Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation, is given. The fractal dimension of a strange attractor and Feigenbaum’s constant of the model are calculated. Moreover, numerical simulations using AUTO and MATLAB are presented to support theoretical results, such as a cascade of period doubling with period-2, 4, 6, 8, 16, 32 orbits, period-10, 20, 19, 38 orbits, invariant curves, codimension-2 bifurcation and chaotic attractor. Chaos in the sense of Marotto is also proved by both analytical and numerical methods. Analyses are displayed to illustrate the effect of magnitude of interference among predators on dynamic behaviors of this model. Further the chaotic orbit is controlled to be a fixed point by using feedback control method.     

Dynamics,circuit realization,control and synchronization of a hyperchaotic hyperjerk system with coexisting attractors

Although different hyperjerk systems have been discovered, a few hyperjerk systems can exhibit hyperchaotic behavior. In this work, we introduce a new hyperjerk system with hyperchaotic attractors. By investigating dynamics of the system, we have observed the different coexisting attractors such as coexistence of period-2 attractors, or coexistence of period-2 attractor and quasiperiodic attractor. It is worth noting that this striking phenomenon is rarely reported in a hyperjerk system. The proposed system has been realized with electronic components. The agreement between the simulation and experimental results indicates the feasibility of the hyperjerk system. Moreover, chaos control and synchronization of such hyperjerk system have been also reported.     

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

多时间尺度问题具有广泛的工程与科学研究背景，慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题，其展现出的振荡形式及内部分岔结构，大多较为单一，此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例，探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线，其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下，存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时，混沌吸引子可能与不稳定不动点相接触，也可能与之相距一定距离.对快子系统吸引域分布的模拟，表明存在着导致边界激变（boundary crisis）的临界值，在这些值附近，经由延迟倍周期分岔演化而来的混沌吸引子可与2ⁿ（n=0，1，2，…）周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后，双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁，从而使系统出现了不同吸引子之间的滞后行为，由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

Phase-Locking and Bifurcations of the Sinusoidally-Driven Double Scroll Circuit

The dynamic alterations of an electronic circuit in a chaotic regime, described by the Double Scroll attractor, subjected to sinusoidal perturbation are numerically investigated. Parameter diagrams of the circuit phase-locking oscillations in terms of the driving amplitude and frequency are computed. These diagrams have highly interleaved and complex structures, part of them Cantor-like fractals. However, a Cantor-like fractal structure is also observed. In addition, the power spectrum analysis is used to find and characterize three ways of phase-locking the Double Scroll circuit, and to determine how this process depends on the driving parameters. Furthermore, the dynamics of bifurcation phenomena, as chaotic attractor entrainment, Arnold's tongues, coexistence of attractors, and hysteresis are identified in the parameter space.     

Bi-parametric topology of subharmonics of an asymmetric bubble oscillator at high dissipation rate

The subharmonic topology of a nonlinear, asymmetric bubble oscillator (Keller–Miksis equation) in glycerine is investigated in the parameter space of its external excitation (frequency and pressure amplitude). The bi-parametric investigation revealed that the exoskeleton of the topology can be described as the composition of U-shaped subharmonics of periodic orbits. The fine substructure (higher-order ultra-subharmonic resonances) usually appearing via the well-known period n-tupling phenomenon is completely missing due to the high dissipation rate of the viscous liquid. Moreover, a complex internal structure of the subharmonics has been found, which are composed by interconnected bifurcation blocks (in a zig-zag pattern) each describing the skeleton of a shrimp-shaped domain. The employed numerical techniques are the combination of an in-house initial value problem solver written in C++/CUDA C to harness the high processing power of professional graphics cards, and the boundary value problem solver AUTO to compute periodic orbits directly regardless of their stability.     

Synthesizing attractors of Hindmarsh–Rose neuronal systems

In this paper a periodic parameter-switching scheme is applied to the Hindmarsh–Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh–Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment.     

