Bursting generation mechanism in a three-dimensional autonomous system,chaos control,and synchronization in its fractional-order form

The bifurcation mechanism of bursting oscillations in a three-dimensional autonomous slow-fast Kingni et al. system (Nonlinear Dyn. 73, 1111–1123, 2013) and its fractional-order form are investigated in this paper. The stability analysis of the system is carried out assuming that the slow subsystem evolves on quasi-static state. It is reveaved that the bursting oscillations found in the system result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. We refer this class of bursting to “source/bursting.” The coexistence of symmetrical bursting limit cycles and chaotic bursting attractors is observed. In addition, the fractional-order chaotic slow-fast system is studied. The lowest order of the commensurate form of this system to exhibit chaotic behavior is found to be 2.199. By tuning the commensurate fractional-order, the chaotic slow-fast system displays Chen- and Lorenz-like chaotic attractors, respectively. The stability analysis of the controlled fractional-order-form of the system to its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Moreover, the synchronization of chaotic bursting oscillations in two identical fractional-order systems is numerically studied using the unidirectional linear error feedback coupling scheme. It is shown that the system can achieve synchronization for appropriate coupling strength. Furthermore, the effect of fractional derivatives orders on chaos control and synchronization is analyzed.     

Multistability in a three-dimensional oscillator: tori,resonant cycles and chaos

The emergence of multistability in a simple three-dimensional autonomous oscillator is investigated using numerical simulations, calculations of Lyapunov exponents and bifurcation analysis over a broad area of two-dimensional plane of control parameters. Using Neimark–Sacker bifurcation of 1:1 limit cycle as the starting regime, many parameter islands with the coexisting attractors were detected in the phase diagram, including the coexistence of torus, resonant limit cycles and chaos; and transitions between the regimes were considered in detail. The overlapping between resonant limit cycles of different winding numbers, torus and chaos forms the multistability.     

Regular oscillations,chaos, and multistability in a system of two coupled van der Pol oscillators: numerical and experimental studies

In this paper, the dynamics of a system of two coupled van der Pol oscillators is investigated. The coupling between the two oscillators consists of adding to each one’s amplitude a perturbation proportional to the other one. The coupling between two laser oscillators and the coupling between two vacuum tube oscillators are examples of physical/experimental systems related to the model considered in this paper. The stability of fixed points and the symmetries of the model equations are discussed. The bifurcations structures of the system are analyzed with particular attention on the effects of frequency detuning between the two oscillators. It is found that the system exhibits a variety of bifurcations including symmetry breaking, period doubling, and crises when monitoring the frequency detuning parameter in tiny steps. The multistability property of the system for special sets of its parameters is also analyzed. An experimental study of the coupled system is carried out in this work. An appropriate electronic simulator is proposed for the investigations of the dynamic behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results yields a very good agreement.     

Regular oscillations or chaos in a fractional order system with any effective dimension

This paper introduces a fractional order system which can generate regular oscillations or create chaos. It shows that this system is capable to create regular or nonregular oscillations under suitable conditions. These necessary conditions are achieved by violation of the no-chaos criteria. The effective dimension of the proposed system can be chosen any order less than three. Therefore, this system is a good example for limit cycle or chaos generation via fractional-order systems with low orders. Numerical simulations illustrate behavior of the proposed system in different situations.     

Antimonotonicity,chaos and multiple attractors in a novel autonomous memristor-based jerk circuit

A novel memristor-based oscillator derived from the autonomous jerk circuit (Sprott in IEEE Trans Circuits Syst II Express Briefs 58:240–243, 2011) is proposed. A first-order memristive diode bridge replaces the semiconductor diode of the original circuit. The complex behavior of the oscillator is investigated in terms of equilibria and stability, phase space trajectories plots, bifurcation diagrams, graphs of Lyapunov exponents, as well as frequency spectra. Antimonotonicity (i.e. concurrent creation and destruction of periodic orbits), chaos, periodic windows and crises are reported. More interestingly, one of the main features of the novel memristive jerk circuit is the presence of a region in the parameters’ space in which the model develops hysteretic behavior. This later phenomenon is marked by the coexistence of four different (periodic and chaotic) attractors for the same values of system parameters, depending solely on the choice of initial conditions. Basins of attractions of various competing attractors display complex basin boundaries thus suggesting possible jumps between coexisting solutions in experiment. Compared to previously published jerk circuits with similar behavior, the novel system distinguishes by the presence of a single equilibrium point and a relatively simpler structure (only off-the-shelf electronic components are involved). Results of theoretical analyses are perfectly traced by laboratory experimental measurements.     

Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation

Nonlinear Dynamics - In this paper, multi-timescale dynamics and the formation mechanism of a 3D non-autonomous system with two slowly varying periodic excitations are systematically investigated....     

A four-wing hyper-chaotic attractor and transient chaos generated from a new 4-D quadratic autonomous system

This paper presents a new four-dimensional (4-D) smooth quadratic autonomous chaotic system, which can present periodic orbit, chaos, and hyper-chaos under the conditions on different parameters. Importantly, the system can generate a four-wing hyper-chaotic attractor and a pair of coexistent double-wing hyper-chaotic attractors with two symmetrical initial conditions. Furthermore, a four-wing transient chaos occurs in the system. The dynamic analysis approach- in the paper involves time series, phase portraits, Poincaré maps, bifurcation diagrams, and Lyapunov exponents, to investigate some basic dynamical behaviors of the proposed 4-D system.     

Nonlinear isochronous oscillations of a fluid in a paraboloid: theory and experiment

Within the framework of the shallow-water model, the nonlinear axisymmetric oscillations of a fluid in a paraboloid of revolution and in an unbounded parabolic channel are investigated. It is established that in the paraboloid of revolution the oscillation period does not depend on the amplitude, that is, the oscillations are isochronous. Experimental investigations of free fluid oscillations in a paraboloid confirm this theoretical result.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, 2004, pp. 131–142. Original Russian Text Copyright © 2004 by Kalashnik, Kakhiani, Lominadze, Patarashvili, Svirkunov, and Tsakadze.     

Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system with slow parametric excitation

Nonlinear Dynamics - For dynamic systems with multiple time scales, its long-term behaviors usually present as mixed mode oscillations (MMOs). The formation of MMOs is closely related to the fast...     

Fractional-order complex T system: bifurcations,chaos control,and synchronization

In this paper, the fractional-order complex T system is proposed. The dynamics of the system including symmetry, the stability of equilibrium points, bifurcations with variation of system parameters, and derivative orders are investigated. Period-doubling and tangent bifurcations with appropriate derivative orders and system parameters are observed. Besides, the control problem of the system is examined by using the feedback control technique. Furthermore, based on the stability theory of fractional-order systems, the scheme of function projective synchronization for the fractional-order complex T system is presented. The function projective synchronization for the system is realized by designing an appropriate synchronization controller. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.     

Periodic solution of a second order, autonomous, nonlinear system

There exist a sufficient condition for the existence of at least one periodic solution for a type of second order autonomous ordinary differential equations. The correctness of the condition has been pointed out by Schauder's fixed point theorem. In order to indicate the validity of the assumptions made, two illustrative examples, showing its application in the nonlinear vibration and relaxation oscillation are presented.     

Chaos in three-dimensional hybrid systems and design of chaos generators

The existence of horseshoes is proved in a class of 3-dim piecewise linear systems, in which a homoclinic orbit connecting the origin to itself is explicitly given. Based on these results, a mathematically rigorous methodology for design of chaos generators is proposed. Implementation of such chaos generators by circuit is easy and the chaotic attractor is robust under small perturbations.     

Temporal large-eddy simulation: theory and implementation

Large-eddy simulation (LES) has relied almost exclusively on spatial filtering to separate resolved and unresolved scales. For many reasons, temporal filtering may be more natural, particularly for flows of engineering interest. The paper develops the theory of temporal LES (TLES) and provides a demonstration of the concept by simulations of viscous Burger’s flow and incompressible plane-channel flow. The latter is accomplished by adapting the approximate deconvolution model (ADM) of Stolz and Adams (Phys. Fluids 11:1699, 1999) to causal, time-domain filtering. The temporal variant of the ADM is termed the TADM.      

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

An experimental study of bifurcation,chaos, and dimensionality in a system forced through a bifurcation parameter

An experimental study of a system that is parametrically excited through a bifurcation parameter is presented. The system consits of a lightly-damped, flexible beam which is buckled and unbuckled magnetically: it is parametrically excited by driving an electromagnet with a low-frequency sine wave. For voltage amplitudes in excess of the static bifurcation value, the beam slowly switches between the one-and two-well configurations. Experimental static and dynamic bifurcation results are presented. Static bifurcatons for the system are shown to involve a butterfly catastrophe. The dynamic bifurcation diagram, obtained with an automated data acquisition system, shows several period-doubling sequences, jump phenomena, and a chaotic region. Poincaré sections of a chaotic steady-state are obtained for various values of the driving phase, and the correlation dimension of the chaotic attractor is estimated over a large scaling region. Singular system analysis is used to demonstrate the effect of delay time on the noise level in delay-reconstructions, and to provide an independent check on the dimension estimate by directly estimating the number of independent coordinates from time series data. The correlation dimension is also estimated using the delay-reconstructed data and shown to be in good agrement with the value obtained from the Poincaré sections. The bifurcation and dimension results are used together with physical sonsiderations to derive the general form of a single-degree-of-freedom model for the experimental system.