103 research outputs found
Bifurcation diagrams and global phase portraits for some hamiltonian systems with rational potentials
In this paper, we study the global dynamical behavior of the Hamiltonian system áș = Hy(x,y), áș=âHx(x,y) with the rational potential Hamiltonian H(x,y) = y2/2 + P(x)/Q(y), where P(x) and Q(y) are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian systems by some linear change of variables. Then we classify all the global phase portraits of these systems in the PoincarĂ© disk and provide their bifurcation diagrams
Normal-internal resonances in quasi-periodically forced oscillators: a conservative approach
We perform a bifurcation analysis of normalâinternal resonances in parametrised families of quasiâperiodically forced
Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the âbackboneâ
system; forced, the system is a skewâproduct flow with a quasiâperiodic driving with basic frequencies. The
dynamics of the forced system are simplified by averaging over the orbits of a linearisation of the unforced system. The
averaged system turns out to have the same structure as in the wellâknown case of periodic forcing ; for a real
analytic system, the nonâintegrable part can even be made exponentially small in the forcing strength. We investigate
the persistence and the bifurcations of quasiâperiodic âdimensional tori in the averaged system, filling normalâinternal
resonance âgapsâ that had been excluded in previous analyses. However, these gaps cannot completely be filled up: secondary
resonance gaps appear, to which the averaging analysis can be applied again. This phenomenon of âgaps within
gapsâ makes the quasiâperiodic case more complicated than the periodic case
Complex Dynamics in One-Dimensional Nonlinear Schrödinger Equations with Stepwise Potential
We prove the existence and multiplicity of periodic solutions as well as solutions presenting a complex behavior for the one-dimensional nonlinear Schrödinger equation -Δ2uâČâČ+V(x)u=f(u), where the potential V(x) approximates a two-step function. The term f(u) generalizes the typical p-power nonlinearity considered by several authors in this context. Our approach is based on some recent developments of the theory of topological horseshoes, in connection with a linked twist maps geometry, which are applied to the discrete dynamics of the PoincarĂ© map. We discuss the periodic and the Neumann boundary conditions. The value of the term Δ>0, although small, can be explicitly estimated
Dynamics of biologically informed neural mass models of the brain
This book contributes to the development and analysis of computational models that help brain function to be understood. The mean activity of a brain area is mathematically modeled in such a way as to strike a balance between tractability and biological plausibility. Neural mass models (NMM) are used to describe switching between qualitatively different regimes (such as those due to pharmacological interventions, epilepsy, sleep, or context-induced state changes), and to explain resonance phenomena in a photic driving experiment. The description of varying states in an ordered sequence gives a principle scheme for the modeling of complex phenomena on multiple time scales. The NMM is matched to the photic driving experiment routinely applied in the diagnosis of such diseases as epilepsy, migraine, schizophrenia and depression. The model reproduces the clinically relevant entrainment effect and predictions are made for improving the experimental setting.Die vorliegende Arbeit stellt einen Beitrag zur Entwicklung und Analyse von
Computermodellen zum VerstÀndnis von Hirnfunktionen dar. Es wird die
mittlere AktivitÀt eines Hirnareals analytisch einfach und dabei
biologisch plausibel modelliert. Auf Grundlage eines Neuronalen
Massenmodells (NMM) werden die Wechsel zwischen Oszillationsregimen (z.B.
durch pharmakologisch, epilepsie-, schlaf- oder kontextbedingte
ZustandsÀnderungen) als geordnete Folge beschrieben und ResonanzphÀnomene
in einem Photic-Driving-Experiment erklÀrt. Dieses NMM kann sehr komplexe
Dynamiken (z.B. Chaos) innerhalb biologisch plausibler Parameterbereiche
hervorbringen. Um das Verhalten abzuschÀtzen, wird das NMM als Funktion
konstanter EingangsgröĂen und charakteristischer Zeitenkonstanten
vollstÀndig auf Bifurkationen untersucht und klassifiziert. Dies
ermöglicht die Beschreibung wechselnder Regime als geordnete Folge durch
spezifische Eingangstrajektorien. Es wird ein Prinzip vorgestellt, um
komplexe PhÀnomene durch Prozesse verschiedener Zeitskalen darzustellen.
Da aufgrund rhythmischer Stimuli und der intrinsischen Rhythmen von
NeuronenverbĂ€nden die EingangsgröĂen hĂ€ufig periodisch sind, wird das
Verhalten des NMM als Funktion der IntensitÀt und Frequenz einer
periodischen Stimulation mittels der zugehörigen Lyapunov-Spektren und der
Zeitreihen charakterisiert. Auf der Basis der gröĂten Lyapunov-Exponenten
wird das NMM mit dem Photic-Driving-Experiment ĂŒberein gebracht. Dieses
Experiment findet routinemĂ€Ăige Anwendung in der Diagnostik verschiedener
Erkrankungen wie Epilepsie, MigrÀne, Schizophrenie und Depression. Durch
die Anwendung des vorgestellten NMM wird der fĂŒr die Diagnostik
entscheidende Mitnahmeeffekt reproduziert und es werden Vorhersagen fĂŒr
eine Verbesserung der Indikation getroffen
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