Approximate analytical solution to oscillations of a conductor in a magnetic field

An approximate analytical solution to a system exhibiting oscillations of a conductor in a magnetic field which is controlled by a discrete waveform is sought by means of multiple scales. The system involves the use of a solenoid driven by a RLC circuit, coupled with a solid state relay (SSRL), to generate large electromagnetic forces acting on a conductor, which oscillates within the solenoid. The steady state response of the metal bar, in terms of oscillations is described. This solution is expressed in terms of system and circuit parameters, valid in the weakly nonlinear region, which is identified to be small oscillatory displacement near the center of the solenoid. By analyzing different cases of resonance, period-1 and period-2 like motions are identified and validated through experimental studies. The solution provides a guideline to design an effective control strategy so as to guide the system to a desirable attractor.     

System identification of the kinematics of an oscillating cylinder using wake velocities

A classical problem in vortex-induced vibration is to know the flow field past an oscillating cylinder. In this paper we use system theory to identify the oscillatory behaviour of a circular cylinder from flow variables in the wake. We use numerical simulations (CFD) of the flow past a cylinder oscillating in the cross-flow direction at different oscillation frequencies and amplitudes to construct a transfer function that relates the displacement of the cylinder and the resulting flow field. This transfer function can then be inverted to ‘predict’ the displacement of the cylinder given the flow field (as determined by simulations or measurements). We investigate this technique in the so-called lock-in region, where the vortex shedding frequency is synchronised with the oscillation frequency of the cylinder.     

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.     

Singular limit of equations for linear viscoelastic fluids with periodic boundary conditions

We consider a one-parameter family of problems, governing, for any fixed parameter, the motion of a linear viscoelastic fluid in a two-dimensional domain with periodic boundary conditions. The asymptotic behavior of each problem is analyzed, by proving the existence of the global attractor. Moreover, letting the parameter go to zero, since the memory effect disappears, we obtain a limiting problem, given by the Navier-Stokes equations. For any fixed parameter, we construct an exponential attractor. The resulting family is robust, meaning that these exponential attractors converge, in an appropriate sense, to an exponential attractor of the limiting problem.     

An electrically actuated imperfect microbeam: Dynamical integrity for interpreting and predicting the device response

In this study we deal with a microelectromechanical system (MEMS) and develop a dynamical integrity analysis to interpret and predict the experimental response. The device consists of a clamped-clamped polysilicon microbeam, which is electrostatically and electrodynamically actuated. It has non-negligible imperfections, which are a typical consequence of the microfabrication process. A single-mode reduced-order model is derived and extensive numerical simulations are performed in a neighborhood of the first symmetric natural frequency, via frequency response diagrams and behavior chart. The typical softening behavior is observed and the overall scenario is explored, when both the frequency and the electrodynamic voltage are varied. We show that simulations based on direct numerical integration of the equation of motion in time yield satisfactory agreement with the experimental data. Nevertheless, these theoretical predictions are not completely fulfilled in some aspects. In particular, the range of existence of each attractor is smaller in practice than in the simulations. This is because these theoretical curves represent the ideal limit case where disturbances are absent, which never occurs under realistic conditions. A reliable prediction of the actual (and not only theoretical) range of existence of each attractor is essential in applications. To overcome this discrepancy and extend the results to the practical case where disturbances exist, a dynamical integrity analysis is developed. After introducing dynamical integrity concepts, integrity profiles and integrity charts are drawn. They are able to describe if each attractor is robust enough to tolerate the disturbances. Moreover, they detect the parameter range where each branch can be reliably observed in practice and where, instead, becomes vulnerable, i.e. they provide valuable information to operate the device in safe conditions according to the desired outcome and depending on the expected disturbances.     

Localized modes in a variety of driven long Josephson junctions with phase shifts

We study both analytically and numerically the localized modes in long Josephson junctions with phase shift formations, so-called \(0{-}\pi {-}0\) and \(0{-}\kappa \) junctions. The system is described by an inhomogeneous sine-Gordon equation with a variety of time-periodic drives. Perturbation technique, together with multiple-scale expansions, is applied to obtain the amplitude of oscillations. It is observed that the obtained amplitude equations decay with time due to radiative damping and emission of high harmonic radiations. It is also observed that the energy taken away from the internal mode by radiation waves can be balanced by applying either direct or parametric drives. The appropriate external drives are applied to re-balance the dissipative and radiative losses. We discuss in detail the excitation by direct and parametric drives with frequencies to be either in the vicinity or double the natural frequency of the system. It is noted that the presence of external applied drives stabilizes the nonlinear damping, producing stable breather modes in long Josephson junctions. It is also noted that in the presence of parametric drives, the amplitudes of the driving forces are much more sensitive than in the case of external ac drives; that is, in the case of parametric drives, a small change in the amplitudes of the driving forces can make a drastic change in the system behavior and the system becomes unstable as compared to the case of the direct ac driving. Furthermore, we noticed that, in the presence of external driving, the driving effect is stronger for the case of driving frequency nearly equal to the system frequency as compared to that of the driving frequency nearly equal to twice the frequency of the oscillatory mode.     

