343 research outputs found
On the periodic orbit bifurcating from a Hopf bifurcation in systems with two slow and one fast variables
The Hopf bifurcation in slow-fast systems with two slow variables and one fast variable has been studied recently, mainly from a numerical point of view. Our goal is to provide an analytic proof of the existence of the zero Hopf bifurcation exhibited for such systems, and to characterize the stability or instability of the periodic orbit which borns in such zero Hopf bifurcation. Our proofs use the averaging theory.The first and third authors are partially supported by a MICINN grant number MTM2011-22877 and by an AGAUR grant number 2009SGR 381. The second author is partially supported by a MICINN/FEDER grant number MTM2008- 03437, by an AGAUR grant number 2009SGR 410 and by ICREA Academia
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
Multimodal oscillations in systems with strong contraction
One- and two-parameter families of flows in near an Andronov-Hopf
bifurcation (AHB) are investigated in this work. We identify conditions on the
global vector field, which yield a rich family of multimodal orbits passing
close to a weakly unstable saddle-focus and perform a detailed asymptotic
analysis of the trajectories in the vicinity of the saddle-focus. Our analysis
covers both cases of sub- and supercritical AHB. For the supercritical case, we
find that the periodic orbits born from the AHB are bimodal when viewed in the
frame of coordinates generated by the linearization about the bifurcating
equilibrium. If the AHB is subcritical, it is accompanied by the appearance of
multimodal orbits, which consist of long series of nearly harmonic oscillations
separated by large amplitude spikes. We analyze the dependence of the
interspike intervals (which can be extremely long) on the control parameters.
In particular, we show that the interspike intervals grow logarithmically as
the boundary between regions of sub- and supercritical AHB is approached in the
parameter space. We also identify a window of complex and possibly chaotic
oscillations near the boundary between the regions of sub- and supercritical
AHB and explain the mechanism generating these oscillations. This work is
motivated by the numerical results for a finite-dimensional approximation of a
free boundary problem modeling solid fuel combustion
On the nature of the torus in the complex Lorenz equations.
The complex Lorenz equations are a nonlinear fifth-order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus. In this paper we build upon previously derived results to examine a connection between this torus at high and low r1 bifurcation parameter) and between zero and nonzero r2(complexity parameter); in so doing, we are able to gain insight on the effect of the rotational invariance of the system, and on how extra weak dispersion (r2 ā 0) affects the chaotic behavior of the real Lorenz system (which describes a weakly dissipative, dispersive instability)
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