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Discrete & Continuous Dynamical Systems - A

2009 , Volume 25 , Issue 4

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Detecting alien limit cycles near a Hamiltonian 2-saddle cycle
Stijn Luca, Freddy Dumortier, Magdalena Caubergh and Robert Roussarie
2009, 25(4): 1081-1108 doi: 10.3934/dcds.2009.25.1081 +[Abstract](126) +[PDF](343.6KB)
This paper aims at providing an example of a cubic Hamiltonian 2-saddle cycle that after bifurcation can give rise to an alien limit cycle; this is a limit cycle that is not controlled by a zero of the related Abelian integral. To guarantee the existence of an alien limit cycle one can verify generic conditions on the Abelian integral and on the transition map associated to the connections of the 2-saddle cycle. In this paper, a general method is developed to compute the first and second derivative of the transition map along a connection between two saddles. Next, a concrete generic Hamiltonian 2-saddle cycle is analyzed using these formula's to verify the generic relation between the second order derivative of both transition maps, and a calculation of the Abelian integral.
Asymptotic equivalence and Kobayashi-type estimates for nonautonomous monotone operators in Banach spaces
Felipe Alvarez and Juan Peypouquet
2009, 25(4): 1109-1128 doi: 10.3934/dcds.2009.25.1109 +[Abstract](88) +[PDF](284.1KB)
We provide a sharp generalization to the nonautonomous case of the well-known Kobayashi estimate for proximal iterates associated with maximal monotone operators. We then derive a bound for the distance between a continuous-in-time trajectory, namely the solution to the differential inclusion $\dot{x} + A(t)x $∋ $ 0$, and the corresponding proximal iterations. We also establish continuity properties with respect to time of the nonautonomous flow under simple assumptions by revealing their link with the function $t \mapsto A(t)$. Moreover, our sharper estimations allow us to derive equivalence results which are useful to compare the asymptotic behavior of the trajectories defined by different evolution systems. We do so by extending a classical result of Passty to the nonautonomous setting.
Polynomial differential equations with small coefficients
M. A. M. Alwash
2009, 25(4): 1129-1141 doi: 10.3934/dcds.2009.25.1129 +[Abstract](70) +[PDF](164.7KB)
Classes of polynomial non-autonomous differential equations of degree $n$ are considered. An explicit bound on the size of the coefficients is given which implies that each equation in the class has exactly $n$ complex periodic solutions. In most of the classes the upper bound can be improved when we consider real periodic solutions. We present a proof to a recent conjecture about the number of periodic solutions. The results are used to give upper bounds for the number of limit cycles of polynomial two-dimensional systems.
On the hyperbolicity of homoclinic classes
Christian Bonatti, Shaobo Gan and Dawei Yang
2009, 25(4): 1143-1162 doi: 10.3934/dcds.2009.25.1143 +[Abstract](119) +[PDF](265.1KB)
We give a sufficient criterion for the hyperbolicity of a homoclinic class. More precisely, if the homoclinic class $H(p)$ admits a partially hyperbolic splitting $T_{H(p)}M=E^s\oplus_{_<}F$, where $E^s$ is uniformly contracting and $\dim E^s= \ $ind$(p)$, and all periodic points homoclinically related with $p$ are uniformly $E^u$-expanding at the period, then $H(p)$ is hyperbolic. We also give some consequences of this result.
KdV cnoidal waves are spectrally stable
Nate Bottman and Bernard Deconinck
2009, 25(4): 1163-1180 doi: 10.3934/dcds.2009.25.1163 +[Abstract](79) +[PDF](305.9KB)
Going back to considerations of Benjamin (1974), there has been significant interest in the question of stability for the stationary periodic solutions of the Korteweg-deVries equation, the so-called cnoidal waves. In this paper, we exploit the squared-eigenfunction connection between the linear stability problem and the Lax pair for the Korteweg-deVries equation to completely determine the spectrum of the linear stability problem for perturbations that are bounded on the real line. We find that this spectrum is confined to the imaginary axis, leading to the conclusion of spectral stability. An additional argument allows us to conclude the completeness of the associated eigenfunctions.
Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations
Dongho Chae, Kyungkeun Kang and Jihoon Lee
2009, 25(4): 1181-1193 doi: 10.3934/dcds.2009.25.1181 +[Abstract](95) +[PDF](213.4KB)
We exclude a type of asymptotically self-similar singularities which are the limiting cases of the results in [5] for the Euler and Navier-Stokes equations in dimension three.
Hypercyclic translation $C_0$-semigroups on complex sectors
José A. Conejero and Alfredo Peris
2009, 25(4): 1195-1208 doi: 10.3934/dcds.2009.25.1195 +[Abstract](105) +[PDF](260.4KB)
We study the hypercyclic behaviour of sequences of operators in a $C_0$-semigroup whose index set is a sector in the complex plane. The hypercyclicity and chaos for the concrete case of the translation semigroup is analyzed. Some examples are provided to complete the results.
On the regularity of global attractors
Monica Conti and Vittorino Pata
2009, 25(4): 1209-1217 doi: 10.3934/dcds.2009.25.1209 +[Abstract](120) +[PDF](158.8KB)
This note is focused on a novel technique to establish the boundedness in more regular spaces for global attractors of dissipative dynamical systems, without appealing to uniform-in-time estimates. As an application, we consider the semigroup generated by the strongly damped wave equation with critical nonlinearity, whose attractor is shown to possess the optimal regularity.
Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions
Francisco Odair de Paiva and Humberto Ramos Quoirin
2009, 25(4): 1219-1227 doi: 10.3934/dcds.2009.25.1219 +[Abstract](76) +[PDF](159.9KB)
We study multiplicity of solutions for a quasilinear elliptic problem related to the $p$-Laplacian operator. Our assumptions rely on the first eigenvalue depending on a weight function. We treat both resonant and non-resonant cases.
Existence and stability of high frequency standing waves for a nonlinear Schrödinger equation
François Genoud
2009, 25(4): 1229-1247 doi: 10.3934/dcds.2009.25.1229 +[Abstract](123) +[PDF](283.4KB)
This article is concerned with the existence and orbital stability of standing waves for a nonlinear Schrödinger equation (NLS) with a nonautonomous nonlinearity. It continues and concludes the series of papers [6, 7, 8]. In [6], the authors make use of a continuation argument to establish the existence in $R \times H^1$(RN$)$ of a smooth local branch of solutions to the stationary elliptic problem associated with (NLS) and hence the existence of standing wave solutions of (NLS) with small frequencies. Complementary conditions on the nonlinearity are found, under which either stability of the standing waves and bifurcation of the branch of solutions from the point $(0,0)\in R \times H^1$(RN$)$ occur, or instability and asymptotic bifurcation occur. The main hypotheses in [6] concern the behaviour of the nonlinearity with respect to the space variable at infinity. The paper [7] extends the results of [6] to (NLS) with more general nonlinearities. In [8], the global continuation of the local branch obtained in [6] is proved under additional hypotheses on the nonlinearity. In particular, spherical symmetry with respect to the space variable is assumed. The aim of the present work is to prove the existence and discuss the orbital stability of standing waves with high frequencies, independently of the results obtained in [6] and [8]. The main hypotheses now concern the behaviour of the nonlinearity with respect to the space variable around the origin. The methods are the same in spirit as that of [6] and permit to discuss the asymptotic behaviour of the global branch of solutions obtained in [8].
Approximating the basin of attraction of time-periodic ODEs by meshless collocation
Peter Giesl and Holger Wendland
2009, 25(4): 1249-1274 doi: 10.3934/dcds.2009.25.1249 +[Abstract](75) +[PDF](446.8KB)
In this paper we study a periodic solution of a general time-periodic ordinary differential equation (ODE) and determine its basin of attraction using a time-periodic Lyapunov function. We show the existence of a Lyapunov function satisfying a certain linear partial differential equation and approximate it using meshless collocation. Therefore, we establish error estimates for the approximate reconstruction and collocation of functions $V(t,x)$ which are periodic with respect to $t$.
