Dynamics of a mass–spring–pendulum system with vastly different frequencies

We investigate the dynamics of a simple pendulum coupled to a horizontal mass?Cspring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass?Cspring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and ??/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.     

Nonlinear dynamics of a non-autonomous network with coupled discrete–continuum oscillators

A network model of a multi-modular floating platform incorporated with a runway structure, viewed as a non-autonomous network with discrete–continuum oscillators, is developed for a general purpose of dynamic analysis. Numerical analysis shows the coupling effect between the two different types of oscillators on various complex dynamics, including sudden leaps, torus motions, beating vibrations, the synergetic effect of phase lock and anti-phase synchronizations. The amplitude death phenomenon, a suppressed weak oscillation state, is studied by using the fundamental solution derived by the averaging method. The parametric domain of the onset of amplitude death is illustrated to show the great significance to the stability design of the floating platform. The effect of the flexural rigidity of the runway on the distribution of amplitude death state is also discussed.     

Adaptive bidirectionally coupled synchronization of?chaotic?systems with unknown parameters

This paper investigates the chaos synchronization of two bidirectionally coupled chaotic systems. In comparison with previous methods (identical bidirectionally coupled synchronization), the present control scheme is different bidirectionally coupled synchronization, which includes different complete bidirectionally coupled synchronization and different partial bidirectionally coupled synchronization. Based on the Lasalle invariance principle, adaptive schemes are designed to make two different bidirectionally coupled chaotic systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Theoretical analysis and numerical simulations are shown to verify the results.     

Stability and bifurcation analysis of delay coupled Van der Pol–Duffing oscillators

In this paper, we aim to investigate the dynamics of a system of Van der Pol–Duffing oscillators with delay coupling. First, taking the time delay as a bifurcation parameter, the stability of the equilibrium, and the existence of Hopf bifurcation are investigated. Then using the center manifold reduction technique and normal form theory, we give the direction of the Hopf bifurcation. And then by means of the symmetric bifurcation theory for delay differential equations and the representation theory of groups, we claim the bifurcation periodic solution induced by time delay is antiphase locked oscillation. Finally, at the end of the paper, numerical simulations are carried out to support our theoretical analysis.     

Multiple scales and normal forms in a ring of delay coupled oscillators with application to chaotic Hindmarsh–Rose neurons

We study the appearance and stability of spatiotemporal periodic patterns like phase-locked oscillations, mirror-reflecting waves, standing waves, in-phase or antiphase oscillations, and coexistence of multiple patterns, in a ring of bidirectionally delay coupled oscillators. Hopf bifurcation, Hopf–Hopf bifurcation, and the equivariant Hopf bifurcation are studied in the viewpoint of normal forms obtained by using the method of multiple scales which is a kind of perturbation technique, thus a clear bifurcation scenario is depicted. We find time delay significantly affects the dynamics and induces rich spatiotemporal patterns. With the help of the unfolding system near Hopf–Hopf bifurcation, it is confirmed in some regions two kinds of stable oscillations may coexist. These phenomena are shown for the delay coupled limit cycle oscillators as well as for the delay coupled chaotic Hindmarsh–Rose neurons.     

Infinitely many solutions of p-Laplacian equations with limit subcritical growth

We discussed a class of p-Laplacian boundary problems on a bounded smooth domain,the nonlinearity is odd symmetric and limit subcritical growing at infinite.A sequence of critical values of the variational functional was constructed after the general- ized Palais-Smale condition was verified.We obtain that the problem possesses infinitely many solutions and corresponding energy levels of the functional pass to positive infinite. The result is a generalization of a similar problem in the case of subcritical.     

On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol–Duffing system with slow-varying periodic excitation

Nonlinear Dynamics - The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When...     

Multiscale Dynamics of Heterogeneous Media in?the?Peridynamic Formulation

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodology is a novel two-scale evolution equation. The rescaled solution of this equation is shown to provide a strong approximation to the actual deformation inside the peridynamic material. The two scale evolution can be split into a microscopic component tracking the dynamics at the length scale of the heterogeneities and a macroscopic component tracking the volume averaged (homogenized) dynamics. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations. The equations show that the forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation.     

