Smooth robustness of parameterized perturbations of exponential dichotomies

We establish the robustness of nonuniform exponential dichotomies in Banach spaces, under sufficiently small C¹-parameterized perturbations. Moreover, we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class C¹ on the parameter, thus yielding an optimal smoothness.     

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

We establish the existence of unique smooth center manifolds for ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that v^′=A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.     

Periodic orbits in complex Abel equations

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z³+B(t)z²+C(t)z, where B and C are smooth, 2π-periodic complex valuated functions, t∈R and z∈C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt=A(t)z³+B(t)z² studied in the literature, where the center variety is located in a finite number of connected components.     

Hopf bifurcation in symmetric configuration of predator-prey-mutualist systems

In this paper we apply the equivariant degree method to study Hopf bifurcations in a system of differential equations describing a symmetric predator-prey-mutualist model with diffusive migration between interacting communities. A topological classification (according to symmetry types), of symmetric Hopf bifurcation in configurations of populations with D₈, D₁₂, A₄ and S₄ symmetries, is presented with estimation on minimal number of bifurcating branches of periodic solutions.     

Embedding smooth diffeomorphisms in flows

In this paper we study the problem on embedding germs of smooth diffeomorphisms in flows in higher dimensional spaces. First we prove the existence of embedding vector fields for a local diffeomorphism with its nonlinear term a resonant polynomial. Then using this result and the normal form theory, we obtain a class of local Ck diffeomorphisms for k∈N∪{∞,ω} which admit embedding vector fields with some smoothness. Finally we prove that for any k∈N∪{∞} under the coefficient topology the subset of local Ck diffeomorphisms having an embedding vector field with some smoothness is dense in the set of all local Ck diffeomorphisms.     

Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

In [Xiang Zhang, The embedding flows of C^∞ hyperbolic diffeomorphisms, J. Differential Equations 250 (5) (2011) 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator.We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterion is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.     

Invariant manifolds around equilibria of Newtonian equations: Some pathological examples

Let the equation be periodic in time, and let the equilibrium x_∗≡0 be a periodic minimizer. If it is hyperbolic, then the set of asymptotic solutions is a smooth curve in the plane ; this is stated by the Stable Manifold Theorem. The result can be extended to nonhyperbolic minimizers provided only that they are isolated and the equation is analytic (Ureña, 2007 [6]). In this paper we provide an example showing that one cannot say the same for C² equations. Our example is pathological both in a global sense (the global stable manifold is not arcwise connected), and in a local sense (the local stable manifolds are not locally connected and have points which are not accessible from the exterior).     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

On the critical periods of perturbed isochronous centers

Consider a family of planar systems having a center at the origin and assume that for ε=0 they have an isochronous center. Firstly, we give an explicit formula for the first order term in ε of the derivative of the period function. We apply this formula to prove that, up to first order in ε, at most one critical period bifurcates from the periodic orbits of isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers. Moreover necessary and sufficient conditions for the existence of this critical period are explicitly given. From the tools developed in this paper we also provide a new characterization of planar isochronous centers.     

Structurally stable perturbations of polynomials in the Riemann sphere

The perturbations of complex polynomials of one variable are considered in a wider class than the holomorphic one. It is proved that under certain conditions on a polynomial p   of the plane, the Cr$C^r$ conjugacy class of a map f   in a C¹$C^1$ neighborhood of p depends only on the geometric structure of the critical set of f. This provides the first class of examples of structurally stable maps with critical points and nontrivial nonwandering set in dimension greater than one.     

Optimal regularity of robustness for parameterized perturbations

For exponential dichotomies defined by nonautonomous linear equations, we show that sufficiently small C¹-parameterized perturbations originate a family of exponential dichotomies of class C¹ in the parameter. We consider the general case of nonuniform exponential dichotomies, and also the general case of arbitrary growth rates of the form eλρ(t) where ρ is an arbitrary function. This includes the usual exponential behavior as a very special case when ρ(t)=t.     

Approximation order of interpolatory nonlinear subdivision schemes

Linear interpolatory subdivision schemes of C^r smoothness have approximation order at least r+1. The present paper extends this result to nonlinear univariate schemes which are in proximity with linear schemes in a certain specific sense. The results apply to nonlinear subdivision schemes in Lie groups and in surfaces which are obtained from linear subdivision schemes. We indicate how to extend the results to the multivariate case.     

On Reeb components of invariant foliations of projectively Anosov flows

We show that if a C² codimension one foliation on a three-dimensional manifold has a Reeb component and is invariant under a projectively Anosov flow, then it must be a Reeb foliation on S²×S¹.     

Existence of periodic solutions for the Lotka-Volterra type systems

In this paper we prove the existence of nonstationary periodic solutions of delay Lotka-Volterra equations. In the proofs we use the S¹-degree due to Dylawerski et al. [G. Dylawerski, K. Geba, J. Jodel, W. Marzantowicz, An S¹-equivariant degree and the Fuller index, Ann. Polon. Math. 63 (1991) 243-280].     

Varieties of local integrability of analytic differential systems and their applications

In this paper we provide a characterization of local integrability for analytic or formal differential systems in Rⁿ${\mathbb{R}}^n$ or Cⁿ${\mathbb{C}}^n$ via the integrability varieties. Our result generalizes the classical one of Poincaré and Lyapunov on local integrability of planar analytic differential systems to any finitely dimensional analytic differential systems. As an application of our theory we study the integrability of a family of four-dimensional quadratic Hamiltonian systems.     

On the Lefschetz periodic point free continuous self-maps on connected compact manifolds

We characterize the Lefschetz periodic point free self-continuous maps on the following connected compact manifolds: CPn the n-dimensional complex projective space, HPn the n-dimensional quaternion projective space, Sn the n-dimensional sphere and Sp×Sq the product space of the p-dimensional with the q-dimensional spheres.     

Topological stability for conservative systems

We prove that the C¹ interior of the set of all topologically stable C¹ incompressible flows is contained in the set of Anosov incompressible flows. Moreover, we obtain an analogous result for the discrete-time case.