Fast-slow analysis for parametrically and externally excited systems with two slow rationally related excitation frequencies

Effect of a modulated acoustic field on the dynamics of a vibrating charged bubble

We investigate the effect of amplitude-modulated acoustic irradiation on the dynamics of a charged bubble vibrating in a liquid. We show that the potential V(x) of the bubble, and the number and stability of its equilibria, depend on the magnitude of the charge it carries. Under high-frequency amplitude-modulation, a modulation threshold, Gth, was found for the onset of increased bubble amplitude oscillations. For some pressure field values, charge can facilitate the control of chaotic dynamics via reversed period-doubling bifurcation sequences. There is evidence for peak-shouldering and shock waves. The Mach number increases rapidly with the drive amplitude G. In the supersonic regime, for G>1.90Pa, the high-frequency modulation raises both Blake's and the transient cavitation thresholds. We found a decrease in the bubble's maximum charge threshold, and threshold modulation amplitude for the occurrence Vibrational resonance (VR). VR occurs due to the modulated oscillatory pressure field, and the influence on VR of the electrostatic charge, and other parameters of the system are investigated. In contrast to the cases of VR reported earlier, where the amplitude G of the high-frequency driving is typically much higher than the amplitude of the low-frequency driving (Ps), the VR resonance peaks occur here at relatively low G values (0 Collapse

Key Words

• Acoustic waves
• Amplitude modulation
• Bubble oscillator
• Electrostatic charges
• Vibrational resonance

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Affiliation(s)

O T Kolebaje Department of Physics, Adeyemi Federal University of Education, Ondo, Ondo State, Nigeria; Department of Physical Sciences, Redeemer's University, P.M.B. 230, Ede, Nigeria
U E Vincent Department of Physical Sciences, Redeemer's University, P.M.B. 230, Ede, Nigeria; Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom.
B E Benyeogor Department of Physical Sciences, Redeemer's University, P.M.B. 230, Ede, Nigeria
P V E McClintock Department of Physics, Lancaster University, Lancaster LA1 4YB, United Kingdom

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Improving the Performance of a Post-Buckled Beam Harvester under Combined External and Parametrical Slow Excitations

In this paper, we consider novel bursting energy harvesting under combined external and parametrical slow excitations, and a harvester is realized by employing an externally and parametrically excited post-buckled beam. Based on the method of fast-slow dynamics analysis, multiple-frequency oscillation, with two slow commensurate excitation frequencies, is used to observe complex bursting patterns, the behaviors of the bursting response are presented, and some novel one-parameter bifurcation patterns are observed. Furthermore, the bursting harvesting performances of the single and the two slow commensurate excitation frequencies are compared, and it was found that the two slow commensurate excitation frequencies can be used to improve the harvesting voltage.

Control of Optoelectronic Scanning and Tracking Seeker by Means the LQR Modified Method with the Input Signal Estimated Using of the Extended Kalman Filter

The paper presents the concept of controlling the designed optoelectronic scanning and tracking seeker. The above device is intended for the so-called passive guidance of short-range anti-aircraft missiles to various types of air maneuvering targets. In the presented control method, the modified linear-quadratic regulator (LQR) and the estimation of input signals using the extended Kalman filter (EKF) were used. The LQR regulation utilizes linearization of the mathematical model of the above-mentioned seeker by means of the so-called Jacobians. What is more, in order to improve the stability of the seeker control, vector selection of signals received by the optoelectronic system was used, which also utilized EKF. The results of the research are presented in a graphical form. Numerical simulations were carried out on the basis of the author’s own program developed in the programming language C++.

Parametric excitation induced extreme events in MEMS and Liénard oscillator

Two paradigmatic nonlinear oscillatory models with parametric excitation are studied. The authors provide theoretical evidence for the appearance of extreme events (EEs) in those systems. First, the authors consider a well-known Liénard type oscillator that shows the emergence of EEs via two bifurcation routes: intermittency and period-doubling routes for two different critical values of the excitation frequency. The authors also calculate the return time of two successive EEs, defined as inter-event intervals that follow Poisson-like distribution, confirming the rarity of the events. Further, the total energy of the Liénard oscillator is estimated to explain the mechanism for the development of EEs. Next, the authors confirmed the emergence of EEs in a parametrically excited microelectromechanical system. In this model, EEs occur due to the appearance of a stick-slip bifurcation near the discontinuous boundary of the system. Since the parametric excitation is encountered in several real-world engineering models, like macro- and micromechanical oscillators, the implications of the results presented in this paper are perhaps beneficial to understand the development of EEs in such oscillatory systems.

