1,910 research outputs found
Periodic solutions of a periodic FitzHugh-Nagumo differential system
Agraïments: The second author is partially supported by Dirección de Investigación DIUBB 1204084/R.Recently some interest has appeared for the periodic FitzHugh-Nagumo differential systems. Here, we provide sufficient conditions for the existence of periodic solutions in such differential systems
Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units
We study a system of two identical FitzHugh-Nagumo units with a mutual
linear coupling in the fast variables. While an attractive coupling always
leads to synchronous behavior, a repulsive coupling can give rise to
dynamical regimes with alternating spiking order, called leap-frogging. We
analyze various types of periodic and chaotic leap-frogging regimes, using
numerical pathfollowing methods to investigate their emergence and stability,
as well as to obtain the complex bifurcation scenario which organizes their
appearance in parameter space. In particular, we show that the stability
region of the simplest periodic leap-frog pattern has the shape of a locking
cone pointing to the canard transition of the uncoupled system. We also
discuss the role of the timescale separation in the coupled FitzHugh-Nagumo
system and the relation of the leap-frog solutions to the theory of
mixed-mode oscillations in multiple timescale systems
Leap-frog patterns in systems of two coupled FitzHugh--Nagumo units
We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical pathfollowing methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems
Symbolic computation and construction of new exact traveling wave solutions to Fitzhugh-Nagumo and Klein-Gordon equations
With the aid of the symbolic computation system Mathematica, many exact solutions for the Fitzhugh-Nagumo equation and the Klein-Gordon equation with a quadratic nonlinearity are constructed by an auxiliary equation method, the so-called (G'/G)-expansion method, where the new and more general forms of solutions are also obtained. Periodic and solitary traveling wave solutions capable of moving in both directions are observed
Phase Dynamics of Nearly Stationary Patterns in Activator-Inhibitor Systems
The slow dynamics of nearly stationary patterns in a FitzHugh-Nagumo model
are studied using a phase dynamics approach. A Cross-Newell phase equation
describing slow and weak modulations of periodic stationary solutions is
derived. The derivation applies to the bistable, excitable, and the Turing
unstable regimes. In the bistable case stability thresholds are obtained for
the Eckhaus and the zigzag instabilities and for the transition to traveling
waves. Neutral stability curves demonstrate the destabilization of stationary
planar patterns at low wavenumbers to zigzag and traveling modes. Numerical
solutions of the model system support the theoretical findings
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
New results on averaging theory and applications
Agraïments: The first author is supported by CNPq 248501/2013-5. CAPES grant 88881.030454 /2013-01 from the Program CSF-PVEThe usual averaging theory reduces the computation of some periodic solutions of a system of ordinary differential equations, to find the simple zeros of an associated averaged function. When one of these zeros is not simple, i.e. the Jacobian of the averaged function in it is zero, the classical averaging theory does not provide information about the periodic solution associated to a non simple zero. Here we provide sufficient conditions in order that the averaging theory can be applied also to non simple zeros for studying their associated periodic solutions. Additionally we do two applications of this new result for studying the zero--Hopf bifurcation in the Lorenz system and in the Fitzhugh--Nagumo system
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