Search results for "Bifurcation diagram"
showing 10 items of 33 documents
Stability analysis of a paramagnetic spheroid in a precessing field
2019
Abstract The stability of a paramagnetic prolate or oblate spheroidal particle in a precessing magnetic field is studied. The bifurcation diagram is calculated analytically as a function of the magnetic field frequency and the precession angle. The orientation of the particle in the synchronous regime is calculated. The rotational dynamics and the mean rotational frequency in the asynchronous regime are also obtained. The theoretical model we describe enables the analytic calculation of the dynamics of the particle in the limiting case when the motion is periodic. The theoretical models were also compared with experimental results of rod like particle dynamics in a precessing magnetic field…
Dynamic complexities in host-parasitoid interaction
1999
In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of…
Non-unique population dynamics: basic patterns
2000
We review the basic patterns of complex non-uniqueness in simple discrete-time population dynamics models. We begin by studying a population dynamics model of a single species with a two-stage, two-habitat life cycle. We then explore in greater detail two ecological models describing host‐macroparasite and host‐parasitoid interspecific interactions. In general, several types of attractors, e.g. point equilibria vs. chaotic, periodic vs. quasiperiodic and quasiperiodic vs. chaotic attractors, may coexist in the same mapping. This non-uniqueness also indicates that the bifurcation diagrams, or the routes to chaos, depend on initial conditions and are therefore non-unique. The basins of attrac…
On the number of solutions of a Duffing equation
1991
The exact number of solutions of a Duffing equation with small forcing term and homogeneous Neumann boundary conditions is given. Several bifurcation diagrams are shown.
Double precision errors in the logistic map: statistical study and dynamical interpretation.
2007
The nature of the round-off errors that occur in the usual double precision computation of the logistic map is studied in detail. Different iterative regimes from the whole panoply of behaviors exhibited in the bifurcation diagram are examined, histograms of errors in trajectories given, and for the case of fully developed chaos an explicit formula is found. It is shown that the statistics of the largest double precision error as a function of the map parameter is characterized by jumps whose location is determined by certain boundary crossings in the bifurcation diagram. Both jumps and locations seem to present geometric convergence characterized by the two first Feigenbaum constants. Even…
Adaptive Fuzzy Control of a Process with Bifurcations
2002
Several fuzzy controllers with different structures have been considered for the control of systems with bifurcation. A mixed feedforward-feedback structure with some additional adaptation mechanisms have been checked by simulation in the control of a non-linear process constituted by a bubble column in which a slow kinetics auto-catalytic reaction takes place. Simulation results show the validity of some of the proposed controllers in avoiding the system reaching bifurcation and instability conditions.
Perturbations of symmetric elliptic Hamiltonians of degree four
2006
AbstractIn this paper four-parameter unfoldings Xλ of symmetric elliptic Hamiltonians of degree four are studied. We prove that in a compact region of the period annulus of X0 the displacement function of Xλ is sign equivalent to its principal part, which is given by a family induced by a Chebychev system; and we describe the bifurcation diagram of Xλ in a full neighborhood of the origin in the parameter space, where at most two limit cycles can exist for the corresponding systems.
Bifurcations of Elementary Graphics
1998
After the regular limit periodic sets, the simplest limit periodic sets are the elementary graphics.
On the construction of lusternik-schnirelmann critical values with application to bifurcation problems
1987
An iterative method to construct Lusternik-Schnirelmann critical values is presented. Examples of its use to obtain numerical solutions to nonlinear eigenvalue problems and their bifurcation branches are given