Cyclic motions near a Hopf bifurcation of a four-dimensional system

We study motions near a Hopf bifurcation of a representative nonconservative four-dimensional autonomous system with quadratic nonlinearities. Special cases of the four-dimensional system represent the envelope equations that govern the amplitudes and phases of the modes of an internally resonant structure subjected to resonant excitations. Using the method of multiple scales, we reduce the Hopf bifurcation problem to two differential equations for the amplitude and phase of the bifurcating cyclic solutions. Constant solutions of these equations provide asymptotic expansions for the frequency and amplitude of the bifurcating limit cycle. The stability of the constant solutions determines the nature of the bifurcation (i.e., subcritical or supercritical). For different choices of the control parameter, the range of validity of the analytical approximation is ascertained using numerical simulations. The perturbation analysis and discussions are also pertinent to other autonomous systems.     

MPS与GPU结合数值模拟LNG液舱晃荡

液舱晃荡是一种在外部激励作用下部分装载的液舱内液体的波动现象,它会对液舱结构强度和运输船舶稳性产生危害.移动粒子半隐式法(moving particle semi-implicit,MPS)是一种典型的无网格粒子类方法,可以有效地模拟剧烈的液舱晃荡问题.但MPS方法存在计算效率低的缺点,难以模拟大规模三维问题,而GPU并行加速技术已广泛应用于科学计算领域.因此,本文将MPS方法与GPU并行加速技术相结合,采用CUDA程序语言编写,自主开发了MPSGPU-SJTU求解器,对三维液化天然气(liquefiednatural gas, LNG)型液舱晃荡进行了数值模拟.通过三种不同粒子间距的数值模拟,验证了求解器的收敛性,其中最大计算粒子数达到了200多万.与其他研究结果相比,MPSGPU-SJTU求解器能够准确地预测壁面砰击压力,并且捕捉晃荡过程中自由面的大幅度变形和强非线性破碎现象.相比CPU求解器的计算时间,GPU并行加速技术可以大幅度地减小计算时长,提高MPS方法的计算效率.本文将LNG型液舱与方型液舱的晃荡进行对比,结果表明在高充液率下LNG型液舱可以有效地减小晃荡幅值和壁面砰击压力.但在中低充液率下,LNG型液舱则会加剧晃荡,自由面呈现明显的三维特征.本文还进一步研究了水和LNG两种不同介质的液舱晃荡现象,数值模拟结果表明二者的流场基本相似,砰击压力则正比于液体密度.      

Inertial waves in a rotating annulus with inclined inner cylinder: comparing the spectrum of wave attractor frequency bands and the eigenspectrum in the limit of zero inclination

We investigate theoretically inertial waves inside a liquid confined between two co-rotating coaxial cylinders of finite length. We consider the case of small viscosity and high angular velocity (i.e., small Ekman numbers), a parameter range of interest for many geophysical applications. In this case, inertial waves propagating in the container show multiple reflections at the walls before the waves can be damped by weak diffusion. We allow for the inner cylinder wall to be parallel or inclined with respect to the annulus’ vector of rotation (truncated cone). For the limit of zero viscosity, the wave propagation is governed by a boundary value problem that is composed of a linear second-order hyperbolic partial differential equation and the impermeability boundary conditions. For the special case of vertical cylinder walls (no inclination of the inner cylinder), this boundary value problem is separable, the corresponding eigenmodes can analytically be found and they are regular. However, when the inner cylinder wall is inclined, the hyperbolicity of the governing equation leads to internal shear layers (corresponding to singularities for the inviscid case). The geometrical structure of the shear layers can be explained by inertial waves, trapped on limit cycles denoted as wave attractors. The shape of the limit cycles depends on the wave frequency. In fact, the spectrum of regular modes, existing for the case of vertical cylinder walls, vanishes almost completely when the inner wall is inclined. Instead of a spectrum of discrete frequencies and regular eigenmodes, a spectrum of wave attractor frequency bands and singular eigenmodes exist. The question addressed here is whether the spectrum of wave attractor intervals collapses to the discrete frequency spectrum when the inclination angle of the inner cylinder goes to zero. To answer this question, the attractor frequency intervals are evaluated numerically for a series of decreasing cylinder inclination angles and are compared with the analytically found eigenspectrum for the case of zero inclination. Goal is to better understand the asymptotic behavior of the problem for decreasing inclination angles. This understanding helps to interpret results from laboratory experiments with geometries that differ from the perfect annulus with parallel cylinder walls.     

