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Discrete & Continuous Dynamical Systems - A
July 2003 , Volume 9 , Issue 4
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2003, 9(4): 793-799
doi: 10.3934/dcds.2003.9.793
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Abstract:
We describe a hybrid method for the numerical solution of the inviscid Burgers equation. The proposed scheme consists of a Fourier Galerkin spectral method for the resolution of the large scales of the solution coupled to a real space method capable of reproducing the small scale features such as shocks. The hybrid scheme captures correctly the decay of the energy of the solution. The underlying idea of the scheme fits in the optimal prediction framework where prior knowledge about an equation is used to improve upon existing algorithms.
We describe a hybrid method for the numerical solution of the inviscid Burgers equation. The proposed scheme consists of a Fourier Galerkin spectral method for the resolution of the large scales of the solution coupled to a real space method capable of reproducing the small scale features such as shocks. The hybrid scheme captures correctly the decay of the energy of the solution. The underlying idea of the scheme fits in the optimal prediction framework where prior knowledge about an equation is used to improve upon existing algorithms.
2003, 9(4): 801-834
doi: 10.3934/dcds.2003.9.801
+[Abstract](1526)
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Abstract:
We study the structure of disconnected polynomial Julia sets. We consider polynomials with an arbitrary number of non-escaping critical points, of arbitrary multiplicity, which interact non-trivially. We use a combinatorial system of a tree with dynamics to give a sufficient condition for the Julia set a polynomial to be an area zero Cantor set. We show that there exist uncountably many combinatorially inequivalent polynomials, which satisfy this condition and have multiple non-escaping critical points, each of which accumulates at all the non-escaping critical points.
We study the structure of disconnected polynomial Julia sets. We consider polynomials with an arbitrary number of non-escaping critical points, of arbitrary multiplicity, which interact non-trivially. We use a combinatorial system of a tree with dynamics to give a sufficient condition for the Julia set a polynomial to be an area zero Cantor set. We show that there exist uncountably many combinatorially inequivalent polynomials, which satisfy this condition and have multiple non-escaping critical points, each of which accumulates at all the non-escaping critical points.
2003, 9(4): 835-858
doi: 10.3934/dcds.2003.9.835
+[Abstract](1302)
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Abstract:
The classical Lagrange inversion formula is extended to analytic and non--analytic inversion problems on non--Archimedean fields. We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations.
We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferentiable germs, closed under composition and derivation, including Gevrey Classes. We prove that Bruno's condition is sufficient for the linearization to belong to the same Class of the germ, whereas new conditions weaker than Bruno's one are introduced if one allows the linearization to be less regular than the germ. This generalizes to dimension $n> 1$ some results of [6]. Our formulation of the Lagrange inversion formula by mean of trees, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. For analytic vector fields of $\mathbb C^2$ we prove a quantitative estimate of a previous qualitative result of [25] and we compare it with a result of [26].
The classical Lagrange inversion formula is extended to analytic and non--analytic inversion problems on non--Archimedean fields. We give some applications to the field of formal Laurent series in $n$ variables, where the non--analytic inversion formula gives explicit formal solutions of general semilinear differential and $q$--difference equations.
We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferentiable germs, closed under composition and derivation, including Gevrey Classes. We prove that Bruno's condition is sufficient for the linearization to belong to the same Class of the germ, whereas new conditions weaker than Bruno's one are introduced if one allows the linearization to be less regular than the germ. This generalizes to dimension $n> 1$ some results of [6]. Our formulation of the Lagrange inversion formula by mean of trees, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. For analytic vector fields of $\mathbb C^2$ we prove a quantitative estimate of a previous qualitative result of [25] and we compare it with a result of [26].
