We consider bifurcation problems associated with some lower semicontinuous functionals that do not satisfy the usual regularity assumptions. For such functionals it is possible to define a generalized "Hessian form" and to show that certain eigenvalues of this one are bifurcation values. The results are applied to a bifurcation problem for elliptic variational inequalities.
In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C ∞, parametrized families {g t ∣t∈ℝ} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative speed as...
In this article, we study germs of holomorphic vector fields which are “higher order” perturbations of a quasihomogeneous vector field in a neighborhood of the origin of , fixed point of the vector fields. We define a “Diophantine condition” on the quasihomogeneous initial part which ensures that if such a perturbation of is formally conjugate to then it is also holomorphically conjugate to it. We study the normal form problem relatively to . We give a condition on that ensures that there...
Let Γ be a finite subgroup of GL(n, C). This subgroup acts on the space of germs of holomorphic vector fields vanishing at the origin in Cn and on the group of germs of holomorphic diffeomorphisms of (Cn, 0). We prove a theorem of invariant conjugacy to a normal form and linearization for the subspace of invariant germs of holomorphic vector fields and we give a description of this type of normal forms in dimension n = 2.
We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.
We explore the convergence/divergence of the normal form for a singularity of a vector field on with nilpotent linear part. We show that a Gevrey- vector field with a nilpotent linear part can be reduced to a normal form of Gevrey- type with the use of a Gevrey- transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.
We establish a Poincaré-Dulac theorem for sequences of holomorphic contractions whose differentials split regularly. The resonant relations determining the normal forms hold on the moduli of the exponential rates of contraction. Our results are actually stated in the framework of bundle maps.Such sequences of holomorphic contractions appear naturally as iterated inverse branches of endomorphisms of . In this context, our normalization result allows to estimate precisely the distortions of ellipsoids...
We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field.
In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, a sequence of period-doubling bifurcations for small-scale oscillations precedes the transition into the spiking regime. For a wide range of values of the timescale separation parameter, this scenario is recovered numerically. Its relation to the singularly perturbed integrable system is discussed.