Smooth Oscillator Research Articles

Transient chaos in smooth memristor oscillator

This paper presents a new smooth memristor oscillator, which is derived from Chua's oscillator by replacing Chua's diode with a flux-controlled memristor and a negative conductance. Novel parameters and initial conditions are dependent upon dynamical behaviours such as transient chaos and stable chaos with an intermittence period and are found in the smooth memristor oscillator. By using dynamical analysis approaches including time series, phase portraits and bifurcation diagrams, the dynamical behaviours of the proposed memristor oscillator are effectively investigated in this paper.

Dynamical analysis of memristor chaotic oscillator

Memristor, which is a nonlinear resistor with memory function, is the fourth fundamental two-terminal circuit element besides the resistor, capacitor and inductor. A memristor based oscillator circuit, directly derived from Chuas oscillator by replacing Chuas diode with a flux-controlled memristor characterized by a smooth monotonically increasing nonlinearity, is presented in this paper. By using conventional dynamical analysis method, the dynamical behaviors of the new smooth memristor oscillator with the variations of circuit parameters and initial conditions are investigated. The research results demonstrate that the dynamical behavior of the smooth memristor oscillator not only depends on the circuit parameters, but also closely depends on the initial conditions of the circuit. Different from general chaotic systems, some novel nonlinear phenomenas, such as the transient chaos and state transitions and so on, can be found in the proposed system.

Low-dimensional maps for piecewise smooth oscillators

Dynamics of the piecewise smooth nonlinear oscillators is considered, for which, general methodology of reducing multidimensional flows to low-dimensional maps is proposed. This includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution. The global map is comprised of local maps, which are constructed in the smooth sub-regions of phase space. To construct this low-dimensional map, it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory. The dimension reduction is directly related to the dimension of the chosen boundary, and the lower its dimension is, the larger dimension reduction can be achieved. Full details are given for a drifting impact oscillator, where the five-dimensional flow is reduced to one-dimensional (1D) approximate analytical map. First an exact two-dimensional map has been formulated and analysed. A further reduction to 1D approximate map is introduced and discussed. Standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos, and co-existence of multiple attractors. Accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters.

A clock and wavefront model for control of the number of repeated structures during animal morphogenesis

Most current models for morphogenesis of repeated patterns, such as vertebrate somites, cannot explain the observed degree of constancy for the number of somites in individuals of a given species. This precision requires a mechanism whereby the lengths of someites (i.e. number of cells per somite) must adjust to the overall size of individual embryos, and one which co-ordinates numbers of somites with position in the whole pattern of body parts. A qualitative model is presented that does admit the observed precision. It is also compatible with experimental observations such as the sequential formation of somites from anterior to posterior in a regular time sequence, the timing of cellular change during development generally, and the increasing evidence for widespread existence of cellular biorhythms. The model involves an interacting “clock” and “wavefront”. The clock is is asmooth cellular oscillator, for which cells throughout the embryo are assumed to be phase-linked. The wavefront is a front of rapid cell change moving slowly down the long axis of the embryo; cells enter a phase of rapid alteration in locomotory and/or adhesive properties at successively later times according to anterior-posterior body position. In the model, the smooth intracellular oscillator itself interacts with the possibility of the rapid primary change or its transmission within cells, thereby gating rhythmically the slow progress of the wavefront. Cells thus enter their rapid change of properties in a succession of separate populations, creating the pattern. It is argued that the elements, a smooth oscillator, a slow wavefront and a rapid cellular change, have biological plausibility. The consequences of combining them were suggested by catastrophe theory. We stress the necessary relation between the present model and the more general concept of positional information (Wolpert, 1969, 1971). Prospective and ongoing experiments stimulated by the model are discussed, and emphasis is placed on how such conceptions of morphogenesis can help reval homology between organisms having developments that are very different to a surface inspection.