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In this paper, an inertial amplifier and a quasi-zero stiffness system are combined to propose an energy harvesting system that can change the dynamic mass of the system. By adjusting the inertial amplifier, it is possible to change the distribution of its chaotic region and the effective range of energy collection. The kinetic equations of the system with an inertial amplifier under a combination of multiple harmonic excitations are developed. Based on this, the system was simulated and tested and the Lyapunov exponent, the RMS value of the induced voltage and the coexisting basins of attraction were plotted. Subsequently, the best optimization solution to improve the energy harvesting efficiency of the system was identified. In addition, the system was tested for impulse excitation based on the given coexisting basins of attraction. The simulation results show that the inertial amplifier can effectively improve the range of the energy harvesting region and the distribution of the chaotic region in the system ground under ultra-low-frequency vibration. The initiation of impulse excitation can change the energy harvesting performance of the system to a great extent.

The paper devotes to the synthesis of local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processors. The discrete model gives rise to a two-dimensional non-invertible map, whose basins of attraction have complicated topological structures due to the intrinsic multi-stability. The influences of joint parameters on the local dynamics are presented in great details. Both theoretical and numerical results are plotted on the two-dimensional parametric planes. To show more detailed bifurcation structure, the isoclines are extended to higher periodic orbits for detecting the cusps of resonant entrainments. Invariant manifolds and critical curves are employed to illustrate the global dynamics of the model vividly. The tangency and intersections of invariant manifolds expound the process of erosions of basins of attraction. The global bifurcations of basins of attraction are deduced dynamically by critical curves.

Chain-rules are maximally chaotic cellular automata (CA) rules that can be constructed at random to provide a huge number of encryption keys – where the CA is run backwards to encrypt, forwards to decrypt. The methods are based on the 1D CA reverse algorithm for directly finding pre-images, and the resulting Z-parameter, and rely on the essential property that chain-rules have minimal in-degree in their basins of attraction, usually an in-degree of just one for larger systems.

We consider the modeling and asymmetry-induced dynamics for a class of chaotic circuits sharing the same feature of an antiparallel diodes pair as the nonlinear component. The simple autonomous jerk circuit of [J. Kengne, Z. T. Njitacke, A. N. Nguomkam, M. T. Fouodji and H. B. Fotsin, Coexistence of multiple attractors and crisis route to chaos in a novel chaotic jerk circuit, Int. J. Bifurcation Chaos Appl. Sci. Eng. 26 (2016) 1650081] is used as the prototype. In contrast to current approaches where the diodes are assumed to be identical (and thus a perfect symmetric circuit), we examine the more realistic situation where the diodes have different electrical properties in spite of unavoidable scattering of parameters. In this case, the nonlinear component formed by the diodes pair displays an asymmetric current–voltage characteristic which induces asymmetry of the whole circuit. The model is described by a continuous-time 3D autonomous system (ODEs) with exponential nonlinearities. We examine the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of operation by using bifurcation diagrams and phase space trajectory plots as the main indicators. Period doubling route to chaos, merging crisis, and multiple coexisting (i.e., two, four, or six) mutually symmetric attractors are reported in the symmetric mode of oscillation. In the asymmetric mode, several unusual nonlinear behaviors arise such as coexisting bifurcations, hysteresis, asymmetric double-band chaotic attractor, crisis, and coexisting multiple (i.e., two, three, four, or five) asymmetric attractors for some suitable ranges of parameters. Theoretical analyses and circuit experiments show a very good agreement. The results obtained in this work let us conjecture that chaotic circuits with antiparallel diodes pair are capable of much more complex dynamics than what is reported in the current literature and thus should be reconsidered accordingly in spite of the approach followed in this work.

In a normal power system, many generators are operating in synchrony. That is, they all have the same speed or frequency, the system frequency. In case some accident occurs, a situation might arise in which one or more generators are running at a different speed, much faster than the system frequency. They are said to be running away or stepping out, or in a state of accelerated stepping out. We have been engaged in a series of studies of this situation, and have found global attractor-basin portraits. In the course of this program, we have observed the phenomenon of decelerated stepping out, in which one or more generators deviate from the system frequency toward lower speeds. These kinds of behavior cannot be explained with the well-known model involving one generator operating on to an infinite bus. Rather, we require a model in which robust subsystems — for example, generator/motor combination, which we call swing pairs — are connected by interconnecting transmission lines. In this more general context, the deviant behaviors we are considering may be regarded as forms of desynchronization of subsystems. We therefore begin this paper with the derivation of a new mathematical model, in which there is no infinite bus nor fixed system frequency. In the simple case of two subsystems (each a swing pair) weakly coupled by an interconnecting transmission line, we develop a system of seven differential equations which include the variation of frequency in a fundamental way. We then go on to study the behavior of this model, using our usual methods of computer simulation to draw the attractor-basin portraits. We have succeeded in finding both accelerated and decelerated stepping out in this new model. In addition, we discovered an unexpected subharmonic swinging of the whole system.

