1,690 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
Limit cycle bifurcations from a nilpotent focus or center of planar systems
In this paper, we study the analytical property of the Poincare return map
and the generalized focal values of an analytical planar system with a
nilpotent focus or center. Then we use the focal values and the map to study
the number of limit cycles of this kind of systems with parameters, and obtain
some new results on the lower and upper bounds of the maximal number of limit
cycles near the nilpotent focus or center.Comment: This paper was submitted to Journal of Mathematical Analysis and
Application
Low-frequency variability and heat transport in a low-order nonlinear coupled ocean-atmosphere model
We formulate and study a low-order nonlinear coupled ocean-atmosphere model
with an emphasis on the impact of radiative and heat fluxes and of the
frictional coupling between the two components. This model version extends a
previous 24-variable version by adding a dynamical equation for the passive
advection of temperature in the ocean, together with an energy balance model.
The bifurcation analysis and the numerical integration of the model reveal
the presence of low-frequency variability (LFV) concentrated on and near a
long-periodic, attracting orbit. This orbit combines atmospheric and oceanic
modes, and it arises for large values of the meridional gradient of radiative
input and of frictional coupling. Chaotic behavior develops around this orbit
as it loses its stability; this behavior is still dominated by the LFV on
decadal and multi-decadal time scales that is typical of oceanic processes.
Atmospheric diagnostics also reveals the presence of predominant low- and
high-pressure zones, as well as of a subtropical jet; these features recall
realistic climatological properties of the oceanic atmosphere.
Finally, a predictability analysis is performed. Once the decadal-scale
periodic orbits develop, the coupled system's short-term instabilities --- as
measured by its Lyapunov exponents --- are drastically reduced, indicating the
ocean's stabilizing role on the atmospheric dynamics. On decadal time scales,
the recurrence of the solution in a certain region of the invariant subspace
associated with slow modes displays some extended predictability, as reflected
by the oscillatory behavior of the error for the atmospheric variables at long
lead times.Comment: v1: 41 pages, 17 figures; v2-: 42 pages, 15 figure
Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations
In this paper we perform the parameter-dependent center manifold reduction
near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf
bifurcations in delay differential equations (DDEs). This allows us to
initialize the continuation of codimension one equilibria and cycle
bifurcations emanating from these codimension two bifurcation points. The
normal form coefficients are derived in the functional analytic perturbation
framework for dual semigroups (sun-star calculus) using a normalization
technique based on the Fredholm alternative. The obtained expressions give
explicit formulas which have been implemented in the freely available numerical
software package DDE-BifTool. While our theoretical results are proven to apply
more generally, the software implementation and examples focus on DDEs with
finitely many discrete delays. Together with the continuation capabilities of
DDE-BifTool, this provides a powerful tool to study the dynamics near
equilibria of such DDEs. The effectiveness is demonstrated on various models
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