Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity

This paper aims at offering an insight into the dynamical behaviors of incommensurate fractional-order singularly perturbed van der Pol oscillators subjected to constant forcing, especially when the forcing is close to Andronov–Hopf bifurcation points. These bifurcation points are predicted thanks to the theorem on stability of incommensurate fractional-order systems, as functions of the forcing and fractional derivative orders. When the forcing is chosen near Andronov–Hopf bifurcation, the dynamics of fractional-order systems show a static-looking transient regime whose length increases exponentially with the closeness to the bifurcation point. This peculiar phenomenon is not common in numerical simulation of dynamical systems. We show that this quasi-static transient behavior is due to the combine action of the slow passage effect at folded saddle-node singularity and fractional derivation memory effect on the slow flow around this singularity; this forces the system to remain for a long time in the vicinity of its equilibrium point, though unstable. The system frees oneself from this quasi-static transient state by spiraling before entering relaxation oscillation. Such a situation results in mixed mode oscillations in the oscillatory regime. One obtains mixed mode oscillations from a very simple system: A two-variable system subjected to constant forcing.     

Numerical calculation of the elastic and plasticbehavior of saturated porous media

In this investigation the field equations governing the mechanical behavior of a fluid-saturated porous media are analyzed and built up for the study of elastic dynamical problems and quasi-static problems in case of elastic–plastic material behavior. The investigations are limited to small deformation in order to apply a geometrical linear approach. The two constituents are assumed to be microscopically incompressible. A numerical solution is derived by means of the standard Galerkin procedure and the finite element method.     

QUASI-STATIC ANALYSIS FOR VISCOELASTIC TIMOSHENKO BEAMS WITH DAMAGE

Based on convolution-type constitutive equations for linear viscoelastic materials with damage and the hypotheses of Timoshenko beams, the equations governing quasi-static and dynamical behavior of Timoshenko beams with damage were first derived. The quasi-static behavior of the viscoelastic Timoshenko beam under step loading was analyzed and the analytical solution was obtained in the Laplace transformation domain. The deflection and damage curves at different time were obtained by using the numerical inverse transform and the influences of material parameters on the quasi-static behavior of the beam were investigated in detail.     

Rate independent hysteresis in a bi-stable chain

The nontrivial behavior of an elastic chain with identical bi-stable elements may be considered prototypical for a large number of nonlinear processes in solids ranging from phase transitions to fracture. The energy landscape of such a chain is extremely wiggly which gives rise to multiple equilibrium configurations and results in a hysteretic evolution and a possibility of trapping. In the present paper, which extends our previous study of the static equilibria in this system (Puglisi and Truskinovsky, J. Mech. Phys. Solids (2000) 1), we analyze the behavior of a bi-stable chain in a soft device under quasi-static loading. We assume that the system is over-damped and explore the variety of available nonequilibrium transformation paths. In particular, we show that the “minimal barrier” strategy leads to the localization of the transformation in a single spring. Loaded periodically, our bi-stable chain exhibits finite hysteresis which depends on the height of the admissible barrier; the cold work/heat ratio in this model is a fixed constant, proportional to the Maxwell stress. Comparison of the computed inner and outer hysteresis loops with recent experiments on shape memory wires demonstrates good qualitative agreement. Finally we discuss a relation between the present model and the Preisach model which is a formal interpolation scheme for hysteresis, also founded on the idea of bi-stability.     

Dynamics on certain sets of stochastic matrices

We study iteration of polynomials on symmetric stochastic matrices. In particular, we focus on a certain one-parameter family of quadratic maps which exhibits chaotic behavior for a wide range of the parameters. The well-known dynamical behavior of the quadratic family on the interval, and its dependence on the parameter, is reproduced on the spectrum of the stochastic matrices. For certain subclasses of stochastic matrices the referred dynamical behavior is also obtained in the matrix entries. Since a stochastic matrix characterizes a Markov chain, we obtain a discrete dynamical system on the space of reversible Markov chains. Therefore, depending on the parameter, there are initial conditions for which the corresponding reversible Markov chains will lead under iteration to a fixed point, to a periodic point, or to an aperiodic point. Moreover, there are sensitivity to initial conditions and the coexistence of infinite repulsive periodic orbits, both features of chaos.     