2003, 9(4): 859-876
doi: 10.3934/dcds.2003.9.859
+[Abstract](1191)
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Abstract:
We consider discretizations $f_N$ of expanding maps $f:I \to I$ in the strict sense: i.e. we assume that the only information available on the map is a finite set of integers. Using this definition for computability, we show that by adding a random perturbation of order $1/N$, the invariant measure corresponding to $f$ can be approximated and we can also give estimates of the error term. We prove that the randomized discrete scheme is equivalent to Ulam's scheme applied to the polygonal approximation of $f$, thus providing a new interpretation of Ulam's scheme. We also compare the efficiency of the randomized iterative scheme to the direct solution of the $N \times N$ linear system.
We consider discretizations $f_N$ of expanding maps $f:I \to I$ in the strict sense: i.e. we assume that the only information available on the map is a finite set of integers. Using this definition for computability, we show that by adding a random perturbation of order $1/N$, the invariant measure corresponding to $f$ can be approximated and we can also give estimates of the error term. We prove that the randomized discrete scheme is equivalent to Ulam's scheme applied to the polygonal approximation of $f$, thus providing a new interpretation of Ulam's scheme. We also compare the efficiency of the randomized iterative scheme to the direct solution of the $N \times N$ linear system.
2003, 9(4): 877-900
doi: 10.3934/dcds.2003.9.877
+[Abstract](1165)
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Abstract:
We study local problems around simple umbilic points of surfaces immersed in $\mathbb R^4$ such as finite determinacy and versal unfoldings.
We study local problems around simple umbilic points of surfaces immersed in $\mathbb R^4$ such as finite determinacy and versal unfoldings.
2003, 9(4): 901-924
doi: 10.3934/dcds.2003.9.901
+[Abstract](1384)
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Abstract:
We study the exact controllability of the one dimensional semilinear wave equation by a control acting on an open subset $(a,b)$ of the domain $(0,1)$. With the aid of d'Alembert's formula and sidewise energy estimates, we obtain sharp conditions in the space-time support of the control, that coincide with the by now well-known geometric control condition.
More precisely, by classical results of J. Lagnese, A. Haraux and E. Zuazua, exact controllability holds in time $T > T_0 $:$= 2 max (a , 1-b)$ and fails if $T < T_0$. We weaken strongly their results: given $T>T_0$, we prove that the control can be chosen so that it is supported only on some special time intervals: they are parts of $(0,T)$, in finite number (depending on $a$ and $b$), and their total length can be arbitrarily small. The only condition is that they have to be "close enough" from each other. If this condition holds, we study the observability cost. If it fails, we prove that exact controllability in time $T$ does not hold, but can still be true in time $T'$ large enough.
We study the exact controllability of the one dimensional semilinear wave equation by a control acting on an open subset $(a,b)$ of the domain $(0,1)$. With the aid of d'Alembert's formula and sidewise energy estimates, we obtain sharp conditions in the space-time support of the control, that coincide with the by now well-known geometric control condition.
More precisely, by classical results of J. Lagnese, A. Haraux and E. Zuazua, exact controllability holds in time $T > T_0 $:$= 2 max (a , 1-b)$ and fails if $T < T_0$. We weaken strongly their results: given $T>T_0$, we prove that the control can be chosen so that it is supported only on some special time intervals: they are parts of $(0,T)$, in finite number (depending on $a$ and $b$), and their total length can be arbitrarily small. The only condition is that they have to be "close enough" from each other. If this condition holds, we study the observability cost. If it fails, we prove that exact controllability in time $T$ does not hold, but can still be true in time $T'$ large enough.
2003, 9(4): 925-936
doi: 10.3934/dcds.2003.9.925
+[Abstract](1646)
+[PDF](168.3KB)
Abstract:
In this paper, we establish the existence of traveling wavefronts for delayed reaction diffusion systems without quasimonotonicity in the reaction term, by using Schauder's fixed point theorem. We show the merit of our result by applying it to the Belousov-Zhabotinskii reaction model with two delays.
