393 research outputs found
Complex oscillations in the delayed Fitzhugh-Nagumo equation
Motivated by the dynamics of neuronal responses, we analyze the dynamics of
the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system
provides a canonical example of a canard explosion for sufficiently small
delays. Beyond this regime, delays significantly enrich the dynamics, leading
to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a
delay-induced subcritical Bogdanov-Takens instability arising at the fold
points of the S-shaped critical manifold. Underlying the transition from
canard-induced to delay-induced dynamics is an abrupt switch in the nature of
the Hopf bifurcation
Direct transition to high-dimensional chaos through a global bifurcation
In the present work we report on a genuine route by which a high-dimensional
(with d>4) chaotic attractor is created directly, i.e., without a
low-dimensional chaotic attractor as an intermediate step. The high-dimensional
chaotic set is created in a heteroclinic global bifurcation that yields an
infinite number of unstable tori.The mechanism is illustrated using a system
constructed by coupling three Lorenz oscillators. So, the route presented here
can be considered a prototype for high-dimensional chaotic behavior just as the
Lorenz model is for low-dimensional chaos.Comment: 7 page
Chaotic saddles in nonlinear modulational interactions in a plasma
A nonlinear model of modulational processes in the subsonic regime involving
a linearly unstable wave and two linearly damped waves with different damping
rates in a plasma is studied numerically. We compute the maximum Lyapunov
exponent as a function of the damping rates in a two-parameter space, and
identify shrimp-shaped self-similar structures in the parameter space. By
varying the damping rate of the low-frequency wave, we construct bifurcation
diagrams and focus on a saddle-node bifurcation and an interior crisis
associated with a periodic window. We detect chaotic saddles and their stable
and unstable manifolds, and demonstrate how the connection between two chaotic
saddles via coupling unstable periodic orbits can result in a crisis-induced
intermittency. The relevance of this work for the understanding of modulational
processes observed in plasmas and fluids is discussed.Comment: Physics of Plasmas, in pres
Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial
A new computational technique based on the symbolic description utilizing
kneading invariants is proposed and verified for explorations of dynamical and
parametric chaos in a few exemplary systems with the Lorenz attractor. The
technique allows for uncovering the stunning complexity and universality of
bi-parametric structures and detect their organizing centers - codimension-two
T-points and separating saddles in the kneading-based scans of the iconic
Lorenz equation from hydrodynamics, a normal model from mathematics, and a
laser model from nonlinear optics.Comment: Journal of Bifurcations and Chaos, 201
Parameter-sweeping techniques for temporal dynamics of neuronal systems: case study of Hindmarsh-Rose model
Background: Development of effective and plausible numerical tools is an imperative task for thorough studies of nonlinear dynamics in life science applications.
Results: We have developed a complementary suite of computational tools for twoparameter screening of dynamics in neuronal models. We test a ‘brute-force’ effectiveness of neuroscience plausible techniques specifically tailored for the examination of temporal characteristics, such duty cycle of bursting, interspike interval, spike number deviation in the phenomenological Hindmarsh-Rose model of a bursting neuron and compare the results obtained by calculus-based tools for evaluations of an entire spectrum of Lyapunov exponents broadly employed in studies of nonlinear systems.
Conclusions: We have found that the results obtained either way agree exceptionally well, and can identify and differentiate between various fine structures of complex dynamics and underlying global bifurcations in this exemplary model. Our future planes are to enhance the applicability of this computational suite for understanding of polyrhythmic bursting patterns and their functional transformations in small networks
Breathers in inhomogeneous nonlinear lattices: an analysis via centre manifold reduction
We consider an infinite chain of particles linearly coupled to their nearest
neighbours and subject to an anharmonic local potential. The chain is assumed
weakly inhomogeneous. We look for small amplitude discrete breathers. The
problem is reformulated as a nonautonomous recurrence in a space of
time-periodic functions, where the dynamics is considered along the discrete
spatial coordinate. We show that small amplitude oscillations are determined by
finite-dimensional nonautonomous mappings, whose dimension depends on the
solutions frequency. We consider the case of two-dimensional reduced mappings,
which occurs for frequencies close to the edges of the phonon band. For an
homogeneous chain, the reduced map is autonomous and reversible, and
bifurcations of reversible homoclinics or heteroclinic solutions are found for
appropriate parameter values. These orbits correspond respectively to discrete
breathers, or dark breathers superposed on a spatially extended standing wave.
Breather existence is shown in some cases for any value of the coupling
constant, which generalizes an existence result obtained by MacKay and Aubry at
small coupling. For an inhomogeneous chain the study of the nonautonomous
reduced map is in general far more involved. For the principal part of the
reduced recurrence, using the assumption of weak inhomogeneity, we show that
homoclinics to 0 exist when the image of the unstable manifold under a linear
transformation intersects the stable manifold. This provides a geometrical
understanding of tangent bifurcations of discrete breathers. The case of a mass
impurity is studied in detail, and our geometrical analysis is successfully
compared with direct numerical simulations
A robust numerical method to study oscillatory instability of gap solitary waves
The spectral problem associated with the linearization about solitary waves
of spinor systems or optical coupled mode equations supporting gap solitons is
formulated in terms of the Evans function, a complex analytic function whose
zeros correspond to eigenvalues. These problems may exhibit oscillatory
instabilities where eigenvalues detach from the edges of the continuous
spectrum, so called edge bifurcations. A numerical framework, based on a fast
robust shooting algorithm using exterior algebra is described. The complete
algorithm is robust in the sense that it does not produce spurious unstable
eigenvalues. The algorithm allows to locate exactly where the unstable discrete
eigenvalues detach from the continuous spectrum. Moreover, the algorithm allows
for stable shooting along multi-dimensional stable and unstable manifolds. The
method is illustrated by computing the stability and instability of gap
solitary waves of a coupled mode model.Comment: key words: gap solitary wave, numerical Evans function, edge
bifurcation, exterior algebra, oscillatory instability, massive Thirring
model. accepted for publication in SIAD
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