582 research outputs found
Resonant normal forms as constrained linear systems
We show that a nonlinear dynamical system in Poincare'-Dulac normal form (in
) can be seen as a constrained linear system; the constraints are given
by the resonance conditions satisfied by the spectrum of (the linear part of)
the system and identify a naturally invariant manifold for the flow of the
``parent'' linear system. The parent system is finite dimensional if the
spectrum satisfies only a finite number of resonance conditions, as implied
e.g. by the Poincare' condition. In this case our result can be used to
integrate resonant normal forms, and sheds light on the geometry behind the
classical integration method of Horn, Lyapounov and Dulac.Comment: 15 pages; revised version (with revised title
Multi-Phase Patterns in Periodically Forced Oscillatory Systems
Periodic forcing of an oscillatory system produces frequency locking bands
within which the system frequency is rationally related to the forcing
frequency. We study extended oscillatory systems that respond to uniform
periodic forcing at one quarter of the forcing frequency (the 4:1 resonance).
These systems possess four coexisting stable states, corresponding to uniform
oscillations with successive phase shifts of . Using an amplitude
equation approach near a Hopf bifurcation to uniform oscillations, we study
front solutions connecting different phase states. These solutions divide into
two groups: -fronts separating states with a phase shift of and
-fronts separating states with a phase shift of . We find a new
type of front instability where a stationary -front ``decomposes'' into a
pair of traveling -fronts as the forcing strength is decreased. The
instability is degenerate for an amplitude equation with cubic nonlinearities.
At the instability point a continuous family of pair solutions exists,
consisting of -fronts separated by distances ranging from zero to
infinity. Quintic nonlinearities lift the degeneracy at the instability point
but do not change the basic nature of the instability. We conjecture the
existence of similar instabilities in higher 2n:1 resonances (n=3,4,..) where
stationary -fronts decompose into n traveling -fronts. The
instabilities designate transitions from stationary two-phase patterns to
traveling 2n-phase patterns. As an example, we demonstrate with a numerical
solution the collapse of a four-phase spiral wave into a stationary two-phase
pattern as the forcing strength within the 4:1 resonance is increased
On the Origin of Traveling Pulses in Bistable Systems
The interaction between a pair of Bloch fronts forming a traveling domain in
a bistable medium is studied. A parameter range beyond the nonequilibrium
Ising-Bloch bifurcation is found where traveling domains collapse. Only beyond
a second threshold the repulsive front interactions become strong enough to
balance attractive interactions and asymmetries in front speeds, and form
stable traveling pulses. The analysis is carried out for the forced complex
Ginzburg-Landau equation. Similar qualitative behavior is found in the bistable
FitzHugh-Nagumo model.Comment: 5 pages, RevTeX. Aric Hagberg: http://t7.lanl.gov/People/Aric/; Ehud
Meron:http://www.bgu.ac.il/BIDR/research/staff/meron.htm
A Phase Front Instability in Periodically Forced Oscillatory Systems
Multiplicity of phase states within frequency locked bands in periodically
forced oscillatory systems may give rise to front structures separating states
with different phases. A new front instability is found within bands where
(). Stationary fronts shifting the
oscillation phase by lose stability below a critical forcing strength and
decompose into traveling fronts each shifting the phase by . The
instability designates a transition from stationary two-phase patterns to
traveling -phase patterns
The evolution and comparative neurobiology of endocannabinoid signalling
CB(1)- and CB(2)-type cannabinoid receptors mediate effects of the endocannabinoids 2-arachidonoylglycerol (2-AG) and anandamide in mammals. In canonical endocannabinoid-mediated synaptic plasticity, 2-AG is generated postsynaptically by diacylglycerol lipase alpha and acts via presynaptic CB(1)-type cannabinoid receptors to inhibit neurotransmitter release. Electrophysiological studies on lampreys indicate that this retrograde signalling mechanism occurs throughout the vertebrates, whereas system-level studies point to conserved roles for endocannabinoid signalling in neural mechanisms of learning and control of locomotor activity and feeding. CB(1)/CB(2)-type receptors originated in a common ancestor of extant chordates, and in the sea squirt Ciona intestinalis a CB(1)/CB(2)-type receptor is targeted to axons, indicative of an ancient role for cannabinoid receptors as axonal regulators of neuronal signalling. Although CB(1)/CB(2)-type receptors are unique to chordates, enzymes involved in biosynthesis/inactivation of endocannabinoids occur throughout the animal kingdom. Accordingly, non-CB(1)/CB(2)-mediated mechanisms of endocannabinoid signalling have been postulated. For example, there is evidence that 2-AG mediates retrograde signalling at synapses in the nervous system of the leech Hirudo medicinalis by activating presynaptic transient receptor potential vanilloid-type ion channels. Thus, postsynaptic synthesis of 2-AG or anandamide may be a phylogenetically widespread phenomenon, and a variety of proteins may have evolved as presynaptic (or postsynaptic) receptors for endocannabinoids
Harmonic forcing of an extended oscillatory system: Homogeneous and periodic solutions
In this paper we study the effect of external harmonic forcing on a
one-dimensional oscillatory system described by the complex Ginzburg-Landau
equation (CGLE). For a sufficiently large forcing amplitude, a homogeneous
state with no spatial structure is observed. The state becomes unstable to a
spatially periodic ``stripe'' state via a supercritical bifurcation as the
forcing amplitude decreases. An approximate phase equation is derived, and an
analytic solution for the stripe state is obtained, through which the
asymmetric behavior of the stability border of the state is explained. The
phase equation, in particular the analytic solution, is found to be very useful
in understanding the stability borders of the homogeneous and stripe states of
the forced CGLE.Comment: 6 pages, 4 figures, 2 column revtex format, to be published in Phys.
Rev.
Frozen spatial chaos induced by boundaries
We show that rather simple but non-trivial boundary conditions could induce
the appearance of spatial chaos (that is stationary, stable, but spatially
disordered configurations) in extended dynamical systems with very simple
dynamics. We exemplify the phenomenon with a nonlinear reaction-diffusion
equation in a two-dimensional undulated domain. Concepts from the theory of
dynamical systems, and a transverse-single-mode approximation are used to
describe the spatially chaotic structures.Comment: 9 pages, 6 figures, submitted for publication; for related work visit
http://www.imedea.uib.es/~victo
Poincare' normal forms and simple compact Lie groups
We classify the possible behaviour of Poincar\'e-Dulac normal forms for
dynamical systems in with nonvanishing linear part and which are
equivariant under (the fundamental representation of) all the simple compact
Lie algebras and thus the corresponding simple compact Lie groups. The
``renormalized forms'' (in the sense of previous work by the author) of these
systems is also discussed; in this way we are able to simplify the
classification and moreover to analyze systems with zero linear part. We also
briefly discuss the convergence of the normalizing transformations.Comment: 17 pages; minor corrections in revised versio
Frequency Locking in Spatially Extended Systems
A variant of the complex Ginzburg-Landau equation is used to investigate the
frequency locking phenomena in spatially extended systems. With appropriate
parameter values, a variety of frequency-locked patterns including flats,
fronts, labyrinths and fronts emerge. We show that in spatially
extended systems, frequency locking can be enhanced or suppressed by diffusive
coupling. Novel patterns such as chaotically bursting domains and target
patterns are also observed during the transition to locking
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