We study a finite uni-directional array of "cascading" or "threshold coupled" chaotic maps. We describe some of the attractors for such systems and prove general results about their basins of attraction. In particular, we show that the basins of attraction have infinitely many path components. We show that these components always accumulate at the corners of the domain of the system. For all threshold parameters above a certain value, we show that they accumulate at a Cantor set in the interior of the domain. For certain ranges of the threshold, we prove that the system has many attractors.

Two-dimensional (Z₁–Z₃–Z₁) maps are such that the plane is divided into three unbounded open regions: a region Z₃, whose points generate three real rank-one preimages, bordered by two regions Z₁, whose points generate only one real rank-one preimage. This paper is essentially devoted to the study of the structures, and the global bifurcations, of the basins of attraction generated by such maps. In particular, the cases of fractal structure of such basins are considered. For the class of maps considered in this paper, a large variety of dynamic situations is shown, and the bifurcations leading to their occurrence are explained.

When two phase oscillators interact with a time-delay, new synchronized and desynchronized regimes appear, in this way giving rise to the phenomenon of multistability. The number of coexisting stable states grows with an increase of the delay, and their frequencies quantize. We show that, while the number of synchronized solutions grows linearly with the delay and/or coupling, the set of the desynchronized solutions, i.e. those with different average frequencies of the individual oscillators, raises quadratically with increasing delay. For the synchronized states, we analyze the mutual arrangement of the basins of attraction, and conclude that the structure and size of the basins are apparently the same for each state. We discuss possible implications for desynchronizing brain stimulation techniques.

This paper is concerned with the computation of the basins of attraction of a simple one degree-of-freedom backlash oscillator using cell-to-cell mapping techniques. This analysis is motivated by the modeling of order vibration in geared systems. We consider both a piecewise-linear stiffness model and a simpler infinite stiffness impacting limit. The basins reveal rich and delicate dynamics, and we analyze some of the transitions in the system's behavior in terms of smooth and discontinuity-induced bifurcations. The stretching and folding of phase space are illustrated via computations of the grazing curve, and its preimages, and manifold computations of basin boundaries using DsTool (Dynamical Systems Toolkit).

We consider the dynamics of the three-dimensional model of the die which can bounce with dissipation on the table. It is shown that for the realistic values of the initial energy the probabilities of the die landing on the face which is the lowest one at the beginning is larger than the probabilities of landing on any other face.

We study synchronization of a number of rotating pendulums mounted on a horizontal beam which can roll on the parallel surface. It has been shown that after the initial transient, different states of pendulums synchronization occur. We derive the analytical equations for the estimation of the phase differences between phase synchronized pendulums. After the study of the basins of attraction of different synchronization states, we argue that the observed phenomena are robust as they occur for a wide range of both initial conditions and system parameters.

This short paper takes a close look at a relatively simple harmonically-excited mechanical oscillator. Throughout the range of forcing frequencies the basins of attraction are investigated by applying strong perturbations to steady-state behavior. In this way, a more general solution space is mapped out. Numerical simulation of the equation of motion agrees very closely with data generated from a laboratory experiment.

The fundamental principle of bistability is widely used across various disciplines, including biology, chemistry, mechanics, physics, electronics and materials science. As the need for more powerful, efficient and sensitive complex-engineered systems grow, networks of coupled bistable systems have gained significant attention in recent years. Modeling and analysis of such higher-dimensional systems is usually focused on finding conditions for the existence and stability of typical invariant sets, i.e. steady states, periodic solutions and chaotic sets. High-dimensionality leads to complex patterns of collective behavior. Which type of behavior is exhibited by a network depends greatly on the initial conditions. Thus, it is also important to study the geometric structure and evolution of the basins of attraction of such patterns. In this manuscript, a complete study of the basins of attraction of a ring of bistable systems, coupled unidirectionally, is presented. 3D visualizations are included to aid the discussion of the changes in the basins of attraction as the coupling parameter varies. The results are broad enough that they can be applied to a wide range of systems with similar coupling topologies.

