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Discrete & Continuous Dynamical Systems - A
October 1998 , Volume 4 , Issue 4
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1998, 4(4): 593-608
doi: 10.3934/dcds.1998.4.593
+[Abstract](1325)
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Abstract:
We investigate the dynamics of systems generalizing interval exchanges to planar mappings. Unlike interval exchanges and translations, our mappings, despite the lack of hyperbolicity, exhibit many features of attractors. The main result states that for a certain class of noninvertible piecewise isometries, orbits visiting both atoms infinitely often must accumulates on the boundaries of the attractor consisting of two maximal invariant discs $D_0 \cup D_1$ fixed by $T$. The key new idea is a dynamical and geometric observation about the monotonic behavior of orbits of a certain first-return map. Our model emerges as the local map for other piecewise isometries and can be the basis for the construction of more complicated molecular attractors.
We investigate the dynamics of systems generalizing interval exchanges to planar mappings. Unlike interval exchanges and translations, our mappings, despite the lack of hyperbolicity, exhibit many features of attractors. The main result states that for a certain class of noninvertible piecewise isometries, orbits visiting both atoms infinitely often must accumulates on the boundaries of the attractor consisting of two maximal invariant discs $D_0 \cup D_1$ fixed by $T$. The key new idea is a dynamical and geometric observation about the monotonic behavior of orbits of a certain first-return map. Our model emerges as the local map for other piecewise isometries and can be the basis for the construction of more complicated molecular attractors.
1998, 4(4): 609-634
doi: 10.3934/dcds.1998.4.609
+[Abstract](1169)
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Abstract:
We consider the two-dimensional Riemann problem for the pressure-gradient equations with four pieces of initial data, so restricted that only one elementary wave appears at each interface. This model comes from the flux-splitting of the compressible Euler system. Lack of the velocity in the eigenvalues, the slip lines have little influence on the structures of solutions. The flow exhibits the simpler patterns than in the Euler system, which makes it possible to clarify the interaction of waves in two dimensions. The present paper is devoted to analyzing the structures of solutions and presenting numerical results to the two-dimensional Riemann problem. Especially, we give the criterion of transition from the regular reflection to the Mach reflection in the interaction of shocks.
We consider the two-dimensional Riemann problem for the pressure-gradient equations with four pieces of initial data, so restricted that only one elementary wave appears at each interface. This model comes from the flux-splitting of the compressible Euler system. Lack of the velocity in the eigenvalues, the slip lines have little influence on the structures of solutions. The flow exhibits the simpler patterns than in the Euler system, which makes it possible to clarify the interaction of waves in two dimensions. The present paper is devoted to analyzing the structures of solutions and presenting numerical results to the two-dimensional Riemann problem. Especially, we give the criterion of transition from the regular reflection to the Mach reflection in the interaction of shocks.
1998, 4(4): 635-640
doi: 10.3934/dcds.1998.4.635
+[Abstract](1034)
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Abstract:
We show the existence and fully characterize a class of dissipative perturbations of the linear wave equation for which the spectrum of the associated linear operator has Hamiltonian symmetry and the energy of finite-energy solutions neither decays to zero nor grows to infinity, but instead oscillates and remains bounded for all time.
We show the existence and fully characterize a class of dissipative perturbations of the linear wave equation for which the spectrum of the associated linear operator has Hamiltonian symmetry and the energy of finite-energy solutions neither decays to zero nor grows to infinity, but instead oscillates and remains bounded for all time.
1998, 4(4): 641-652
doi: 10.3934/dcds.1998.4.641
+[Abstract](1044)
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Abstract:
We study the convergence (as $h \rightarrow \infty$) of solutions of control problems $(CP_h)$ governed by inclusions $A_h(y) \in B_h(u)$, where the sequence of abstract operators $A_h$ is $G$-convergent. We are especially interested in finding an explicit form of the limit problem for $(CP_h)$. This is done by means of the theory of $\Gamma$-convergence.