On a regular convex solver potential for an elastic-damage constitutive model: a theoretical analysis

This work is related with the proposition of a so-called regular or convex solver potential to be used in numerical simulations involving a certain class of constitutive elastic-damage models. All the mathematical aspects involved are based on convex analysis, which is employed aiming a consistent variational formulation of the potential and its conjugate one. It is shown that the constitutive relations for the class of damage models here considered can be derived from the solver potentials by means of sub-differentials sets. The optimality conditions of the resulting minimisation problem represent in particular a linear complementarity problem. Finally, a simple example is present in order to illustrate the possible integration errors that can be generated when finite step analysis is performed.     

Towards numerical simulations of supersonic liquid jets using ghost fluid method

A computational tool based on the ghost fluid method (GFM) is developed to study supersonic liquid jets involving strong shocks and contact discontinuities with high density ratios. The solver utilizes constrained reinitialization method and is capable of switching between the exact and approximate Riemann solvers to increase the robustness. The numerical methodology is validated through several benchmark test problems; these include one-dimensional multiphase shock tube problem, shock–bubble interaction, air cavity collapse in water, and underwater-explosion. A comparison between our results and numerical and experimental observations indicate that the developed solver performs well investigating these problems. The code is then used to simulate the emergence of a supersonic liquid jet into a quiescent gaseous medium, which is the very first time to be studied by a ghost fluid method. The results of simulations are in good agreement with the experimental investigations. Also some of the famous flow characteristics, like the propagation of pressure-waves from the liquid jet interface and dependence of the Mach cone structure on the inlet Mach number, are reproduced numerically. The numerical simulations conducted here suggest that the ghost fluid method is an affordable and reliable scheme to study complicated interfacial evolutions in complex multiphase systems such as supersonic liquid jets.     

On the physical interpretation of material parameters in phase field models for ferroelectrics

This publication intends to clarify the relations between the free energy expansion and the choice of the order parameter in phase field models for ferroelectrics. The free energy is expanded to include the piezoelectric coupling terms, reflecting the symmetry properties of the material below the Curie temperature. The order parameter can then be interpreted as the material polarization less than the dielectric and piezoelectric contributions. Furthermore, a 1d reduction of the problem is used to identify the physical meaning of the remaining model parameters. The theoretical results are verified by numerical simulations of 180^? and 90^? domain walls. Both the static case, where the emphasis lies on the specific energy and the interface width, and the dynamic case, where interfaces are subjected to an electric driving field, are considered in the simulations.     

时变小扰动Hamilton系统的Hopf分岔

运用Melnikov方法研究了时变小扰动Ｈamilton系统周期轨道发生Ｈopf分岔的条件,并将这些条件应用到一类三维时变小扰动非自治系统,数值结果验证了本文理论的正确性,进一步数值积分表明,所研究的系统还存在复杂而有规律的环面分岔行为。     

基于近似技术的涡轮叶片气动优化设计

根据五次多项式方法进行三维涡轮叶片的参数化建模,采用N-S方程和湍流模型进行三维流场分析计算,以K-S函数法作为优化方法,利用近似技术加速循环优化速度,建立了一种基于近似技术的涡轮叶片的气动优化方法。将气动效率和总压比作为目标函数,对涡轮叶片进行多目标气动优化、形状优化。算例表明本文提出的涡轮叶片优化设计方法是有效的。