2,141 research outputs found
Multiple Time Scale Dynamics With Two Fast Variables And One Slow Variable
This thesis considers dynamical systems that have multiple time scales. The focus lies on systems with two fast variables and one slow variable. The twoparameter bifurcation structure of the FitzHugh-Nagumo (FHN) equation is analyzed in detail. A singular bifurcation diagram is constructed and invariant manifolds of the problem are computed. A boundary-value approach to compute slow manifolds of saddle-type is developed. Interactions of classical invariant manifolds and slow manifolds explain the exponentially small turning of a homoclinic bifurcation curve in parameter space. Mixed-mode oscillations and maximal canards are detected in the FHN equation. An asymptotic formula to find maximal canards is proved which is based on the first Lyapunov coefficient at a singular Hopf bifurcation
Rigorous Enclosures of a Slow Manifold
Slow-fast dynamical systems have two time scales and an explicit parameter
representing the ratio of these time scales. Locally invariant slow manifolds
along which motion occurs on the slow time scale are a prominent feature of
slow-fast systems. This paper introduces a rigorous numerical method to compute
enclosures of the slow manifold of a slow-fast system with one fast and two
slow variables. A triangulated first order approximation to the two dimensional
invariant manifold is computed "algebraically". Two translations of the
computed manifold in the fast direction that are transverse to the vector field
are computed as the boundaries of an initial enclosure. The enclosures are
refined to bring them closer to each other by moving vertices of the enclosure
boundaries one at a time. As an application we use it to prove the existence of
tangencies of invariant manifolds in the problem of singular Hopf bifurcation
and to give bounds on the location of one such tangency
From First Lyapunov Coefficients to Maximal Canards
Hopf bifurcations in fast-slow systems of ordinary differential equations can
be associated with surprising rapid growth of periodic orbits. This process is
referred to as canard explosion. The key step in locating a canard explosion is
to calculate the location of a special trajectory, called a maximal canard, in
parameter space. A first-order asymptotic expansion of this location was found
by Krupa and Szmolyan in the framework of a "canard point"-normal-form for
systems with one fast and one slow variable. We show how to compute the
coefficient in this expansion using the first Lyapunov coefficient at the Hopf
bifurcation thereby avoiding use of this normal form. Our results connect the
theory of canard explosions with existing numerical software, enabling easier
calculations of where canard explosions occur.Comment: preprint version - for final version see journal referenc
A mathematical framework for critical transitions: normal forms, variance and applications
Critical transitions occur in a wide variety of applications including
mathematical biology, climate change, human physiology and economics. Therefore
it is highly desirable to find early-warning signs. We show that it is possible
to classify critical transitions by using bifurcation theory and normal forms
in the singular limit. Based on this elementary classification, we analyze
stochastic fluctuations and calculate scaling laws of the variance of
stochastic sample paths near critical transitions for fast subsystem
bifurcations up to codimension two. The theory is applied to several models:
the Stommel-Cessi box model for the thermohaline circulation from geoscience,
an epidemic-spreading model on an adaptive network, an activator-inhibitor
switch from systems biology, a predator-prey system from ecology and to the
Euler buckling problem from classical mechanics. For the Stommel-Cessi model we
compare different detrending techniques to calculate early-warning signs. In
the epidemics model we show that link densities could be better variables for
prediction than population densities. The activator-inhibitor switch
demonstrates effects in three time-scale systems and points out that excitable
cells and molecular units have information for subthreshold prediction. In the
predator-prey model explosive population growth near a codimension two
bifurcation is investigated and we show that early-warnings from normal forms
can be misleading in this context. In the biomechanical model we demonstrate
that early-warning signs for buckling depend crucially on the control strategy
near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio
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