Nonlinear dynamics of delay systems: an overview

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2013

2000

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Chemical Engineering Science, 1989

We study the dynamics of a general system under linear feedback control subject to system, measuremctit and actuator delays. There are two general classes of dynamic behavior, resonant and nonresonant, which are pertinent to the tuning of the controllers and to the nonlinear closed-loop dynamics. For non-resonant systems, the cross-over frequency and ultimate gain are monotonic and continuous functions of the delay and conventional Ziegler-Nichols type tuning methods are appropriate. For resonant systems, however, the cross-over frequency is a discontinuous function of the delay and at certain critical values of the delay, two cross-over frequencies coexist at the same ultimate gain. Such systems cannot be tuned with the Ziegler-Nichols methods. These critical delays of a resonant system cot-respond to double-Hopf bifurcation points with two distinct pairs of purely imaginary eigenvalues (closed-loop poles). Using center manifold projection and normal form analyses, we are able to predict all nonlinear dynamics in the vicinity of these singularities. In particular, a two-frequency torus is always found. This is verified experimentally in a cascaded liquid level control system. There is also evidence that the torus degenerates into a chaotic attractor as the feedback gain is increased.

Physical Review E, 2004

The characteristics of a time-delayed system with time-dependent delay time is investigated. We demonstrate the nonlinearity characteristics of the time-delayed system are significantly changed depending on the properties of time-dependent delay time and especially that the reconstructed phase trajectory of the system is not collapsed into simple manifold, differently from the delayed system with fixed delay time. We discuss the possibility of a phase space reconstruction and its applications.

Automatica, 2004

Nonlinear Phenomena in Complex Systems, 2020

Time-delay naturally arises in many real-world systems, due to the fact that the instantaneous rate of change of such systems does not only depend on their current time but rather on their previous history as well. Hence, time-delays are ubiquitous, their introduction often leads to suppression of oscillations, multistability and chaotic motion in the dynamical systems. This review presents some models with different kinds of time-delays such as discrete, distributed and combination of both discrete and distributed time-delays with special emphasis on the reason of incorporating such delays into the system.

International Journal of Bifurcation and Chaos, 2001

We report numerical evidence of the effects of a periodic modulation in the delay time of a delayed dynamical system. By referring to a Mackey–Glass equation and by adding a modulation in the delay time, we describe how the solution of the system passes from being chaotic to shadow periodic states. We analyze this transition for both sinusoidal and sawtooth wave modulations, and we give, in the latter case, the relationship between the period of the shadowed orbit and the amplitude of the modulation. Future goals and open questions are highlighted.

Chaos, Solitons & Fractals, 1995

Oscillatory dynamic behaviour emerging from Hopf bifurcation phenomenon in nonlinear feedback systems with delay is presented. The periodic branches are recovered after using Harmonic Balance Approximations (HBA) and the theory of feedback control systems. The results are plotted and compared against those obtained by simulations. Furthermore, other static, dynamic and degenerate bifurcations are pointed out using this methodology in the so-called frequency domain. Two examples for illustrating the application of this technique are included.

2011

The aim of this note is to discuss conditions on delay derivatives, frequently encountered in the literature, from a systems theory point of view. First an overview of potential problems when the delay rate exceeds one is given, including the violation of causality and the violation of the principe of existence and uniqueness of solutions. Second, the required assumptions on the delay variation to overcome these problems are stated, allowing to define a state space in a rigorous way. Finally, it is shown that a stability analysis of systems with a fast varying delay, where no assumptions on the delay rate are made, can be performed in a meaningful and practically relevant way in some cases, but should be interpreted in terms of relaxations of solutions. A look at fast varying and state dependent delays from a systems theory point of view Wim Michiels∗ and Erik I. Verriest† Abstract The aim of this note is to discuss conditions on delay derivatives, frequently encountered in the lite...