Random Dynamics of Stochastic Reaction–Diffusion Systems with Additive Noise

For a typical autocatalytic stochastic reaction–diffusion system with additive noises, the multicomponent reversible Gray–Scott reaction–diffusion system on a two-dimensional bounded domain, the existence of a random attractor and its attracting regularity are proved through the sharp uniform estimates showing respectively the pullback absorbing, asymptotically compact, and flattening properties.     

Coordinated motion of Lagrangian systems with auxiliary oscillators under cooperative and cooperative–competitive interactions

Nonlinear Dynamics - The present paper investigates coordinated oscillatory motion of networked Lagrangian systems under diverse interactions. Based on two new auxiliary oscillator systems, we...     

Dynamics of a delayed diffusive predator–prey model with hyperbolic mortality

This paper is devoted to consider a time-delayed diffusive prey–predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction–diffusion model without time delay.     

Two Problems of the Semi-Conductor Physics Discussed with the Point of View of the Fluid Dynamics

In this article,we discuss two problems of the semi-conductor physics from the point of view ofthe fluid dynamics.Firstly,we discuss the problem of the p-n junction,and find that the previoustreatment and the previous conclusion of the problem are somewhat erroneous.Secondly,we discussthe coefficient C of the block resistance,and find that the mathematical method of the previoustreatment is erroneous.     

Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations

In this study, the nonlinear vibrations of an axially moving beam are investigated by considering the coupling of the longitudinal and transversal motion. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. By detuning the axially velocity, the exact parameters with which the system may turn to internal resonance are detected. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. The saturation and jump phenomena of such system have been reported by investigating the nonlinear amplitude–response curves with respect to external excitation, internal, and external detuning parameters. The longitudinal external excitation may trigger only longitudinal response when excitation amplitude is weak. However, beyond the critical excitation amplitude, the response energy will be transferred from the longitudinal motion to the transversal motion even the excitation is employed on the longitudinal direction. Such energy transfer due to saturation has the potential to be used in the vibration suppression.     

Leading-Order Dynamics of Gravity-Driven Flows in?Free-Standing Soap Films

We derive the leading-order equations that govern the dynamics of the flow in a falling, free-standing soap film. Starting with the incompressible Navier?CStokes equations, we carry out an asymptotic analysis using parameters that correspond to a common experimental setup. We account for the effects of inertia, surface elasticity, pressure, viscous stresses, gravity, and air drag. We find that the dynamics of the flow is dominated by the effects of inertia, surface elasticity, gravity, and air drag. We solve the leading-order equations to compute the steady-state profiles of velocity, thickness, and pressure in an experiment in which the film is in the Marangoni elasticity regime. The computational results, which include a Marangoni shock, are in good accord with the experimental measurements.     

Frequency shifts and analysis of AFM accompanying with coupled flexural–torsional motions

In a conventional dynamic atomic force microscopy (AFM), observing the flexural characteristics of a cantilever subjected to the tip–sample interaction is for extracting the topography and the material properties of a sample’s surface. Recently, Sahin et al. (2007) found that it is essential for understanding surface properties to design a cantilever with an eccentric tip and observe its coupled flexural–torsional characteristics. For effectively analyzing the flexural and torsional signals simultaneously, one has to find out the mode of a cantilever that the ratio of the tip gradient of flexural deformation and the tip torsional angle is comparable. Moreover, the development of an analytical model that can accurately simulate the surface-coupled dynamics of the cantilever is important for quantitative and qualitative understanding of measured results. In this paper, an analytical model of a cantilever with an eccentric tip and subjected to a nonlinear tip–sample force is established. The analytical solution is derived. It is found that the first two modes are the flexural motion and the third mode is the coupled flexural–torsional motion. Finally, the influences of several parameters on the tip angle ratio and frequency shift are investigated.     

Dynamics of diverse data-driven solitons for the three-component coupled nonlinear Schrödinger model by the MPS-PINN method

Nonlinear Dynamics - We improve the physical information neural network by adding multiple parallel subnets to predict seven types of soliton dynamics, such as one soliton, two solitons and soliton...