We study the convergence (as $h \rightarrow \infty$) of solutions of control problems $(CP_h)$ governed by inclusions $A_h(y) \in B_h(u)$, where the sequence of abstract operators $A_h$ is $G$-convergent. We are especially interested in finding an explicit form of the limit problem for $(CP_h)$. This is done by means of the theory of $\Gamma$-convergence.
1998, 4(4): 653-670
doi: 10.3934/dcds.1998.4.653
+[Abstract](1363)
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Abstract:
In this paper, we consider a path following algorithm for solving infinite quadratic programming problems. The convergence properties of a smoothly parametrized curve, known as the central trajectory, is studied. We show that the points of this curve converge to the optimal solution of the problem, so by approximating this curve, solutions arbitrarily close to the optimal solution can be calculated. As an example, we consider the linear-quadratic optimal control problem with state inequality constraints at every time instant.
In this paper, we consider a path following algorithm for solving infinite quadratic programming problems. The convergence properties of a smoothly parametrized curve, known as the central trajectory, is studied. We show that the points of this curve converge to the optimal solution of the problem, so by approximating this curve, solutions arbitrarily close to the optimal solution can be calculated. As an example, we consider the linear-quadratic optimal control problem with state inequality constraints at every time instant.
1998, 4(4): 671-690
doi: 10.3934/dcds.1998.4.671
+[Abstract](1092)
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Abstract:
In this paper, we are concerned with Crank-Nicolson like schemes for:
In this paper, we are concerned with Crank-Nicolson like schemes for:
$ (NLW_\omega ) \frac{1}{\omega^2} \partial_t^2 E_\omega -i\partial_t E_\omega -\D E_\omega =\lambda | E_\omega |^{2\sigma} E_\omega. $
We present two schemes for which we give some convergence results. On of the scheme is dissipative and we describe precisely the dissipation. We prove that the solution of the second scheme fits that of $(NLW_\omega )$ while the first one compute a average value of the solution.
1998, 4(4): 691-708
doi: 10.3934/dcds.1998.4.691
+[Abstract](949)
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Abstract:
Given a solution of a symmetric reaction-diffusion system of the form $\frac d{ dt} u_k = \lambda \frac{d^2}{dx^2} u_k + u_k \hat{g} (t,x,u_1,u_2,\frac d{dx}u_1,\frac d{dx}u_2, \frac{d^2}{ dx^2}u_1,\frac{d^2}{ dx^2}u_2,\sqrt{u_1^2+u_2^2} )$, $k=1,2$, with Dirichlet boundary conditions on the interval $(0,1)$, we introduce a non-negative number called torsion number which vanishes iff the solution is planar, where we call the solution $(u_1,u_2)$ planar, if the curves $\gamma_t: x\mapsto (u_1(t,x),u_2(t,x))\in\mathbb{R}^2$, for $x \in [0,1]$ and $t>0$, are contained in a space $\{\xi(\cos\a,\sin\a):\xi\in\mathbb{R}\}\subset\mathbb{R}^2$, for some $\a\in [0,2\pi)$ and all $t>0$. Loosely speaking, the torsion number measures the torsion of the curve $x\mapsto (x,u_1(t,x),u_2(t,x))\in\mathbb{R}^3$, for $x \in [0,1]$.
We introduce a function called angle function $\varphi(t,x)$ which is a continuous and coincides with the polar angle $\arctan $ $u_2(t,x)$/$u_1(t,x))$ wherever $(u_1(t,x), u_2(t,x))\ne (0,0)$. Then the torsion number is given by the difference between the supremum and the infimum of $\varphi(t,\cdot)$. Under certain conditions, which are, in particular, satisfied if the underlying solution is stationary, we show that this torsion number is either strictly decreasing in time or it vanishes identically. Torsion numbers are designed to play a role in the investigation of reaction-diffusion systems. Their role is comparable to the role of oscillation numbers which are a useful tool for the examination of solutions of one single reaction-diffusion equation.
