Upper bounds for the number of limit cycles of some planar polynomial differential systems
Armengol Gasull Hector Giacomini
We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples.
keywords: Dulac function. limit cycle Polynomial differential system
Characterizing asymptotic stability with Dulac functions
Marc Chamberland Anna Cima Armengol Gasull Francesc Mañosas
This paper studies questions regarding the local and global asymptotic stability of analytic autonomous ordinary differential equations in $\mathbb{R}^n$. It is well-known that such stability can be characterized in terms of Liapunov functions. The authors prove similar results for the more geometrically motivated Dulac functions. In particular it holds that any analytic autonomous ordinary differential equation having a critical point which is a global attractor admits a Dulac function. These results can be used to give criteria of global attraction in two-dimensional systems.
keywords: Global and local asymptotic stability curvature of orbits. Liapunov function Markus-Yamabe Jacobian conjecture Dulac function
Global linearization of periodic difference equations
Anna Cima Armengol Gasull Francesc Mañosas
We deal with $m$-periodic, $n$-th order difference equations and study whether they can be globally linearized. We give an affirmative answer when $m=n+1$ and for most of the known examples appearing in the literature. Our main tool is a refinement of the Montgomery-Bochner Theorem.
keywords: recurrences periodic difference equations Periodic maps linearization.
Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations
Armengol Gasull Hector Giacomini Joan Torregrosa
It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy. These results allow one to construct analytical approximate expressions for the traveling wave solutions with a rigorous control of the errors for arbitrary values of the independent variables. These explicit expressions are very simple and tractable for practical purposes. They are constructed with exponential and rational functions.
keywords: reaction-diffusion partial differential equation invariant manifold. heteroclinic orbit Fisher-Kolmogorov equation Traveling wave
Center problem for systems with two monomial nonlinearities
Armengol Gasull Jaume Giné Joan Torregrosa
We study the center problem for planar systems with a linear center at the origin that in complex coordinates have a nonlinearity formed by the sum of two monomials. Our first result lists several centers inside this family. To the best of our knowledge this list includes a new class of Darboux centers that are also persistent centers. The rest of the paper is dedicated to try to prove that the given list is exhaustive. We get several partial results that seem to indicate that this is the case. In particular, we solve the question for several general families with arbitrary high degree and for all cases of degree less or equal than 19. As a byproduct of our study we also obtain the highest known order for weak-foci of planar polynomial systems of some given degrees.
keywords: Poincaré--Lyapunov constants Nondegenerate center Darboux center Holomorphic center Reversible center Persistent center.
Local and global phase portrait of equation $\dot z=f(z)$
Antonio Garijo Armengol Gasull Xavier Jarque
This paper studies the differential equation $\dot z=f(z)$, where $f$ is an analytic function in $\mathbb C$ except, possibly, at isolated singularities. We give a unify treatment of well known results and provide new insight into the local normal forms and global properties of the solutions for this family of differential equations.
keywords: polycycle. conformal conjugacy Phase portrait holomorphic map
Stability of singular limit cycles for Abel equations
José Luis Bravo Manuel Fernández Armengol Gasull
We obtain a criterion for determining the stability of singular limit cycles of Abel equations $x'=A(t)x^3+B(t)x^2$. This stability controls the possible saddle-node bifurcations of limit cycles. Therefore, studying the Hopf-like bifurcations at $x=0$, together with the bifurcations at infinity of a suitable compactification of the equations, we obtain upper bounds of their number of limit cycles. As an illustration of this approach, we prove that the family $x'=a t(t-t_A)x^3+b (t-t_B)x^2$, with $a ,b>0$, has at most two positive limit cycles for any $t_B,t_A$.
keywords: Abel equation periodic solution limit cycle. closed solution
Bifurcation values for a family of planar vector fields of degree five
Johanna D. García-Saldaña Armengol Gasull Hector Giacomini
We study the number of limit cycles and the bifurcation diagram in the Poincaré sphere of a one-parameter family of planar differential equations of degree five $\dot {\bf x}=X_b({\bf x})$ which has been already considered in previous papers. We prove that there is a value $b^*>0$ such that the limit cycle exists only when $b\in(0,b^*)$ and that it is unique and hyperbolic by using a rational Dulac function. Moreover we provide an interval of length $27/1000$ where $b^*$ lies. As far as we know the tools used to determine this interval are new and are based on the construction of algebraic curves without contact for the flow of the differential equation. These curves are obtained using analytic information about the separatrices of the infinite critical points of the vector field. To prove that the Bendixson--Dulac Theorem works we develop a method for studying whether one-parameter families of polynomials in two variables do not vanish based on the computation of the so called double discriminant.
keywords: Polynomial planar system double discriminant. phase portrait on the Poincaré sphere Dulac function bifurcation uniqueness of limit cycles
On the stability of periodic orbits for differential systems in $\mathbb{R}^n$
Armengol Gasull Héctor Giacomini Maite Grau
We consider an autonomous differential system in $\mathbb{R}^n$ with a periodic orbit and we give a new method for computing the characteristic multipliers associated to it. Our method works when the periodic orbit is given by the transversal intersection of $n-1$ codimension one hypersurfaces and is an alternative to the use of the first order variational equations. We apply it to study the stability of the periodic orbits in several examples, including a periodic solution found by Steklov studying the rigid body dynamics.
keywords: rigid body dynamics Periodic orbit characteristic multipliers Steklov periodic orbit. invariant curve Mathieu's equation
Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points
Anna Cima Armengol Gasull Víctor Mañosa

We show that for periodic non-autonomous discrete dynamical systems, even when a common fixed point for each of the autonomous associated dynamical systems is repeller, this fixed point can became a local attractor for the whole system, giving rise to a Parrondo's dynamic type paradox.

keywords: Periodic discrete dynamical systems non-hyperbolic points local and global asymptotic stability Parrondo's dynamic paradox

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