There are many ways that a person can encounter chaos, such as through a time series from a lab experiment, a basin of attraction with fractal boundaries, a map with a crossing of stable and unstable manifolds, a fractal attractor, or in a system for which uncertainty doubles after some time period. These encounters appear so diverse, but the chaos is the same in all of the underlying systems; it is just observed in different ways. We describe these different types of chaos. We then give two conjectures about the types of dynamical behavior that is observable if one randomly picks out a dynamical system without searching for a specific property. In particular, we conjecture that from picking a system at random, one observes (1) only three types of basic invariant sets: periodic orbits, quasiperiodic orbits, and chaotic sets; and (2) that all the definitions of chaos are in agreement.

A crisis of amplitude control can occur when a system is multistable. This paper proposes a new chaotic system with a line of equilibria to demonstrate the threat to amplitude control from multistability. The new symmetric system has two coefficients for amplitude control, one of which is a partial amplitude controller, while the other is a total amplitude controller that simultaneously controls the frequency. The amplitude parameter rescales the basins of attraction and triggers a state switch among different states resulting in a failure of amplitude control to the desired state.

The Newton–Raphson basins of attraction, associated with the libration points (attractors), are revealed in the generalized Hill problem. The parametric variation of the position and the linear stability of the equilibrium points is determined, when the value of the perturbation parameter $𝜖$ varies. The multivariate Newton–Raphson iterative scheme is used to determine the attracting domains on several types of two-dimensional planes. A systematic and thorough numerical investigation is performed in order to demonstrate the influence of the perturbation parameter on the geometry as well as of the basin entropy of the basins of convergence. The correlations between the basins of attraction and the corresponding required number of iterations are also illustrated and discussed. Our numerical analysis strongly indicates that the evolution of the attracting regions in this dynamical system is an extremely complicated yet very interesting issue.

Supercavity can increase the velocity of underwater vehicles greatly, however the launching state of vehicle and systematic parameters often lead to unstable motion. To solve the problem, the effect of parameters and initial conditions on the stability of vehicles is studied. With two variable parameters, namely cavitation number and feedback control gain of fin deflection angle, a simple dynamic model of supercavity system is studied. The multistability is verified through simulation. Robustness of the system is also analyzed based on its basins of attraction. There are various coexisting attractors in a relatively large region of parameter space of the supercavity system, namely coexistence of a stable equilibrium point and a periodic attractor, coexistence of various periodic attractors, coexistence of a periodic attractor with a chaotic attractor and so on, which explain the effect of parameters and initial values on stability of vehicles qualitatively. In addition, without major change in cavitation number, there is a negative correlation between the robustness of the vehicle and feedback control gain of fin deflection angle. The robustness can be improved through optimization of parameters.

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

Tooth surface contact deformation is part of the main causes for gear system vibration. The safety region of the single-stage spur system vibration is established based on the tooth surface contact deformation. Safe basin and its erosion are calculated numerically according to the safety region as the system control parameters are varied. The basin of attraction in the safe basin is computed by combining the simple cell-to-cell method. Vibration safety and global dynamics of the gear system are investigated. Some bifurcations and mechanisms of the erosion of safe basin are studied by using phase portraits, Poincaré maps, bifurcation diagrams under multi-initial values and top Lyapunov exponents (TLE). The sensitivity of system behavior and bifurcation to initial values are discussed as well. Hidden bifurcating points and attractors are revealed. To get a better understanding of the sensitivity of the system behavior to initial values, a bifurcation dendrogram under multi-initial values is designed. It is found that there is the erosion of safe basin existing in the examining area. Both vibration amplitude changing of coexisting attractors and appearing or disappearing of coexisting attractors are the main causes for the erosion of the safe basin. Bifurcation of the system behavior is selective to initial values. With varying frequency, backlash and comprehensive transmission error, the period-1 response gradually bifurcates under multi-initial conditions and evolves into other new periodic responses that coexist with the period-1 response. The new coexisting response has a key effect on the erosion of the safe basin. The study is helpful to the optimization design of the gear system and the control of the system behavior.

The stability and the two-parameter bifurcation of a two-dimensional discrete Gierer–Meinhardt system are investigated in this paper. The analysis is carried out both theoretically and numerically. It is found that the model can exhibit codimension-two bifurcations ($1:2$, $1:3$, and $1:4$ strong resonances) for certain critical values at the positive fixed point. The normal forms are obtained by using a series of affine transformations and bifurcation theory. Numerical simulations including bifurcation diagrams, phase portraits and basins of attraction are conducted to validate the theoretical predictions, which can also display some interesting and complex dynamical behaviors.