Bursting and mixed mode oscillations during the transition to limit cycle oscillations in a matrix burner

We investigate the route to self-excited thermoacoustic instability in a laminar flow multiple flame matrix burner. With an increase in the equivalence ratio, the thermoacoustic system that is initially quiet (stable operation) transitions to limit cycle oscillations through two distinct dynamical states, namely, bursting oscillations and mixed mode oscillations. The acoustic pressure oscillations transition from quiescence to large amplitudes during bursting oscillations. Such high amplitude bursting oscillations that occur well ahead of the onset of limit cycle oscillations can potentially cause structural damage. The thermoacoustic system exhibits hysteresis. The transition to limit cycle oscillations is replicated in a phenomenological model containing slow-fast time scales.

Natural extension of fast-slow decomposition for dynamical systems

Modeling and parameter estimation to capture the dynamics of physical systems are often challenging because many parameters can range over orders of magnitude and are difficult to measure experimentally. Moreover, selecting a suitable model complexity requires a sufficient understanding of the model's potential use, such as highlighting essential mechanisms underlying qualitative behavior or precisely quantifying realistic dynamics. We present an approach that can guide model development and tuning to achieve desired qualitative and quantitative solution properties. It relies on the presence of disparate time scales and employs techniques of separating the dynamics of fast and slow variables, which are well known in the analysis of qualitative solution features. We build on these methods to show how it is also possible to obtain quantitative solution features by imposing designed dynamics for the slow variables in the form of specified two-dimensional paths in a bifurcation-parameter landscape.

Route to bursting via pulse-shaped explosion

This Rapid Communication reports on the discovery of a route to bursting, called a pulse-shaped explosion (PSE), for a paradigmatic class of nonlinear oscillators. We find that both an equilibrium and a limit cycle can exhibit pulse-shaped sharp quantitative changes in relation to the variation of system parameters, which are interesting explosive behaviors, the PSE. It leads to large-amplitude oscillations in the rest phase (i.e., small-amplitude oscillations) of bursting, giving rise to additional active phases alternating with the rest phase, and finally determines compound bursting structures. This way, the route to complex bursting dynamics by PSE is explained and its robustness is shown. PSE opens different ways for the control dynamics of complex systems.

Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations

This paper aims to report two novel bursting patterns, the turnover-of-pitchfork-hysteresis-induced bursting and the compound pitchfork-hysteresis bursting, demonstrated for the Duffing system with multiple-frequency parametric excitations. Typically, a hysteresis behavior between the origin and non-zero equilibria of the fast subsystem can be observed due to delayed pitchfork bifurcation. Based on numerical analysis, we show that the stable equilibrium branches, related to the non-zero equilibria resulted from the pitchfork bifurcation, may become the ones with twists and turns. Then, the novel bursting pattern turnover-of-pitchfork-hysteresis-induced bursting is revealed accordingly. In particular, we show that additional pitchfork bifurcation points may appear in the fast subsystem under certain parameter conditions. This creates multiple delay-induced hysteresis behavior and helps us to reveal the other novel bursting pattern, the compound pitchfork-hysteresis bursting. Besides, effects of parameters on the bursting patterns are studied to explore the relation of these two novel bursting patterns.

Inverse period-doubling bifurcations determine complex structure of bursting in a one-dimensional non-autonomous map

We propose a simple one-dimensional non-autonomous map, in which some novel bursting patterns (e.g., "fold/double inverse flip" bursting, "fold/multiple inverse flip" bursting, and "fold/a cascade of inverse flip" bursting) can be observed. Typically, these bursting patterns exhibit complex structures containing a chain of inverse period-doubling bifurcations. The active states related to these bursting can be period-2(n) (n = 1, 2, 3,…) attractors or chaotic attractors, which may evolve to quiescence by a chain of inverse period-doubling bifurcations when the slow excitation decreases through period-doubling bifurcation points of the map. This accounts for the complex inverse period-doubling bifurcation structures observed in bursting patterns. Our findings enrich the possible routes to bursting as well as the underlying mechanisms of bursting.