Exact solutions,conservation laws,bifurcation of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver equation

In the present work, we observe the dynamical behavior of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver (STO) equation. Exact solutions are derived using \({1}/{G^{^{\prime }}}\) expansion and modified Kudryashov methods. The wave transformation is used to transform STO equation into an ordinary differential equation. Combining Runge–Kutta fourth-order and Fourier spectral technique, we use a mixed scheme for the numerical study of STO equation. Since spectral methods expand the solution in trigonometric series resulting into higher-order technique and Runge–Kutta produces improved accuracy, we extract these qualities for a mixed scheme. Results so produced are presented graphically which provide a useful information about the dynamical behavior. Bifurcation behavior of nonlinear and supernonlinear traveling waves of STO equation is studied with the help of bifurcation theory of planar dynamical systems. It is observed that STO equation supports nonlinear solitary wave, periodic wave, shock wave, stable oscillatory wave and most important supernonlinear periodic wave.     

不可压饱和多孔弹性梁、杆动力响应的数学模型

基于多孔介质理论,首先建立了饱和多孔弹性杆件弯曲与轴向变形时动力响应的数学模型.其次,基于多孔弹性梁弯曲变形的数学模型,利用Laplace变换,分析了两端可渗透的饱和多孔弹性悬臂梁在自由端受阶梯载荷作用下的动静力响应,给出了梁弯曲时挠度、弯矩以及孔隙流体压力等效力偶等物理量随时间的响应曲线.发现不可压多孔弹性梁的拟静态响应亦存在Mandel-Cryer现象,多孔弹性梁的挠度具有与粘弹性梁挠度类似的蠕变特征,然而,其应力响应不同于粘弹性梁,随着时间的增加,梁拟静态响应的弯矩逐渐增加,并达到一个稳态值.这些结果有助于揭示植物根茎等力学行为的机理.     

Emergence of diverse dynamical responses in a fractional-order slow–fast pest–predator model

To explore the impact of pest-control strategy on integrated pest management, a three-dimensional (3D) fractional- order slow–fast prey–predator model is introduced in this article. The prey community (assumed as pest) represents fast dynamics and two predators exhibit slow dynamical variables in the three-species interacting prey–predator model. In addition, common enemies of that pest are assumed as predators of two different species. Pest community causes serious damage to the economy. Fractional-order systems can better describe the real scenarios than classical-order dynamical systems, as they show previous history-dependent properties. We establish the ability of a fractional-order model with Caputo’s fractional derivative to capture the dynamics of this prey–predator system and analyze its qualitative properties. To investigate the importance of fractional-order dynamics on the behavior of the pest, we perform the local stability analysis of possible equilibrium points, using certain assumptions for different sets of parameters and reveal that the fractional-order exponent has an impact on the stability and the existence of Hopf bifurcations in the prey–predator model. Next, we discuss the existence, uniqueness and boundedness of the fractional-order system. We also observe diverse oscillatory behavior of different amplitude modulations including mixed mode oscillations (MMOs) for the fractional-order prey–predator model. Higher amplitude pest periods are interspersed with the outbreaks of small pest concentration. With the decrease of fractional-order exponent, small pest concentration increases with decaying long pest periods. We further notice that the reduced-order model is biologically significant and sensitive to the fractional-order exponent. Additionally, the dynamics captures adaptation that occurs over multiple timescales and we find consistent differences in the characteristics of the model for various fractional exponents.

     

Dynamic Stability of Equilibrium Capillary Drops

We investigate a model for contact angle motion of quasi-static capillary drops resting on a horizontal plane. We prove global in time existence and long time behavior (convergence to equilibrium) in a class of star-shaped initial data for which we show that topological changes of drops can be ruled out for all times. Our result applies to any drop which is initially star-shaped with respect to a small ball inside the drop, given that the volume of the drop is sufficiently large. For the analysis, we combine geometric arguments based on the moving-plane type method with energy dissipation methods based on the formal gradient flow structure of the problem.     

Transition waves in bistable structures. I. Delocalization of damage

We consider chains of dimensionless masses connected by breakable bistable links. A non-monotonic piecewise linear constitutive relation for each link consists of two stable branches separated by a gap of zero resistance. Mechanically, this model can be envisioned as a ”twin-element” structure which consists of two links (rods or strands) of different lengths joined by the ends. The longer link does not resist to the loading until the shorter link breaks. We call this construction the waiting link structure. We show that the chain of such strongly non-linear elements has an increased in-the-large stability under extension in comparison with a conventional chain, and can absorb a large amount of energy. This is achieved by two reasons. One is an increase of dissipation in the form of high-frequency waves transferring the mechanical energy to heat; this is a manifestation of the inner instabilities of the bonds. The other is delocalization of the damage of the chain. The increased stability is a consequence of the distribution of a partial damage over a large volume of the body instead of its localization, as in the case of a single neck formation in a conventional chain. We optimize parameters of the structure in order to improve its resistance to a slow loading and show that it can be increased significantly by delocalizing a damage process. In particular, we show that the dissipation is a function of the gap between the stable branches and find an optimal gap corresponding to maximum energy consumption under quasi-static extension. The results of numerical simulations of the dynamic behavior of bistable chains show that these chains can withstand without breaking the force which is several times larger than the force sustained by a conventional chain. The formulation and results are also related to the modelling of compressive destruction of a porous material or a frame construction which can be described by a two-branched diagram with a large gap between the branches. We also consider an extension of the model to multi-link chain that could imitate plastic behavior of material.     

Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation

In this paper, we consider a (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation, which is a higher-order generalization of the celebrated Kadomtsev–Petviashvili (KP) equation. By considering the Hirota bilinear form of the CDGKS equation, we study a type of exact interaction waves by the way of vector notations. The interaction solutions, which possess extensive applications in the nonlinear system, are composed by lump wave parts and soliton wave parts, respectively. Under certain conditions, this kind of solutions can be transformed into the pure lump waves or the stripe solitons. Moreover, we provide the graphical analysis of such solutions in order to better understand their dynamical behavior.     

On Dirichlet Problem for a Class of Delayed Reaction–Diffusion Equations with Spatial Non-locality

We consider a very general class of delayed reaction–diffusion equations in which the reaction term can be non-monotone as well as spatially non-local. By employing comparison technique and a dynamical system approach, we study the global asymptotic behavior of solutions to the equation subject to the homogeneous Dirichlet condition. Established are threshold results and global attractiveness of the trivial steady state, as well as the existence, uniqueness and global attractiveness of a positive steady state solution to the problem. As illustrations, we apply our main results to the local delayed diffusive Mackey–Glass equation and the nonlocal delayed diffusive Nicholson blowfly equation, leading to some very sharp results for these two particular models.     

Experimental investigation of the stress–stretch behavior of EPDM rubber with loading rate effects

The tensile stress–stretch behavior of an ethylene–propylene–diene terpolymer (EPDM) was experimentally investigated, both in a quasi-static stretching rate range (<0.4/s) with a conventional material test machine and in a dynamic stretching rate range (2800/s–3200/s) with a split Hopkinson tension bar (SHTB) technique. Experimental data were then analyzed using the Ogden and Roxburgh’s idealized Mullins effect modeling theory. Results show that the stress–stretch behavior is significantly dependent on stretching rate and the Mullins effect exists under dynamic loading. Furthermore, stretching rate only affects the material properties. The degree of damage in a stretched specimen is a function of only the maximum stretch ratio the specimen experienced.     

On the encounter of an acoustic shear pulse with a phase boundary in an elastic material: energy and dissipation

The fully dynamical motion of a phase boundary is examined for a specific class of elastic materials whose stress-strain relation in simple shear is nonmonotone. Previous work has shown that a preexisting stationary phase boundary in such a material can be set in motion by a finite amplitude shear pulse and that an infinity of solutions is possible according to the present theory. In this work, these solutions are examined in detail from the perspective of energy and dissipation. It is shown that there exists at most two solutions which involve no dissipation (corresponding to conservation of mechanical energy). It is also shown that there exists one solution that maximizes the mechanical energy dissipation rate. The total mechanical energy remaining in the dynamical fields after one such pulse-phase boundary encounter is shown to exceed the total methanical energy after either an energy minimal quasi-static motion or a maximally dissipative quasi-static motion.     

Autoparametric amplification of two nonlinear coupled mass–spring systems

The gearboxes of machines generally operate under a time-varying state rather than under steady-state conditions. However, it is difficult to investigate the nonlinear dynamics of a time-varying gear system. A gear system model of a railway vehicle was proposed in consideration of its time-varying mesh stiffness, nonlinear backlash, transmission error, time-varying external excitation, and rail irregularity. To obtain the nonlinear behaviors of a time-varying stochastic gear system, a quasi-static analysis was performed to observe its doubling-periodic bifurcation, chaotic motion, and transition from a lower to a higher power periodic motion. Based on the energy comparison results, the time-varying stochastic gear system was degraded to a time-varying system to simplify the calculation. Furthermore, the nonlinear response of the time-varying system was computed using the Runge–Kutta method and was compared with the results of a quasi-static analysis that employed a short-time Fourier transform method. The results of the quasi-static analysis were consistent with the results of the time–frequency analysis for the time-varying gear system except for the result at 3180 r/min, which represented a short period wherein the process transitioned to chaos. Hence, the comparison demonstrates the applicability of the quasi-static analysis for the nonlinear behavior analysis of a time-varying stochastic system.     