In this paper, we establish the existence of traveling wavefronts for delayed reaction diffusion systems without quasimonotonicity in the reaction term, by using Schauder's fixed point theorem. We show the merit of our result by applying it to the Belousov-Zhabotinskii reaction model with two delays.
2003, 9(4): 937-948
doi: 10.3934/dcds.2003.9.937
+[Abstract](1208)
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Abstract:
In this work we prove the existence of a global attractor for the non local evolution equation $ \frac { \partial m ( r , t ) } { \partial t } = - m ( r , t ) + \tanh ( \beta J $*$ m ( r , t ) ) $ in the space of $\tau$-periodic functions, for $\tau$ sufficiently large. We also show the existence of non constant (unstable) equilibria in these spaces.
In this work we prove the existence of a global attractor for the non local evolution equation $ \frac { \partial m ( r , t ) } { \partial t } = - m ( r , t ) + \tanh ( \beta J $*$ m ( r , t ) ) $ in the space of $\tau$-periodic functions, for $\tau$ sufficiently large. We also show the existence of non constant (unstable) equilibria in these spaces.
2003, 9(4): 949-968
doi: 10.3934/dcds.2003.9.949
+[Abstract](1387)
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Abstract:
We analyze a conserved phase-field system, characterized by heat memory terms: memory kernels $a$ and $b$ account for relaxation effects in the energy constitutive equation and in the Gurtin-Pipkin heat conduction law, respectively. This model consists of a hyperbolic integrodifferential equation for the temperature $\theta$ coupled with a nonlinear fourth order evolution equation for the phase variable $\chi$. With appropriate initial and boundary data, we prove that the system can be interpreted as a process. We show that it possesses an absorbing set, under three -thermodynamically consistent- conditions on the memory kernel $a$.
We analyze a conserved phase-field system, characterized by heat memory terms: memory kernels $a$ and $b$ account for relaxation effects in the energy constitutive equation and in the Gurtin-Pipkin heat conduction law, respectively. This model consists of a hyperbolic integrodifferential equation for the temperature $\theta$ coupled with a nonlinear fourth order evolution equation for the phase variable $\chi$. With appropriate initial and boundary data, we prove that the system can be interpreted as a process. We show that it possesses an absorbing set, under three -thermodynamically consistent- conditions on the memory kernel $a$.
2003, 9(4): 969-978
doi: 10.3934/dcds.2003.9.969
+[Abstract](1240)
+[PDF](143.3KB)
Abstract:
Let $f$ be a continuous map $f:X\to X$ of a metric space $X$ into itself. Often the information about the map is presented in the following form: for a finite collection of compact sets $A_1, \ldots, A_n$ it is known which sets have the images containing other sets, and which sets are disjoint. We study similar but weaker than usual conditions on compact sets $A_1, \ldots, A_n$ assuming that the common intersection of all sets $A_1,\ldots, A_n$ is empty (or making even weaker but more technical assumptions). As we show, this implies that the map is chaotic in the sense that it has positive topological entropy, and moreover, there exists an invariant compact set on which $f$ is semiconjugate to a full one-sided shift.
Let $f$ be a continuous map $f:X\to X$ of a metric space $X$ into itself. Often the information about the map is presented in the following form: for a finite collection of compact sets $A_1, \ldots, A_n$ it is known which sets have the images containing other sets, and which sets are disjoint. We study similar but weaker than usual conditions on compact sets $A_1, \ldots, A_n$ assuming that the common intersection of all sets $A_1,\ldots, A_n$ is empty (or making even weaker but more technical assumptions). As we show, this implies that the map is chaotic in the sense that it has positive topological entropy, and moreover, there exists an invariant compact set on which $f$ is semiconjugate to a full one-sided shift.
2003, 9(4): 979-984
doi: 10.3934/dcds.2003.9.979
+[Abstract](1264)
+[PDF](131.0KB)
Abstract:
We show that the linearizing homeomorphism in the Hartman--Grobman Theorem is differentiable at the fixed point.
