International Journal of Bifurcation and Chaos

This paper discusses the time-dependent dynamics of electrostatic vibration energy harvesters (eVEHs) with linear and nonlinear mechanical resonators. These eVEHs are fundamentally nonlinear regardless of whether a linear or nonlinear resonator is being used. The model of the system under investigation has the form of a piecewise-smooth dynamical system of a Filippov type that has a specific discontinuity in the form of a hold-on term. We use a perturbation technique called the multiple scales method to develop a theory to analyze the steady-state dynamics of the system, be it with a linear or a nonlinear resonator. We then analyze the stability of the steady-state orbit to determine when the first doubling bifurcation occurs in the system. This gives an upper bound on the region of steady-state oscillations which allows us to determine a theoretical limit on the power convertible by the eVEH. We then turn our discussion to the nonlinear behavior we see in the system's transition to chaos. Since the eVEH studied here is a Filippov type system, sliding modes and sliding bifurcations are possible in the system. We discuss the evolution of the sliding region and give particular examples of sliding phenomena and sliding bifurcations. An understanding of sliding phenomena is required for analyzing the transition to chaos since segments of sliding motion appear on trajectories that undergo period-doubling bifurcations. The transition to chaos is explained in detail by the example of the system with a linear resonator, however we discuss examples of the system with mechanical nonlinearities and discuss the difference between the linear and nonlinear cases.

The role of the number of breakpoints N in the sawtooth form of the characteristic function in the modified Chua's circuit model equation on vibrational and ghost-vibrational resonances is investigated in this paper. To observe vibrational resonance, the system should be driven by two periodic forces of frequencies ω and Ω, with Ω ≫ ω. Resonance occurs at the frequency ω when the amplitude of the high-frequency force is varied. When the system is subjected to an input signal containing multifrequencies which are of higher-order than a certain (missing) fundamental frequency, then a resonance at the missing fundamental frequency is induced by the high-frequency input signal and is called ghost-vibrational resonance. In both types of resonances, the number of resonances is N and hysteresis occurs in each resonance region. There are some similarities and differences in these two resonance phenomena. We report in detail the influence of the role of number of breakpoints N on the features of vibrational and ghost-vibrational resonances.

In this paper, we develop a parameter matching technique using adaptive synchronization with a chaotic attractor. Two Chua's oscillators, identical in every parameter except for one, are set-up in a master/slave configuration where the slave's mismatched parameter is adaptable. Using Lyapunov functions and incorporating the presence of chaotic signals, adaptive control laws are designed to ensure exact parameter matching. One of the derived adaptive controllers is experimentally validated by using an adaptive inductor-gyrator composed of current feedback op-amps (CFOAs). The experimental results are compared to high-fidelity SPICE simulations, and performance of adaptive controllers are compared over a wide range of parameter values using MATLAB.

This paper deals with the nonlocal hydrodynamic models based on Brenner's modification of Navier–Stokes equations. Traveling wave solutions of the nonlocal models are studied in more detail. The models are shown to admit the multiperiodic, chaotic, as well as homoclinic solutions. The influence of spatiotemporal nonlocalities upon the solutions' features is studied by the Poincaré section technique.

In this paper, we examine the localization problem of compact invariant sets of Raychaudhuri equations with nonzero parameters. The main attention is attracted to the localization of periodic/homoclinic orbits and homoclinic cycles: we prove that there are neither periodic/homoclinic orbits nor homoclinic cycles; we find heteroclinic orbits connecting distinct equilibrium points. We describe some unbounded domain such that nonescaping to infinity positive semitrajectories which are contained in this domain have the omega-limit set located in the boundary of this domain. We find a locus of other types of compact invariant sets respecting three-dimensional and two-dimensional invariant planes. Besides, we describe the phase portrait of the system obtained from the Raychaudhuri equations by the restriction on the two-dimensional invariant plane.

We generalize some properties of (f, δ)-recurrence introduced in [Szała, 2013] to n-dimensional case and show that (f, δ)-recurrence and uniform (f, δ)-recurrence are equivalent. We also study properties of one-dimensional nonautonomous dynamical systems with randomly perturbed trajectories.

This paper studies the global bifurcations and multipulse chaotic dynamics of a four-edge simply supported honeycomb sandwich rectangular plate under combined in-plane and transverse excitations. Based on the von Karman type equation for the geometric nonlinearity and Reddy's third-order shear deformation theory, the governing equations of motion are derived for the four-edge simply supported honeycomb sandwich rectangular plate. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional nonautonomous nonlinear system is simplified to a three-order standard form by using the normal form method. The extended Melnikov method is improved to investigate the six-dimensional nonautonomous nonlinear dynamical system in a mixed coordinate. The global bifurcations and multipulse chaotic dynamics of the four-edge simply supported honeycomb sandwich rectangular plate are studied by using the improved extended Melnikov method. The multipulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.

