools will be illustrated with applications in control problems, time delay autosynchronisation, coherence resonance, and the computation and structure of power spectra in noisy time delay systems.
KeywordsPeriodic Orbit Time Delay System Dynamical System Theory Slow Manifold Coherence Resonance
This work was supported by DFG in the framework of SFB 910.
- 9.L. Arnold, Random Dynamical Systems (Springer, Berlin, 2002)Google Scholar
- 15.W. Just, A. Pelster, M. Schanz, E. Schöll (eds.), Delayed Complex Systems, Theme Issue of Phil. Trans. R. Soc. A 368, 301–513 (2010)Google Scholar
- 16.V. Flunkert, I. Fischer, E. Schöll (eds.), Dynamics, control and information in delay-coupled systems, Theme Issue of Phil. Trans. R. Soc. A 371, 20120465 (2013)Google Scholar
- 17.J.Q. Sun, G. Ding (eds.), Advances in Analysis and Control of Time-Delayed Dynamical Systems (World Scientific, Singapore, 2013)Google Scholar
- 18.M. Soriano, J. Garca-Ojalvo, C. Mirasso, I. Fischer, Rev. Mod. Phys. 85, 421 (2013)Google Scholar
- 19.J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations (Springer, New York, 1993)Google Scholar
- 20.K. Engelborghs, DDE-BIFTOOL: a Mathlab package for bifurcation analysis of delay differential equations. http://www.cs.kuleuven.ac.be/koen/delay/ddebiftool.shtml
- 55.J.F.M. Avila, H.L.D. de S. Cavalcante, J.R.R. Leite. Phys. Rev. Lett. 93, 144101 (2004)Google Scholar