13,120 research outputs found
Generalized models as a universal approach to the analysis of nonlinear dynamical systems
We present a universal approach to the investigation of the dynamics in
generalized models. In these models the processes that are taken into account
are not restricted to specific functional forms. Therefore a single generalized
models can describe a class of systems which share a similar structure. Despite
this generality, the proposed approach allows us to study the dynamical
properties of generalized models efficiently in the framework of local
bifurcation theory. The approach is based on a normalization procedure that is
used to identify natural parameters of the system. The Jacobian in a steady
state is then derived as a function of these parameters. The analytical
computation of local bifurcations using computer algebra reveals conditions for
the local asymptotic stability of steady states and provides certain insights
on the global dynamics of the system. The proposed approach yields a close
connection between modelling and nonlinear dynamics. We illustrate the
investigation of generalized models by considering examples from three
different disciplines of science: a socio-economic model of dynastic cycles in
china, a model for a coupled laser system and a general ecological food web.Comment: 15 pages, 2 figures, (Fig. 2 in color
Fractional Dynamical Systems
In this paper the author presents the results of the preliminary
investigation of fractional dynamical systems based on the results of numerical
simulations of fractional maps. Fractional maps are equivalent to fractional
differential equations describing systems experiencing periodic kicks. Their
properties depend on the value of two parameters: the non-linearity parameter,
which arises from the corresponding regular dynamical systems; and the memory
parameter which is the order of the fractional derivative in the corresponding
non-linear fractional differential equations. The examples of the fractional
Standard and Logistic maps demonstrate that phase space of non-linear
fractional dynamical systems may contain periodic sinks, attracting slow
diverging trajectories, attracting accelerator mode trajectories, chaotic
attractors, and cascade of bifurcations type trajectories whose properties are
different from properties of attractors in regular dynamical systems. The
author argues that discovered properties should be evident in the natural
(biological, psychological, physical, etc.) and engineering systems with
power-law memory.Comment: 6 pages, 4 figure
Bifurcation and chaos in the double well Duffing-van der Pol oscillator: Numerical and analytical studies
The behaviour of a driven double well Duffing-van der Pol (DVP) oscillator
for a specific parametric choice () is studied. The
existence of different attractors in the system parameters () domain
is examined and a detailed account of various steady states for fixed damping
is presented. Transition from quasiperiodic to periodic motion through chaotic
oscillations is reported. The intervening chaotic regime is further shown to
possess islands of phase-locked states and periodic windows (including period
doubling regions), boundary crisis, all the three classes of intermittencies,
and transient chaos. We also observe the existence of local-global bifurcation
of intermittent catastrophe type and global bifurcation of blue-sky catastrophe
type during transition from quasiperiodic to periodic solutions. Using a
perturbative periodic solution, an investigation of the various forms of
instablities allows one to predict Neimark instablity in the plane
and eventually results in the approximate predictive criteria for the chaotic
region.Comment: 15 pages (13 figures), RevTeX, please e-mail Lakshmanan for figures,
to appear in Phys. Rev. E. (E-mail: [email protected]
On the Relationship between Equilibrium Bifurcations and Ideal MHD Instabilities for Line-Tied Coronal Loops
For axisymmetric models for coronal loops the relationship between the
bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the
points of linear ideal MHD instability is investigated imposing line-tied
boundary conditions. Using a well-studied example based on the Gold-Hoyle
equilibrium, it is demonstrated that if the equilibrium sequence is calculated
using the Grad-Shafranov equation, the instability corresponds to the second
bifurcation point and not the first bifurcation point because the equilibrium
boundary conditions allow for modes which are excluded from the linear ideal
stability analysis. This is shown by calculating the bifurcating equilibrium
branches and comparing the spatial structure of the solutions close to the
bifurcation point with the spatial structure of the unstable mode. If the
equilibrium sequence is calculated using Euler potentials the first bifurcation
point of the Grad-Shafranov case is not found, and the first bifurcation point
of the Euler potential description coincides with the ideal instability
threshold. An explanation of this results in terms of linear bifurcation theory
is given and the implications for the use of MHD equilibrium bifurcations to
explain eruptive phenomena is briefly discussed.Comment: 22 pages, 6 figures, accepted by Solar Physic
Snaking and isolas of localised states in bistable discrete lattices
We consider localised states in a discrete bistable Allen-Cahn equation. This
model equation combines bistability and local cell-to-cell coupling in the
simplest possible way. The existence of stable localised states is made
possible by pinning to the underlying lattice; they do not exist in the
equivalent continuum equation. In particular we address the existence of
'isolas': closed curves of solutions in the bifurcation diagram. Isolas appear
for some non-periodic boundary conditions in one spatial dimension but seem to
appear generically in two dimensions. We point out how features of the
bifurcation diagram in 1D help to explain some (unintuitive) features of the
bifurcation diagram in 2D.Comment: 14 page
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