Delay-induced multistability near a global bifurcation

We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems.Comment: Int. J. Bif. Chaos (2007), in prin

Nonlinear response of a forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two

Non-resonant bifurcations of codimension two may appear in the controlled van der Pol-Duffing oscillator when two critical time delays corresponding to a double Hopf bifurcation have the same value. With the aid of centre manifold theorem and the method of multiple scales, the non-resonant response and two types of primary resonances of the forced van der Pol-Duffing oscillator at non-resonant bifurcations of codimension two are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. It is shown that the non-resonant response of the forced oscillator may exhibit quasi-periodic motions on a two- or three-dimensional (2D or 3D) torus. The primary resonant responses admit single and mixed solutions and may exhibit periodic motions or quasi-periodic motions on a 2D torus. Illustrative examples are presented to interpret the dynamics of the controlled system in terms of two dummy unfolding parameters and exemplify the periodic and quasi-periodic motions. The analytical predictions are found to be in good agreement with the results of numerical integration of the original delay differential equation. © 2008 Elsevier Ltd. All rights reserved

Additive resonances of a controlled van der Pol-Duffing oscillator

The trivial equilibrium of a controlled van der Pol-Duffing oscillator with nonlinear feedback control may lose its stability via a non-resonant interaction of two Hopf bifurcations when two critical time delays corresponding to two Hopf bifurcations have the same value. Such an interaction results in a non-resonant bifurcation of co-dimension two. In the vicinity of the non-resonant Hopf bifurcations, the presence of a periodic excitation in the controlled oscillator can induce three types of additive resonances in the forced response, when the frequency of the external excitation and the frequencies of the two Hopf bifurcations satisfy a certain relationship. With the aid of centre manifold theorem and the method of multiple scales, three types of additive resonance responses of the controlled system are investigated by studying the possible solutions and their stability of the four-dimensional ordinary differential equations on the centre manifold. The amplitudes of the free-oscillation terms are found to admit three solutions; two non-trivial solutions and the trivial solution. Of two non-trivial solutions, one is stable and the trivial solution is unstable. A stable non-trivial solution corresponds to a quasi-periodic motion of the original system. It is also found that the frequency-response curves for three cases of additive resonances are an isolated closed curve. It is shown that the forced response of the oscillator may exhibit quasi-periodic motions on a three-dimensional torus consisting of three frequencies; the frequencies of two bifurcating solutions and the frequency of the excitation. Illustrative examples are given to show the quasi-periodic motions. © 2008 Elsevier Ltd. All rights reserved

Parametric resonance in the Rayleigh-Duffing oscillator with time-delayed feedback

We investigate the principal parametric resonance of a Rayleigh–Duffing oscillator with time-delayed feedback position and linear velocity terms. Using the asymptotic perturbation method, we obtain two slow flow equations on the amplitude and phase of the oscillator. We study the effects of the frequency detuning, the deterministic amplitude, and the time-delay on the dynamical behaviors, such as stability and bifurcation associated with the principal parametric resonance. Moreover, the appropriate choice of the feedback gain and the time-delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time-delay can broaden the stable region of the non-trivial steady-state solutions and enhance the control performance. Theoretical stability analysis is verified through a numerical simulation.The University of Pretoriahttp://www.elsevier.com/locate/cnsnsai201

Delayed effects and critical transitions in climate models

There is a continuous demand for new and improved methods of understanding our climate system. The work in this thesis focuses on the study of delayed feedback and critical transitions. There is much room to develop upon these concepts in their application to the climate system. We explore the two concepts independently, but also note that the two are not mutually exclusive. The thesis begins with a review of delay differential equation (DDE) theory and the use of delay models in climate, followed by a review of the literature on critical transitions and examples of critical transitions in climate. We introduce various methods of deriving delay models from more complex systems. Our main results center around the Saltzman and Maasch (1988) model for the Pleistocene climate (`Carbon cycle instability as a cause of the late Pleistocene ice age oscillations: modelling the asymmetric response.' Global biogeochemical cycles, 2(2):177-185, 1988). We observe that the model contains a chain of first-order reactions. Feedback chains of this type limits to a discrete delay for long chains. We can then approximate the chain by a delay, resulting in scalar DDE for ice mass. Through bifurcation analysis under varying the delay, we discover a previously unexplored bistable region and consider solutions in this parameter region when subjected to periodic and astronomical forcing. The astronomical forcing is highly quasiperiodic, containing many overlapping frequencies from variations in the Earth's orbit. We find that under the astronomical forcing, the model exhibits a transition in time that resembles what is seen in paleoclimate records, known as the Mid-Pleistocene Transition. This transition is a distinct feature of the quasiperiodic forcing, as confi rmed by the change in sign of the leading nite-time Lyapunov exponent. Additional results involve a box model of the Atlantic meridional overturning circulation under a future climate scenario and time-dependent freshwater forcing. We find that the model exhibits multiple types of critical transitions, as well as recovery from potential critical transitions. We conclude with an outlook on how the work presented in this thesis can be utilised for further studies of the climate system and beyond.European Commissio

Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space $X$ which is acted on by any continuous semigroup $\{S(t)\}_{t \geq 0}$. Suppose that $\S(t)\}_{t \geq 0}$ possesses a global attractor $\mathcal{A}$. We show that, for any generalized Banach limit $\underset{T \rightarrow \infty}{\rm{LIM}}$ and any distribution of initial conditions $\mathfrak{m}_0$, that there exists an invariant probability measure $\mathfrak{m}$, whose support is contained in $\mathcal{A}$, such that $$\int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t,$$ for all observables $\phi$ living in a suitable function space of continuous mappings on $X$. This work is based on a functional analytic framework simplifying and generalizing previous works in this direction. In particular our results rely on the novel use of a general but elementary topological observation, valid in any metric space, which concerns the growth of continuous functions in the neighborhood of compact sets. In the case when $\{S(t)\}_{t \geq 0}$ does not possess a compact absorbing set, this lemma allows us to sidestep the use of weak compactness arguments which require the imposition of cumbersome weak continuity conditions and limits the phase space $X$ to the case of a reflexive Banach space. Two examples of concrete dynamical systems where the semigroup is known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic

Utilizing the time delayed PPF controller to suppress vibrations of a nonlinear system containing real power exponents in damping and restoring forces

The time delayed Positive Position Feedback (PPF) controller is utilized to suppress the primary resonance of vibrations of an excited base oscillator by real power exponents of the restoring and damping forces. Multiple scales method is conducted to get the frequency response equations. The stability of the system is studied by using the Lyapunov first method. The influences of system parameters and time delay on the system response are investigated to avoid the jump phenomenon for better system performance. Time margin is deduced for most possible values of controller gain. Analytic results are verified by numerical integration of the original system equations

The physics of climate variability and climate change

The climate system is a forced, dissipative, nonlinear, complex and heterogeneous system that is out of thermodynamic equilibrium. The system exhibits natural variability on many scales of motion, in time as well as space, and it is subject to various external forcings, natural as well as anthropogenic. This paper reviews the observational evidence on climate phenomena and the governing equations of planetary-scale flow, as well as presenting the key concept of a hierarchy of models as used in the climate sciences. Recent advances in the application of dynamical systems theory, on the one hand, and of nonequilibrium statistical physics, on the other, are brought together for the first time and shown to complement each other in helping understand and predict the system's behavior. These complementary points of view permit a self-consistent handling of subgrid-scale phenomena as stochastic processes, as well as a unified handling of natural climate variability and forced climate change, along with a treatment of the crucial issues of climate sensitivity, response, and predictability

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics