Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system

This paper presents a new four-dimensional autonomous system having complex hyperchaotic dynamics. Basic properties of this new system are analyzed, and the complex dynamical behaviors are investigated by dynamical analysis approaches, such as time series, Lyapunov exponents’ spectra, bifurcation diagram, phase portraits. Moreover, when this new system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of hyperchaotic systems reported before, which implies the new system has strong hyperchaotic dynamics in itself. The Kaplan–Yorke dimension, Poincaré sections and the frequency spectra are also utilized to demonstrate the complexity of the hyperchaotic attractor. It is also observed that the system undergoes an intermittent transition from period directly to hyperchaos. The statistical analysis of the intermittency transition process reveals that the mean lifetime of laminar state between bursts obeys the power-law distribution. It is shown that in such four-dimensional continuous system, the occurrence of intermittency may indicate a transition from period to hyperchaos not only to chaos, which provides a possible route to hyperchaos. Besides, the local bifurcation in this system is analyzed and then a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincaré–Andronov–Hopf bifurcation theorem. Numerical simulation results show consistency with our theoretical analysis.     

Slope modulation of ring waves governed by two-dimensional sine–Gordon equation

A theory of slope modulation of waves governed by the two-dimensional sine–Gordon equation is proposed. A large time asymptotic solution describing the slope modulation of trains of ring kinks is obtained. The comparison with the numerical solution of two-dimensional sine–Gordon equation shows excellent agreement.     

Effect of strain hardening in elastic–plastic transition behavior in a hemisphere in contact with a rigid flat

An axisymmetrical hemispherical asperity in contact with a rigid flat is modeled for an elastic–plastic material on the lines of the Kogut–Etsion Model (KE Model) and the Jackson–Green Model (JG Model). The present work extends the previous KE and JG works, accounting for the effect of realistic material behavior in terms of the varying yield strengths and the isotropic strain hardening behavior. The predicted results show that the transition behavior of the materials from the elastic–plastic to the fully plastic case is influenced by the yield strength and the tangent modulus (E_t) and such transition do not take place at specific values of interference ratios as suggested by the KE model. New empirical relations are proposed to determine the contact load and the contact area based on the analysis. Numerical results from the finite element modeling are also validated with an experimental ball on flat configuration approach.     

Stress–softening effect of SBR/nanocomposites by a phenomenological Gent–Zener viscoelastic model

An experimental study of a tensile loading–unloading procedure, as well as multi-cyclic response in a strain-controlled program of a Styrene-Butadiene (SBR) elastomer reinforced with four different weight fractions of carbon nanotubes (CNTs) has been performed. The Mullins effect features, namely hysteresis, damage and residual strain, exhibited by the SBR/nanocomposites were analyzed by a modified Gent–Zener rheological model, and a damage function. Especially for the multi-cyclic stress–strain curves, phenomenological equation of the model parameters evolution with strain were also introduced. The same loading procedure was applied in pre-stressed materials, revealing a different stress–strain response due to strain prehistory. The model has been proven to accurately capture the loading–unloading behavior, the residual strain, hysteresis loops as well as the multi-cyclic behavior of the SBR/CNT nanocomposites.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

Study of laminar–turbulent flow transition under pulsatile conditions in a constricted channel

In this paper, direct numerical simulation is performed to investigate a pulsatile flow in a constricted channel to gain physical insights into laminar–turbulent–laminar flow transitions. An in-house computer code is used to conduct numerical simulations based on available high-performance shared memory parallel computing facilities. The Womersley number tested is fixed to 10.5 and the Reynolds number varies from 500 to 2000. The influences of the degree of stenosis and pulsatile conditions on flow transitions and structures are investigated. In the region upstream of the stenosis, the flow pattern is primarily laminar. Immediately after the stenosis, the flow recirculates under an adverse streamwise pressure gradient, and the flow pattern transitions from laminar to turbulent. In the region far downstream of the stenosis, the flow becomes re-laminarised. The physical characteristics of the flow field have been thoroughly analysed in terms of the mean streamwise velocity, turbulence kinetic energy, viscous wall shear stresses, wall pressure and turbulence kinetic energy spectra.     

Instability of a liquid–vapor phase transition front in inhomogeneous wettable porous media

The stability of vertical flows through a horizontally extended two-dimensional region of a porous medium is considered in the case of presence of a phase transition front. It is shown that the plane steady-state phase transition front may have several steady-state positions in the wettable porous medium and the necessary condition of their existence is obtained. The spectral stability of the plane phase transition interface is investigated. It is found that in the presence of capillary forces exerted on the phase transition front in the wettable medium the plane front can be destabilized on the mode with both infinite and zero wavenumbers (short- and long-wave instabilities); the short-wave instability can then exist even in the case of the sole steady-state position of the front.     