We show that the linearizing homeomorphism in the Hartman--Grobman Theorem is differentiable at the fixed point.
2003, 9(4): 985-992
doi: 10.3934/dcds.2003.9.985
+[Abstract](1179)
+[PDF](121.1KB)
Abstract:
The Blasius equation is a well-known third-order nonlinear ordinary equation, which arises in certain boundary layer problems in the fluid mechanics. In this note, we investigate the behavior of blowing-up solutions for related initial value problems.
The Blasius equation is a well-known third-order nonlinear ordinary equation, which arises in certain boundary layer problems in the fluid mechanics. In this note, we investigate the behavior of blowing-up solutions for related initial value problems.
2003, 9(4): 993-1028
doi: 10.3934/dcds.2003.9.993
+[Abstract](1914)
+[PDF](339.7KB)
Abstract:
Existence and $C^N$-smoothness of a local unstable manifold at $0$ are shown for the delay differential equation $\dot x(t)=F(x_t)$ with $F:C([-h,0],\mathbb R^n)\to \mathbb R^n$, $h>0$, $F(0)=0$, under the hypotheses: There exist a linear continuous $L$ and a continuous $g$ with $F=L+g$; $0$ is a hyperbolic equilibrium of $\dot y(t)=Ly_t$; the restriction $g|_{C^k([-h,0],\mathbb R^n)}:C^k([-h,0],\mathbb R^n)\to \mathbb R^n$ is $C^k$-smooth for each $k\in$ {$1,\ldots,N$}; $D(g|_{C^1([-h,0],\mathbb R^n)})(0)=0$; in addition, for the derivatives $D^k(g|_{C^k([-h,0],\mathbb R^n)}) $, $k\in${$1,\ldots,N$}, certain extension properties hold. The conditions on $F$ are motivated and are satisfied by a wide class of differential equations with state-dependent delay.
Existence and $C^N$-smoothness of a local unstable manifold at $0$ are shown for the delay differential equation $\dot x(t)=F(x_t)$ with $F:C([-h,0],\mathbb R^n)\to \mathbb R^n$, $h>0$, $F(0)=0$, under the hypotheses: There exist a linear continuous $L$ and a continuous $g$ with $F=L+g$; $0$ is a hyperbolic equilibrium of $\dot y(t)=Ly_t$; the restriction $g|_{C^k([-h,0],\mathbb R^n)}:C^k([-h,0],\mathbb R^n)\to \mathbb R^n$ is $C^k$-smooth for each $k\in$ {$1,\ldots,N$}; $D(g|_{C^1([-h,0],\mathbb R^n)})(0)=0$; in addition, for the derivatives $D^k(g|_{C^k([-h,0],\mathbb R^n)}) $, $k\in${$1,\ldots,N$}, certain extension properties hold. The conditions on $F$ are motivated and are satisfied by a wide class of differential equations with state-dependent delay.
2003, 9(4): 1029-1048
doi: 10.3934/dcds.2003.9.1029
+[Abstract](1466)
+[PDF](244.8KB)
Abstract:
We determine the order of integer matrices $A \in SL_2(\mathbb Z)$ on lattices $L_N=\frac{1}{N}\mathbb Z^2/\mathbb Z^2$ of $\mathbb Q^2/\mathbb Z^2$, for $N=P_n \equiv $ the number of n-periodic points ( for the particular matrix-action on the rational 2-torus ). The arguments lean heavily on arithmetical properties of ( integer specializations of ) certain Chebychev polynomials.
We determine the order of integer matrices $A \in SL_2(\mathbb Z)$ on lattices $L_N=\frac{1}{N}\mathbb Z^2/\mathbb Z^2$ of $\mathbb Q^2/\mathbb Z^2$, for $N=P_n \equiv $ the number of n-periodic points ( for the particular matrix-action on the rational 2-torus ). The arguments lean heavily on arithmetical properties of ( integer specializations of ) certain Chebychev polynomials.
