# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

April 2007 , Volume 17 , Issue 2

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2007, 17(2): i-ii doi: 10.3934/dcds.2007.17.2i +[Abstract](1483) +[PDF](31.6KB)
Abstract:
This special issue of DCDS is dedicated to Carlos Gutierrez and Marco Antonio Teixeira, on the occasion of their 60th birthday.
Born in Peru, C. Gutierrez obtained his Ph.D. degree from IMPA in 1974, under the supervision of Jorge Sotomayor. In the same year he began to serve as a researcher there. He retired from IMPA and now is a full professor at ICMC-University of São Paulo (USP), where his leadership has been crucial to the development of the research on dynamical systems.

2007, 17(2): 223-245 doi: 10.3934/dcds.2007.17.223 +[Abstract](1407) +[PDF](316.0KB)
Abstract:
We prove that the shadowing property does not hold for diffeomorphisms in an open and dense subset of the set of $C^1$-robustly non-hyperbolic transitive diffeomorphisms (i.e., diffeomorphisms with a $C^1$-neighborhood consisting of non-hyperbolic transitive diffeomorphisms).
2007, 17(2): 247-258 doi: 10.3934/dcds.2007.17.247 +[Abstract](2029) +[PDF](186.3KB)
Abstract:
We study, from a new point of view, families of planar vector fields without singularities $\{ X_{\mu}$   :   $-\varepsilon < \mu < \varepsilon\}$ defined on the complement of an open ball centered at the origin such that, at $\mu=0$, infinity changes from repellor to attractor, or vice versa. We also study a sort of local stability of some $C^1$ planar vector fields around infinity.
2007, 17(2): 259-270 doi: 10.3934/dcds.2007.17.259 +[Abstract](1672) +[PDF](181.8KB)
Abstract:
The statistical analysis of the structurally stable quadratic vector fields made in [4] shows that the phase portrait 7.1 (see Figure 1) appears without limit cycles, when the other three phase portraits in the same family with low probability sometimes appear with limit cycles. Here we prove that quadratic vector fields having the phase portrait 7.1 have no limit cycles.
2007, 17(2): 271-280 doi: 10.3934/dcds.2007.17.271 +[Abstract](1519) +[PDF](199.4KB)
Abstract:
We consider the set $\F$ of the $C^1$-maps $f:\S^1 \to \S^1$ which are of degree 2, uniformly expanding except for a small interval and such that the origin is a fixed critical point. Fixing a $f \in \F$, we show that, for a generic $\alpha$-Hölder potential $A:\S^1 \to \RR$, the $f$-invariant probability measure that maximizes the action given by the integral of $A$ is unique and uniquely ergodic on its support. Furthermore, restricting $f$ to a suitable subset of $\S^1$, we also show that this measure is supported on periodic orbits. Our main tool is the existence of a $\alpha$-Hölder sub-action function for each fixed $f \in \F$ and each fixed potential $A$. We also show that these results can be applied to the special potential $A=\log|f'|$.
2007, 17(2): 281-292 doi: 10.3934/dcds.2007.17.281 +[Abstract](1416) +[PDF](176.1KB)
Abstract:
We extend the Hartman-Grobman theorems for discrete random dynamical systems (RDS), proved in [7], in two directions: for continuous RDS and for hyperbolic stationary trajectories. In this last case there exists a conjugacy between travelling neighborhoods of trajectories and neighborhoods of the origin in the corresponding tangent bundle. We present applications to deterministic dynamical systems.
2007, 17(2): 293-308 doi: 10.3934/dcds.2007.17.293 +[Abstract](1182) +[PDF](258.5KB)
Abstract:
This paper studies the behavior of lines of curvature near umbilic points that appear generically on surfaces depending on two parameters.
2007, 17(2): 309-329 doi: 10.3934/dcds.2007.17.309 +[Abstract](1403) +[PDF](374.1KB)
Abstract:
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.
2007, 17(2): 331-347 doi: 10.3934/dcds.2007.17.331 +[Abstract](1574) +[PDF](421.5KB)
Abstract:
We present an algorithm which determines global conditions for a class of discontinuous vector fields in 4D (called polynomial relay systems) to have periodic orbits. We present explicit results relying on constructive proofs, which involve classical Effective Algebraic Geometry algorithms.
2007, 17(2): 349-364 doi: 10.3934/dcds.2007.17.349 +[Abstract](1481) +[PDF](339.4KB)
Abstract:
We study geometric properties of the integral curves of an implicit differential equation in a neighbourhood of a codimension $\le 1$ singularity. We also deal with the way these singularities bifurcate in generic families of equations and the changes in the associated geometry. The main tool used here is the Legendre transformation.
