2,880 research outputs found
Zero-Hopf bifurcation in the FitzHugh-Nagumo system
We characterize the values of the parameters for which a zero--Hopf
equilibrium point takes place at the singular points, namely, (the origin),
and in the FitzHugh-Nagumo system. Thus we find two --parameter
families of the FitzHugh-Nagumo system for which the equilibrium point at the
origin is a zero-Hopf equilibrium. For these two families we prove the
existence of a periodic orbit bifurcating from the zero--Hopf equilibrium point
. We prove that exist three --parameter families of the FitzHugh-Nagumo
system for which the equilibrium point at and is a zero-Hopf
equilibrium point. For one of these families we prove the existence of , or
, or periodic orbits borning at and
Heating and thermal squeezing in parametrically-driven oscillators with added noise
In this paper we report a theoretical model based on Green functions, Floquet
theory and averaging techniques up to second order that describes the dynamics
of parametrically-driven oscillators with added thermal noise. Quantitative
estimates for heating and quadrature thermal noise squeezing near and below the
transition line of the first parametric instability zone of the oscillator are
given. Furthermore, we give an intuitive explanation as to why heating and
thermal squeezing occur. For small amplitudes of the parametric pump the
Floquet multipliers are complex conjugate of each other with a constant
magnitude. As the pump amplitude is increased past a threshold value in the
stable zone near the first parametric instability, the two Floquet multipliers
become real and have different magnitudes. This creates two different effective
dissipation rates (one smaller and the other larger than the real dissipation
rate) along the stable manifolds of the first-return Poincare map. We also show
that the statistical average of the input power due to thermal noise is
constant and independent of the pump amplitude and frequency. The combination
of these effects cause most of heating and thermal squeezing. Very good
agreement between analytical and numerical estimates of the thermal
fluctuations is achieved.Comment: Submitted to Phys. Rev. E, 29 pages, 12 figures. arXiv admin note:
substantial text overlap with arXiv:1108.484
Mixed-mode oscillations in a multiple time scale phantom bursting system
In this work we study mixed mode oscillations in a model of secretion of GnRH
(Gonadotropin Releasing Hormone). The model is a phantom burster consisting of
two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The
forcing system (Regulator) evolves on the slowest scale and acts by moving the
slow nullcline of the forced system (Secretor). There are three modes of
dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady
state) and small oscillations related to the passage of the slow nullcline
through a fold point of the fast nullcline. We derive a variety of reductions,
taking advantage of the mentioned features of the system. We obtain two
results; one on the local dynamics near the fold in the parameter regime
corresponding to the presence of small oscillations and the other on the global
dynamics, more specifically on the existence of an attracting limit cycle. Our
local result is a rigorous characterization of small canards and sectors of
rotation in the case of folded node with an additional time scale, a feature
allowing for a clear geometric argument. The global result gives the existence
of an attracting unique limit cycle, which, in some parameter regimes, remains
attracting and unique even during passages through a canard explosion.Comment: 38 pages, 16 figure
Time-Scale and Noise Optimality in Self-Organized Critical Adaptive Networks
Recent studies have shown that adaptive networks driven by simple local rules
can organize into "critical" global steady states, providing another framework
for self-organized criticality (SOC). We focus on the important convergence to
criticality and show that noise and time-scale optimality are reached at finite
values. This is in sharp contrast to the previously believed optimal zero noise
and infinite time scale separation case. Furthermore, we discover a noise
induced phase transition for the breakdown of SOC. We also investigate each of
the three new effects separately by developing models. These models reveal
three generically low-dimensional dynamical behaviors: time-scale resonance
(TR), a new simplified version of stochastic resonance - which we call steady
state stochastic resonance (SSR) - as well as noise-induced phase transitions.Comment: 4 pages, 6 figures; several changes in exposition and focus on
applications in revised versio
Intrinsic unpredictability of strong El Ni\~no events
The El Ni\~no-Southern Oscillation (ENSO) is a mode of interannual
variability in the coupled equatorial ocean/atmosphere Pacific. El Ni\~no
describes a state in which sea surface temperatures in the eastern Pacific
increase and upwelling of colder, deep waters diminishes. El Ni\~no events
typically peak in boreal winter, but their strength varies irregularly on
decadal time scales. There were exceptionally strong El Ni\~no events in
1982-83, 1997-98 and 2015-16 that affected weather on a global scale. Widely
publicized forecasts in 2014 predicted that the 2015-16 event would occur a
year earlier. Predicting the strength of El Ni\~no is a matter of practical
concern due to its effects on hydroclimate and agriculture around the world.
This paper presents a new robust mechanism limiting the predictability of
strong ENSO events: the existence of an irregular switching between an
oscillatory state that has strong El Ni\~no events and a chaotic state that
lacks strong events, which can be induced by very weak seasonal forcing or
noise.Comment: 4 pages, 6 figure
The phase-space of generalized Gauss-Bonnet dark energy
The generalized Gauss-Bonnet theory, introduced by Lagrangian F(R,G), has
been considered as a general modified gravity for explanation of the dark
energy. G is the Gauss-Bonnet invariant. For this model, we seek the situations
under which the late-time behavior of the theory is the de-Sitter space-time.
This is done by studying the two dimensional phase space of this theory, i.e.
the R-H plane. By obtaining the conditions under which the de-Sitter space-time
is the stable attractor of this theory, several aspects of this problem have
been investigated. It has been shown that there exist at least two classes of
stable attractors : the singularities of the F(R,G), and the cases in which the
model has a critical curve, instead of critical points. This curve is R=12H^2
in R-H plane. Several examples, including their numerical calculations, have
been discussed.Comment: 19 pages, 11 figures, typos corrected, a reference adde
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