Investigation on effect of drag models on flow behavior of power-law fluid–solid two-phase flow in fluidized bed

In this study, a Eulerian-Eulerian two-fluid model combined with the kinetic theory of granular flow is adopted to simulate power-law fluid–solid two-phase flow in the fluidized bed. Two new power-law liquid–solid drag models are proposed based on the rheological equation of power-law fluid and pressure drop. One called model A is a modified drag model considering tortuosity of flow channel and ratio of the throat to pore, and the other called model B is a blending drag model combining drag coefficients of high and low particle concentrations. Predictions are compared with experimental data measured by Lali et al., where the computed porosities from model B are closer to the measured data than other models. Furthermore, the predicted pressure drop rises as liquid velocity increases, while it decreases with the increase of particle size. Simulation results indicate that the increases of consistency coefficient and flow behavior index lead to the decrease of drag coefficient, and particle concentration, granular temperature, granular pressure, and granular viscosity go down accordingly.     

Chaos break and synchrony enrichment within Hindmarsh–Rose-type memristive neural models

Nonlinear Dynamics - The fluctuation of ions concentration across the cell membrane of neuron can generate a time varying electromagnetic field. Thus, memristors are used to realize the coupling...     

On the traveling waves governed by the Camassa–Holm equation

The Camassa–Holm equation admits undistorted traveling waves that are either smooth or exhibit peaks or cusps. All three wave types can be periodic or solitary. Also waves of different types may be combined. In the present paper it is shown that, apart from peaks and cusps, the traveling waves governed by the Camassa–Holm equation can be found from some simpler equation. In the case of peaked solutions, this reduced equation is even linear. The governing equation of traveling waves in its original form can be interpreted as a nonlinear combination of the reduced equation and its first integral. For a small range of the integration constant, the reduced equation admits bounded solutions, which then are directly inherited by the Camassa–Holm equation. In general, the solutions of the reduced equation are unbounded and cannot be considered to represent traveling waves. The full equation, however, has a nonlinearity in the highest derivative, which is characteristic for the Camassa–Holm and some other equations. This nonlinear term offers the possibility of constructing bounded traveling waves from the unbounded solutions of the reduced equation. These waves necessarily have discontinuities in the slope and are, therefore, solutions only in a generalized sense.     

Lag synchronization of multiple identical Hindmarsh–Rose neuron models coupled in a ring structure

Lag synchronization of multiple identical Hindmarsh–Rose neuron systems coupled in a ring structure is investigated. In the coupled systems, each neuron receives signals only via synaptic strength from the nearest neighbors. Based on the Lyapunov stability theory, the sufficient conditions for synchronization of the multiple systems with chaotic bursting behavior can be obtained. The synchronization condition about the control parameter g is also obtained by numerical method. Finally, numerical simulations are provided to show the effectiveness of the developed methods.     

Chaos control of a φ 6-Van der Pol oscillator driven by external excitation

This paper studies the dynamics of a ?? ⁶-Van der Pol oscillator subjected to an external excitation. Numerical analysis is presented to observe its periodic and chaotic motions, and a method called Multiple-prediction Delayed Feedback Control is proposed to control chaos effectively via periodic feedback gain. The controller is designed based on plural Poincaré maps which are defined to regard the nonautonomous system as a T-periodic discrete time system, therefore, the stability of the closed-loop system can be evaluated from the theory of monodromy matrix. Numerical simulations are provided to illustrate the validity of the proposed control strategy.     

Impulsive stabilization of mechanical systems in Takagi–Sugeno models

Takagi–Sugeno fuzzy impulsive systems are analyzed for Lyapunov stability. Lyapunov’s second method is used to establish sufficient stability conditions for such systems. It is shown that these conditions are expressed by a system of matrix inequalities. Impulsive fuzzy control of two coupled pendulums is considered as an example     

Is the suppression of short waves by a swell a three-dimensional effect?

We consider the phenomenon of suppression of short waves by a long wave, observed by Mitsuyasu in 1966. The recently proposed [1] essentially 3-D explanation of this phenomenon is reviewed and compared with more traditional 2-D explanations. Several physical implications of this 3-D explanation are suggested and the experimental verification is discussed.     

Pattern formation in a nonequilibrium phase transition for a generalized Burgers–Fisher equation

A nonequilibrium phase transition of a generalized Burgers–Fisher equation describing biological pattern formation with a periodic boundary condition is examined. In the presence of a weak external force, some approximate bifurcation solutions near a critical point and new spatially periodic patterns are obtained by using the perturbation method in an infinite-dimensional space. The result shows that the external force delays the bifurcation.     

Chaos in a tethered satellite system induced by atmospheric drag and Earth’s oblateness

Nonlinear Dynamics - This paper describes the chaos behavior of an in-plane tethered satellite system induced by atmospheric drag and the Earth’s oblateness. A commonly used model, the...