Search

The occurrence of synchronization and amplitude death phenomena due to the coupled interaction of limit cycle oscillators (LCO) has received increased attention over the last few decades in various fields of science and engineering. Studies pertaining to these coupled oscillators are often performed by studying the effect of various coupling parameters on their mutual interaction. However, the effect of system parameters (i.e., the amplitude and frequency) on the coupled interaction of such LCO has not yet received much attention, despite their practical importance. In this paper, we investigate the dynamical behavior of time-delay coupled Stuart–Landau (SL) oscillators exhibiting subcritical Hopf bifurcation for the variation of amplitude and frequency of these oscillators in their uncoupled state. For identical SL oscillators, a gradual increase in the amplitude of LCO shrinks the amplitude death regions observed between the regions of in-phase and anti-phase synchronization leading to its eventual disappearance, resulting in the occurrence of phase-flip bifurcations at higher amplitudes of LCO. We also observe an alternate existence of in-phase and anti-phase synchronization regions for higher values of time delay, whose prevalence of occurrence increases with an increase in the frequency of the oscillator. With the introduction of frequency mismatch, the region of amplitude death. The forced response of SL oscillator shows an asymmetry in the Arnold tongue and the manifestation of asynchronous quenching of LCO. An increase in the amplitude of LCO narrows the Arnold tongue and reduces the region of asynchronous quenching observed in the system. Finally, we compare the coupled and forced response of SL oscillators with the corresponding experimental results obtained from laminar thermoacoustic oscillators and the numerical results from van der Pol (VDP) oscillators. We show that the SL model qualitatively displays many features observed experimentally in coupled and forced thermoacoustic oscillators. In contrast, the VDP model does not capture most of the experimental results due to the limitation in the independent variation of system parameters.

This article studies routes to chaos occurring within a resonance wedge for a 3-parametric family of differential equations acting on a 3-sphere. Our starting point is an autonomous vector field whose flow exhibits a weakly attracting heteroclinic network made by two 1-dimensional connections and a 2-dimensional separatrix between two equilibria with different Morse indices. After changing the parameters, while keeping the 1-dimensional connections unaltered, we concentrate our study in the case where the 2-dimensional invariant manifolds of the equilibria do not intersect. We derive the first return map near the ghost of the attractor and we reduce the analysis of the system to a 2-dimensional map on the cylinder. Complex dynamical features arise from a discrete-time Bogdanov-Takens singularity, which may be seen as the organizing center by which one can obtain infinitely many attracting tori, strange attractors, infinitely many sinks and non-trivial contracting wandering domains. These dynamical phenomena occur within a structure that we call resonance wedge. As an application, we may see the "classical" Arnold tongue as a projection of a resonance wedge. The results are general, extend to other contexts and lead to a fine-tuning of the theory., Comment: 32 pages, 14 figures

Towards controlling the frequency of limit cycle locomotion as well as adapting to rough terrain and performing specific tasks, a novel and indirect method has been recently introduced using an active wobbling mass attached to limit cycle walkers. One of the strongest advantages of the method is the easiness of its implementation, prompting its applicability to a wide variety of locomotion systems. To deeply understand the nonlinear dynamics for further enhancement of the methodology, we use a combined rimless wheel with an active wobbling mass as an example to perform nonlinear analysis in this paper. First, we introduce the simplified equation of motion and the gait frequency control method. Second, we obtain Arnold tongue, which represents region of entrained locomotion. In regions where the locomotion is not entrained, we explore chaotic and quasi-periodic gaits. To characterize bistability of two different locomotions that underlie hysteresis phenomena, basins of attraction for the two locomotions were computed. The present nonlinear analysis may help understanding the detailed mechanism of indirectly controlled limit cycle walkers.

The occurrence of strange non-chaotic attractors (SNA) in quasiperiodically forced systems has attracted considerable interest over the last two decades, in particular since it provides a rich class of examples for the possibility of complicated dynamics in the absence of chaos. Their existence was first described by Millionscikov (and later by Vinograd and also Herman) for quasiperiodic \({\rm SL(2, {\mathbb R})}\) -cocycles and by Grebogi et al (and later Keller) for so-called pinched skew products. However, except for these two particular classes there are still hardly any rigorous results on the topic, despite a large number of numerical studies confirming the widespread existence of SNA in quasiperiodically forced systems. Here, we prove the existence of SNA in quasiperiodically forced circle maps under rather general conditions, which can be stated in terms of \({{\mathcal C}^1}\) -estimates. As a consequence, we obtain the existence of SNA for parameter sets of positive measure in suitable parameter families. These SNA carry the unique physical measure of the system, which determines the behaviour of Lebesgue-almost all initial conditions. Finally, we show that the dynamics are minimal in the considered situations. The results apply in particular to a forced version of the Arnold circle map. For this example, we also describe how the first Arnold tongue collapses and looses its regularity due to the presence of strange non-chaotic attractors and a related unbounded mean motion property.

We extend a few well-known results about orientation preserving homeomorphisms of the circle to orientation preserving circle maps, allowing even an infinite number of discontinuities. We define a set-valued map associated to the lift by filling the gaps in the graph, that shares many properties with continuous functions. Using elementary set-valued analysis, we prove existence and uniqueness of the rotation number, periodic limit orbit in the case when the latter is rational, and Cantor structure of the unique limit set when the rotation number is irrational. Moreover, the rotation number is found to be continuous with respect to the set-valued extension if we endow the space of such maps with the Haussdorff topology on the graph. For increasing continuous families of such maps, the set of parameter values where the rotation number is irrational is a Cantor set (up to a countable number of points).

In this paper, we show the existence of the Arnold diffusion in the elliptic restricted three-body problem. This gives one of the very few examples of Arnold diffusion in real physical systems. The construction is based on the transversal homoclinic orbits in the circular restricted three-body problem ([6, 7, 14]). We prove that the small perturbations to the horseshoe maps in the neighborhood of the homoclinic orbits creates the Arnold diffusion. The existence of the Arnold diffusion also shows that the elliptic restricted three-body problem is non-integrable.