We investigate combinatorial issues relating to the use of random orbit approximations to the attractor of an iterated function system with the aim of clarifying the role of the stochastic process during the generation of the orbit. A Baire category counterpart of almost sure convergence is presented.

The dynamic feedback control of the cardiac pacing interval has been widely used to suppress alternans. In this paper, temporally and spatially suppressing the alternans for cardiac tissue consisting of a one-dimensional chain of cardiac units is investigated. The model employed is a nonlinear partial difference equation. The model's fixed points and their stability conditions are determined, and bifurcations and chaos phenomenon have been studied by numerical simulations. The main objective of this paper is to stabilize the unstable fixed point of the model. The proposed approach is nonlinear spatiotemporal delayed feedback, and the appropriate interval for controller feedback gain is calculated using the linear stability analysis. It is proven that the proposed approach is robust with respect to all bifurcation parameter variations. Also, set point tracking is achieved by employing delayed feedback with an integrator. Finally, simulation results are provided to show the effectiveness of the proposed methodology.

Degenerate grazing bifurcation occurring in a simple bilinear oscillator, namely the limit discontinuous case of the smooth and discontinuous (SD) oscillator, is investigated by numerical simulations. The unperturbed system has a saddle-like singularity at the origin with two periodic orbits grazing at the same point. The matrix in the leading-order truncation of the Poincaré map at a grazing bifurcation for either of the two periodic orbits has a zero eigenvalue. Our numerical experiments suggest that the near-grazing dynamics are much more complicated than previously found. The results obtained in this paper is expected to be useful in developing analytical methods of this type of degenerate grazing bifurcations.

Photovoltaic (PV) systems with a battery back-up form an integral part of distributed generation systems and therefore have recently attracted a lot of interest. In this paper, we consider a system of charging a battery from a PV panel through a current mode controlled boost dc-dc converter. We analyze its complete nonlinear/nonsmooth dynamics, using a piecewise model of the converter and realistic nonlinear v–i characteristics of the PV panel. Through this study, it is revealed that system design without taking into account the nonsmooth dynamics of the converter combined with the nonlinear v–i characteristics of the PV panel can lead to unpredictable responses of the overall system with high current ripple and other undesirable phenomena. This analysis can lead to better designed converters that can operate under a wide variation of the solar irradiation and the battery's state of charge. We show that the v–i characteristics of the PV panel combined with the battery's output voltage variation can increase or decrease the converter's robustness, both under peak current mode control and average current mode control. We justify the observation in terms of the change in the discrete-time map caused by the nonlinear v–i characteristics of the PV panel. The theoretical results are validated experimentally.

A generalized memristor consisting of a memristive diode bridge with a first order parallel RC filter is proposed in this letter. The mathematical model of the circuit is established and its fingerprints are analyzed by the pinched hysteresis loops with different periodic stimuli. The results verified by experimental measurements indicate that the proposed circuit is a simple voltage-controlled generalized memristor.

Bifurcations in a cubic system with a degenerate saddle point are investigated using the technique of blow-up, the method of planar perturbation theory and qualitative analysis. It has been found that after appropriate perturbations, at least 12 limit cycles can bifurcate from a degenerate saddle point in a type of cubic systems.

Nonlinear dynamical flows are often solved approximately by numerical integration based on advancing the independent variable, typically time. Here, we present a unified framework for numerical integration of differential equations based on advancing either the dependent variable, often space, or the independent variable, or any combination of the two. Spacetime stepping unifies and extends previous work. In particular, dependent variable stepping can produce better numerical results and also more faithfully describe the underlying physics, thereby providing both practical and conceptual benefits. We indicate extensions to higher dimensions.

In this letter we investigate the role of complex fixed-points in finding hidden attractors in chaotic flows with no equilibria. If these attractors could be found by starting the trajectory in the neighborhood of complex fixed-points, maybe it would be better not to call them hidden.

In this work, we study a model of a one-dimensional magnetic metamaterial formed by a discrete array of nonlinear resonators. We focus on periodic and localized traveling waves of the model, in the presence of loss and an external drive. Employing a Melnikov analysis we study the existence and persistence of such traveling waves, and study their linear stability. We show that, under certain conditions, the presence of dissipation and/or driving may stabilize or destabilize the solutions. Our analytical results are found to be in good agreement with direct numerical computations.

The Browder nonejective fixed point theorem [Browder, 1965] will be generalized in two directions. One generalization will be then applied to the existence of nonejective topological multivalued fractals on Peano's continua. An illustrative example will be supplied.

In this paper, we construct the polytope which contains all compact ω-limit sets of the four-dimensional Rössler system which is a generalization of the hyperchaotic Rössler system for the case of positive parameters. Further, we find a few three-dimensional planes containing all compact ω-limit sets for bounded positive half-trajectories located in some subdomains in the half-space z > 0. Besides, we analyze one case in which all compact ω-limit sets in the half-space z > 0 are contained in one three-dimensional plane. Our approach is based on a combination of the LaSalle theorem and the extreme-based localization method of compact invariant sets.