Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model

We investigate complex dynamics occurring in a non-smooth model of a Jeffcott rotor with a bearing clearance. A bifurcation analysis of the rotor system is carried out by means of the software TC-HAT [25], a toolbox of AUTO 97 [6] allowing path-following and detection of bifurcations of periodic trajectories of non-smooth dynamical systems. The study reveals a rich variety of dynamics, which includes grazing-induced fold and period-doubling bifurcations, as well as hysteresis loops produced by a cusp singularity. Furthermore, an analytical expression predicting grazing incidences is derived.     

A new method for characterizing patterns of neural spike trains and its application

A method for characterizing and identifying firing patterns of neural spike trains is presented. Based on the time evolution of a neural spike train, the counting process is constructed as a time-dependent stair-like function. Three characteristic variables defined at sequential moments, including two formal derivatives and the integration of the counting process, are introduced to reflect the temporal patterns of a spike train. The reconstruction of a spike train with these variables verify the validity of this method. And a model of cold receptor is used as an example to investigate the temporal patterns under different temperature conditions. The most important contribution of our method is that it not only can reflect the features of spike train patterns clearly by using their geometrical properties, but also it can reflect the trait of time, especially the change of bursting of spike train. So it is a useful complementarity to conventional method of averaging.     

Analysis of the periodic solutions of a smooth and discontinuous oscillator

In this paper, the periodic solutions of the smooth and discontinuous (SD) oscillator, which is a strongly irrational nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the unperturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equation using Lyapunov method. The results presented herein this paper are valid for both smooth ( α > 0) and discontinuous ( α = 0) stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.     

Chatter in a transverse grinding process

In transverse grinding, the wheel moves along the workpiece, which induces unique grinding dynamics. To understand these dynamic phenomena, specifically the grinding chatter, a new dynamical model of the process is proposed, in which the wheel position is assumed to be quasi-static since the transverse wheel velocity is small. From the stability and bifurcation analyses of the chatter vibration, it appears that the dynamics of the process is governed by the quasi-static interactions. Moreover, the obtained results also show that the wheel and workpiece chatters are quite different, having continuous and intermittent characters respectively.     

Complete and phase synchronization in a heterogeneous small-world neuronal network

Synchronous firing of neurons is thought to be important for information communication in neuronal networks. This paper investigates the complete and phase synchronization in a heterogeneous small-world chaotic Hindmarsh--Rose neuronal network. The effects of various network parameters on synchronization behaviour are discussed with some biological explanations. Complete synchronization of small-world neuronal networks is studied theoretically by the master stability function method. It is shown that the coupling strength necessary for complete or phase synchronization decreases with the neuron number, the node degree and the connection density are increased. The effect of heterogeneity of neuronal networks is also considered and it is found that the network heterogeneity has an adverse effect on synchrony.     

The limit case response of the archetypal oscillator for smooth and discontinuous dynamics

In this paper, the limit case of the SD (smooth and discontinuous) oscillator is studied. This system exhibits standard dynamics governed by the hyperbolic structure associated with the stationary state of the double-well. The substantial deviation from the standard dynamics is the non-smoothness of the velocity in crossing from one well to another, caused by the loss of local hyperbolicity due to the discontinuity. Without dissipation, the KAM structure on the Poincaré section is constructed with generic KAM curves and a series of fixed points associated with surrounded islands of quasi-periodic orbits and the chaotic connection orbits. It is found that, for a fixed set of parameters, a special chaotic orbit exits there which fills a finite region and connects a series of islands dominated by different chains of fixed points. As one adds weak dissipation, the periodic solutions in this finite region remain unchanged while the quasi-periodic solutions (isolated islands) are converted to the corresponding periodic solutions. The relevant dynamics for the system with weak dissipation under external excitation is shown having period doubling bifurcation leading to chaos, and multi-stable solutions.     

Non-linear modal analysis for beams subjected to axial loads: Analytical and finite-element solutions

A rigorous derivation of non-linear equations governing the dynamics of an axially loaded beam is given with a clear focus to develop robust low-dimensional models. Two important loading scenarios were considered, where a structure is subjected to a uniformly distributed axial and a thrust force. These loads are to mimic the main forces acting on an offshore riser, for which an analytical methodology has been developed and applied. In particular, non-linear normal modes (NNMs) and non-linear multi-modes (NMMs) have been constructed by using the method of multiple scales. This is to effectively analyse the transversal vibration responses by monitoring the modal responses and mode interactions. The developed analytical models have been crosschecked against the results from FEM simulation. The FEM model having 26 elements and 77 degrees-of-freedom gave similar results as the low-dimensional (one degree-of-freedom) non-linear oscillator, which was developed by constructing a so-called invariant manifold. The comparisons of the dynamical responses were made in terms of time histories, phase portraits and mode shapes.