2007, 17(2): 365-370 doi: 10.3934/dcds.2007.17.365 +[Abstract](1071) +[PDF](121.3KB)
Abstract:
In this note we present a simple computable criteria that assures the existence of hyperbolic horseshoes for certain diffeomorphisms of the torus. The main advantage of our method is that it is very easy to check numerically whether the criteria is satisfied or not.
2007, 17(2): 371-386 doi: 10.3934/dcds.2007.17.371 +[Abstract](1422) +[PDF](240.6KB)
Abstract:
We obtain results on existence and continuity of physical measures through equilibrium states and apply these to non-uniformly expanding transformations on compact manifolds with non-flat critical sets, deducing sufficient conditions for continuity of physical measures and, for local diffeomorphisms, necessary and sufficient conditions for stochastic stability. In particular we show that, under certain conditions, stochastically robust non-uniform expansion implies existence and continuous variation of physical measures.
2007, 17(2): 387-395 doi: 10.3934/dcds.2007.17.387 +[Abstract](1787) +[PDF](145.0KB)
Abstract:
We present some results and one open question on the existence of polynomial inverse integrating factors for polynomial vector fields.
2007, 17(2): 397-402 doi: 10.3934/dcds.2007.17.397 +[Abstract](1725) +[PDF](135.5KB)
Abstract:
Using the half-Reeb component technique as introduced in [10], we try to clarify the intrinsic relation between the injectivity of differentiable local homeomorphisms $X$ of $R^2$ and the asymptotic behavior of real eigen-values of derivations $DX(x)$. The main result shows that a differentiable local homeomorphism $X$ of $R^2$ is injective and that its image $X(R^2)$ is a convex set if $X$ satisfies the following condition: (*) There does not exist a sequence $R^2$ ∋ $x_i\rightarrow \infty$ such that $X(x_i)\rightarrow a\in \R^2$ and $DX(x_i)$ has a real eigenvalue $\lambda _i\rightarrow 0$. When the graph of $X$ is an algebraic set, this condition becomes a necessary and sufficient condition for $X$ to be a global diffeomorphism.
2007, 17(2): 403-422 doi: 10.3934/dcds.2007.17.403 +[Abstract](1708) +[PDF](232.6KB)
Abstract:
We show that there are examples of expansive, non-Anosov geodesic flows of compact surfaces with non-positive curvature, where the Livsic Theorem holds in its classical (continuous, Hölder) version. We also show that such flows have continuous subaction functions associated to Hölder continuous observables.
2007, 17(2): 423-439 doi: 10.3934/dcds.2007.17.423 +[Abstract](1321) +[PDF](224.6KB)
Abstract:
In this paper we obtain some non-linear analogues of Schur's theorem asserting that a finitely generated subgroup of a linear group all of whose elements have finite order is, in fact, finite. The main result concerns groups of symplectomorphisms of certain manifolds of dimension $4$ including the torus $T^4$.
2007, 17(2): 441-448 doi: 10.3934/dcds.2007.17.441 +[Abstract](1710) +[PDF](141.9KB)
Abstract:
A lot of partial results are known about the Liénard differential equations : $\dot x= y -F_a^n(x),\ \ \dot y =-x.$ Here $F_a^n$ is a polynomial of degree $2n+1,\ \ F_a^n(x)= \sum_{i=1}^{2n}a_ix^i+x^{2n+1},$ where $a = (a_1,\cdots,a_{2n}) \in \R^{2n}.$ For instance, it is easy to see that for any $a$ the related vector field $X_a$ has just a finite number of limit cycles. This comes from the fact that $X_a$ has a global return map on the half-axis $Ox=\{x \geq 0\},$ and that this map is analytic and repelling at infinity. It is also easy to verify that at most $n$ limit cycles can bifurcate from the origin. For these reasons, Lins Neto, de Melo and Pugh have conjectured that the total number of limit cycles is also bounded by $n,$ in the whole plane and for any value $a.$
In fact it is not even known if there exists a finite bound $L(n)$ independent of $a,$ for the number of limit cycles. In this paper, I want to investigate this question of finiteness. I show that there exists a finite bound $L(K,n)$ if one restricts the parameter in a compact $K$ and that there is a natural way to put a boundary to the space of Liénard equations. This boundary is made of slow-fast equations of Liénard type, obtained as singular limits of the Liénard equations for large values of the parameter. Then the existence of a global bound $L(n)$ can be related to the finiteness of the number of limit cycles which bifurcate from slow-fast cycles of these singular equations.

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