Analog circuit design and optimal synchronization of a modified Rayleigh system

This paper addresses the problem of optimization of the synchronization of a chaotic modified Rayleigh system. We first introduce a four-dimensional autonomous chaotic system which is obtained by the modification of a two-dimensional Rayleigh system. Some basic dynamical properties and behaviors of this system are investigated. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the proposed system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. Furthermore, we propose an optimal robust adaptive feedback which accomplishes the synchronization of two modified Rayleigh systems using the controllability functions method. The advantage of the proposed scheme is that it takes into account the energy wasted by feedback coupling and the closed loop performance on synchronization. Also, a finite horizon is explicitly computed such that the chaos synchronization is achieved at an established time. Numerical simulations are presented to verify the effectiveness of the proposed synchronization strategy. Pspice analog circuit implementation of the complete master–slave controller system is also presented to show the feasibility of the proposed scheme.     

A General Approach to the Asymptotic Behavior of Traveling Waves in a Class of Three-Component Lattice Dynamical Systems

In this paper, we develop a general approach to deal with the asymptotic behavior of traveling wave solutions in a class of three-component lattice dynamical systems. Then we demonstrate an application of these results to construct entire solutions which behave as two traveling wave fronts moving towards each other from both sides of x-axis for a three-species competition system with Lotka–Volterra type nonlinearity in a lattice.     

The Melnikov method for detecting chaotic dynamics in a planar hybrid piecewise-smooth system with a switching manifold

This paper discusses stability conditions and a chaotic behavior of the Lorenz dynamical system involving the Caputo fractional derivative of orders between 0 and 1. Contrary to some existing results on the topic, we study these problems with respect to a general (not specified) value of the Rayleigh number as a varying control parameter. Such a bifurcation analysis is known for the classical Lorenz system; we show that analysis of its fractional extension can yield different conclusions. In particular, we theoretically derive (and numerically illustrate) that nontrivial equilibria of the fractional Lorenz system become locally asymptotically stable for all values of the Rayleigh number large enough, which contradicts the behavior known from the classical case. As a main proof tool, we derive the optimal Routh–Hurwitz conditions of fractional type, i.e., necessary and sufficient conditions guaranteeing that all zeros of the corresponding characteristic polynomial are located inside the Matignon stability sector. Beside it, we perform other bifurcation investigations of the fractional Lorenz system, especially those documenting its transition from stability to chaotic behavior.     

A nonlinear symmetry breaking effect in shear cracks

Shear cracks propagation is a basic dynamical process that mediates interfacial failure. We develop a general weakly nonlinear elastic theory of shear cracks and show that these experience tensile-mode crack tip deformation, including possibly opening displacements, in agreement with Stephenson's prediction. We quantify this nonlinear symmetry breaking effect, under two-dimensional deformation conditions, by an explicit inequality in terms of the first and second order elastic constants in the quasi-static regime and by semi-analytic calculations in the fully dynamic regime. Our general results are applied to various materials. Finally, we discuss related works in the literature and note the potential relevance of elastic nonlinearities for various problem, including frictional sliding.     

Analysis of worm-like locomotion driven by the sine-squared strain wave in a linear viscous medium

In this paper, a worm-like locomotion in a linear resistive medium is studied to achieve controlled shape changes of the worm-like body by choosing a kind of driving with low energy expended and high-velocity locomotion in certain condition. To this end, we first develop the full dynamic model of the system under consideration to obtain the mean velocity related to friction coefficient, wave speed, linear density, body length and wave width. Correspondingly, a quasi-static model is also given from which the velocity can be expressed analytically. In the case of the shape change driven by the sine-squared strain wave (SSSW), it is seen that these two velocities will tend to uniformity with the friction coefficient or length of the body increasing or the wave speed decreasing when keeping the other parameters unchanged. Thus, the inertia term is ignorable for a large friction, a long body-length but a small wave-speed of the SSSW, which implies that the dynamical model can be reduced to the quasi-static one. The relative criterion is approximately given. As a result, the corresponding quasi-static model is employed to consider two typical drives, namely, the SSSW and the square strain wave (SSW). The result shows the shape change driven by the SSSW has an advantage in both the mean velocity and the average energy expended over that by the SSW when the necessary condition is satisfied. The analytical results are verified by numerical simulation.