2003, 9(4): 1049-1061
doi: 10.3934/dcds.2003.9.1049
+[Abstract](1301)
+[PDF](206.7KB)
Abstract:
In this paper, we discuss the positive steady-state existence for predator-prey and competing interaction systems between two species with linear self-cross diffusions. The methods employed are the decomposing operators and the theory of fixed point index on cones in a Banach space. We give sufficient conditions for the existence of positive solutions. The conditions are given in terms of the signs of the principal eigenvalues of certain differential operators.
In this paper, we discuss the positive steady-state existence for predator-prey and competing interaction systems between two species with linear self-cross diffusions. The methods employed are the decomposing operators and the theory of fixed point index on cones in a Banach space. We give sufficient conditions for the existence of positive solutions. The conditions are given in terms of the signs of the principal eigenvalues of certain differential operators.
2003, 9(4): 1063-1071
doi: 10.3934/dcds.2003.9.1063
+[Abstract](1516)
+[PDF](144.1KB)
Abstract:
We study positive solutions for the system
We study positive solutions for the system
$-\Delta_p u = \lambda f(v)$ in $\quad \Omega $
$-\Delta_p v = \lambda g(u)$ in $ \quad \Omega $
$u = 0 = v$ on $ \quad \partial \Omega$
where $ \lambda > 0 $ is a parameter, $ \Delta_p $ denotes the p-Laplacian operator defined by $ \Delta_p(z)$:=div$(|\nabla z|^{p-2}\nabla z) $ for $ p> 1 $ and $ \Omega $ is a bounded domain with smooth boundary. Here $ f,g \in C[0,\infty) $ belong to a class of functions satisfying $ \lim_{z \to \infty}\frac{f(z)}{z^{p-1}}=0, \lim_{z \to \infty}\frac{g(z)}{z^{p-1}}=0 $. In particular, we discuss the existence of radial solutions for large $ \lambda $ when $ \Omega $ is an annulus. For a general bounded region $ \Omega, $ we also discuss a non-existence result when $ f(0) < 0 $ and $ g(0) < 0. $
2003, 9(4): 1073-1077
doi: 10.3934/dcds.2003.9.1073
+[Abstract](1390)
+[PDF](125.7KB)
Abstract:
In this paper, we are concerned with the structural stability of a Morse-Smale gradient-like flow $\varphi^t$ and show that if $\{\varphi_\epsilon^t\}$ is a smooth one-parameter family of $C^3$ flows with $\varphi_0^t=\varphi^t$, and {$\psi_\epsilon^t$} is another one-parameter family of $C^3$ flows such that $\psi_\epsilon^t$ is $C^0$ $O(\epsilon^3)$-close and $C^1$ $O(\epsilon^2)$-close to $\varphi_\epsilon^t$, then for all small $|\epsilon|$, there is a homeomorphism $h_\epsilon$, which is $C^0$ $O(\epsilon^2)$-near the identity map, such that $h_\epsilon$ takes the trajectories of $\varphi^t_\epsilon$ to the ones of $\psi^t_\epsilon$.
In this paper, we are concerned with the structural stability of a Morse-Smale gradient-like flow $\varphi^t$ and show that if $\{\varphi_\epsilon^t\}$ is a smooth one-parameter family of $C^3$ flows with $\varphi_0^t=\varphi^t$, and {$\psi_\epsilon^t$} is another one-parameter family of $C^3$ flows such that $\psi_\epsilon^t$ is $C^0$ $O(\epsilon^3)$-close and $C^1$ $O(\epsilon^2)$-close to $\varphi_\epsilon^t$, then for all small $|\epsilon|$, there is a homeomorphism $h_\epsilon$, which is $C^0$ $O(\epsilon^2)$-near the identity map, such that $h_\epsilon$ takes the trajectories of $\varphi^t_\epsilon$ to the ones of $\psi^t_\epsilon$.
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