Strong traces for degenerate parabolic-hyperbolic equations
Young-Sam Kwon
2009, 25(4): 1275-1286 doi: 10.3934/dcds.2009.25.1275 +[Abstract](97) +[PDF](202.9KB)
In this paper we consider bounded weak solutions $u$ of degenerate parabolic-hyperbolic equations defined in a subset $]0,T[\times\Omega\subset \R^{+}\times \R^d$. We define a strong notion of trace at the boundary $]0,T[\times\partial\Omega$ reached by $L^1$ convergence for a large class of functionals of $u$ and at $0 \times \Omega$ reached by $L^1$ convergence for solution $u$. This result develops the strong trace results of Kwon, Vasseur [13] and Panov [19, 20] for more general equations, namely, degenerate parabolic-hyperbolic equations.
$3$ - dimensional Hopf bifurcation via averaging theory of second order
Jaume Llibre, Amar Makhlouf and Sabrina Badi
2009, 25(4): 1287-1295 doi: 10.3934/dcds.2009.25.1287 +[Abstract](74) +[PDF](133.6KB)
We study the Hopf bifurcation occurring in polynomial quadratic vector fields in $\R^3$. By applying the averaging theory of second order to these systems we show that at most $3$ limit cycles can bifurcate from a singular point having eigenvalues of the form $\pm bi$ and $0$. We provide an example of a quadratic polynomial differential system for which exactly $3$ limit cycles bifurcate from a such singular point.
Fixed point shifts of inert involutions
Nicholas Long
2009, 25(4): 1297-1317 doi: 10.3934/dcds.2009.25.1297 +[Abstract](83) +[PDF](279.2KB)
Given a mixing shift of finite type $X$, we consider which subshifts of finite type $Y \subset X$ can be realized as the fixed point shift of an inert involution of $X$. We establish a condition on the periodic points of $X$ and $Y$ that is necessary for $Y$ to be the fixed point shift of an inert involution of $X$. We show that this condition is sufficient to realize $Y$ as the fixed point shift of an involution, up to shift equivalence on $X$, if $X$ is a shift of finite type with Artin-Mazur zeta function equivalent to 1 mod 2. Given an inert involution $f$ of a mixing shift of finite type $X$, we characterize what $f$-invariant subshifts can be realized as the fixed point shift of an inert involution.
Zero-relaxation limit of non-isentropic hydrodynamic models for semiconductors
Jiang Xu and Wen-An Yong
2009, 25(4): 1319-1332 doi: 10.3934/dcds.2009.25.1319 +[Abstract](71) +[PDF](210.3KB)
This paper deals with non-isentropic hydrodynamic models for semiconductors with short momentum and energy relaxation times. With the help of the Maxwell iteration, we construct a new approximation and show that periodic initial-value problems of certain scaled non-isentropic hydrodynamic models have unique smooth solutions in a time interval independent of the two relaxation times. Furthermore, it is proved that as the two relaxation times both tend to zero, the smooth solutions converge to solutions of the corresponding semilinear drift-diffusion models.
Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms
Xiaojing Xu
2009, 25(4): 1333-1347 doi: 10.3934/dcds.2009.25.1333 +[Abstract](94) +[PDF](192.0KB)
In this paper, we consider the local existence and a blow-up criterion for smooth solutions to the 2-D isentropic compressible Boussinesq equations, and obtain some new commutator estimates. In particular, we show that the time integral of the spatial maximum of gradient of velocity controls the breakdown of smooth solutions.
Minimal non-hyperbolicity and index-completeness
Dawei Yang, Shaobo Gan and Lan Wen
2009, 25(4): 1349-1366 doi: 10.3934/dcds.2009.25.1349 +[Abstract](104) +[PDF](256.8KB)
We study a problem raised by Abdenur et. al. [3] that asks, for any chain transitive set $\Lambda$ of a generic diffeomorphism $f$, whether the set $I(\Lambda)$ of indices of hyperbolic periodic orbits that approach $\Lambda$ in the Hausdorff metric must be an "interval", i.e., whether $\alpha\in I(\Lambda)$ and $\beta\in I(\Lambda)$, $\alpha<\beta$, must imply $\gamma\in I(\Lambda)$ for every $\alpha<\gamma<\beta$. We prove this is indeed the case if, in addition, $f$ is $C^1$ away from homoclinic tangencies and if $\Lambda$ is a minimally non-hyperbolic set.

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