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Reduced dynamics for delayed systems with harmonic or stochastic forcing

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10.1063/1.4760250

### Abstract

The analysis of nonlinear delay-differential equations (DDEs) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimension. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDEs. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes.

© 2012 American Institute of Physics

Received 07 May 2012
Accepted 03 October 2012
Published online 26 November 2012

Lead Paragraph:

Delayed feedback systems have a widespread use across the engineering, physical, and biological sciences. In their applications, these are often subjected to temporally varying forcing to implement the active role played by external influences on the dynamics. However, due to the complexity of infinite dimensional systems, the way additive temporal fluctuations interact with retarded dynamics is far from understood. In particular, a statement on the exact role played by non-homogeneous components on the stability of solutions in delayed feedback models is still missing. Here, the authors propose a novel approach to handle that problem using center manifold (CM) theory formulated for delay equations, where an explicit time-dependence is taken into account. Their method captures the effect of forcing on the stability of a delayed feedback system in the vicinity of a bifurcation, and serves as a new strategy to highlight novel non-linear phenomena.

Acknowledgments: The authors would like to thank NSERC, FQRNT and INRIA for support. In addition, Axel Hutt has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 257253.

Article outline:

I. INTRODUCTION

II. CENTER MANIFOLD REDUCTION

A. Analysis of autonomous DDEs

B. Spectral analysis and phase space decomposition

C. The autonomous center manifold reduction

D. Analysis of non-autonomous DDEs

III. APPLICATION TO THE TRANSCRITICAL BIFURCATION

A. Autonomous case: *I*(*t*) = 0

1. Quadratic terms

2. Cubic terms

B. Non-autonomous case:

1. Periodic driving

2. Driving at multiple frequencies

C. Insights about stochastic systems stability

IV. DISCUSSION

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2012-11-26

2014-04-24

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