Given a solution of a symmetric reaction-diffusion system of the form $\frac d{ dt} u_k = \lambda \frac{d^2}{dx^2} u_k + u_k \hat{g} (t,x,u_1,u_2,\frac d{dx}u_1,\frac d{dx}u_2, \frac{d^2}{ dx^2}u_1,\frac{d^2}{ dx^2}u_2,\sqrt{u_1^2+u_2^2} )$, $k=1,2$, with Dirichlet boundary conditions on the interval $(0,1)$, we introduce a non-negative number called torsion number which vanishes iff the solution is planar, where we call the solution $(u_1,u_2)$ planar, if the curves $\gamma_t: x\mapsto (u_1(t,x),u_2(t,x))\in\mathbb{R}^2$, for $x \in [0,1]$ and $t>0$, are contained in a space $\{\xi(\cos\a,\sin\a):\xi\in\mathbb{R}\}\subset\mathbb{R}^2$, for some $\a\in [0,2\pi)$ and all $t>0$. Loosely speaking, the torsion number measures the torsion of the curve $x\mapsto (x,u_1(t,x),u_2(t,x))\in\mathbb{R}^3$, for $x \in [0,1]$.
We introduce a function called angle function $\varphi(t,x)$ which is a continuous and coincides with the polar angle $\arctan $ $u_2(t,x)$/$u_1(t,x))$ wherever $(u_1(t,x), u_2(t,x))\ne (0,0)$. Then the torsion number is given by the difference between the supremum and the infimum of $\varphi(t,\cdot)$. Under certain conditions, which are, in particular, satisfied if the underlying solution is stationary, we show that this torsion number is either strictly decreasing in time or it vanishes identically. Torsion numbers are designed to play a role in the investigation of reaction-diffusion systems. Their role is comparable to the role of oscillation numbers which are a useful tool for the examination of solutions of one single reaction-diffusion equation.
1998, 4(4): 709-720
doi: 10.3934/dcds.1998.4.709
+[Abstract](1136)
+[PDF](210.9KB)
Abstract:
Given a connected compact $C^\infty$ manifold M of dimension $n\ge2$ without boundary, a Riemannian metric $g$ on M and an eigenvalue $\lambda^*(M,g)$ of multiplicity $\nu\ge2$ of the Laplace-Beltrami operator $\Delta_g,$ we provide a sufficient condition such that the set of the deformations of the metric $g,$ which preserve the multiplicity of the eigenvalue, is locally a manifold of codimension $1/2 \nu(\nu+1)-1$ in the space of $C^k$ symmetric covariant 2-tensors on M. Furthermore we prove that such a condition is fulfilled when $n=2$ and $\nu=2.$
Given a connected compact $C^\infty$ manifold M of dimension $n\ge2$ without boundary, a Riemannian metric $g$ on M and an eigenvalue $\lambda^*(M,g)$ of multiplicity $\nu\ge2$ of the Laplace-Beltrami operator $\Delta_g,$ we provide a sufficient condition such that the set of the deformations of the metric $g,$ which preserve the multiplicity of the eigenvalue, is locally a manifold of codimension $1/2 \nu(\nu+1)-1$ in the space of $C^k$ symmetric covariant 2-tensors on M. Furthermore we prove that such a condition is fulfilled when $n=2$ and $\nu=2.$
1998, 4(4): 721-734
doi: 10.3934/dcds.1998.4.721
+[Abstract](1205)
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Abstract:
We consider a clamped Rayleigh beam subject to a positive viscous damping. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the underlying system. We next identify the optimal energy decay rate of the system with the supremum of the real part of the spectrum of the infinitesimal generator of the associated semigroup.
We consider a clamped Rayleigh beam subject to a positive viscous damping. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the underlying system. We next identify the optimal energy decay rate of the system with the supremum of the real part of the spectrum of the infinitesimal generator of the associated semigroup.
1998, 4(4): 735-764
doi: 10.3934/dcds.1998.4.735
+[Abstract](1385)
+[PDF](299.1KB)
Abstract:
If $B$ is the generator of an increasing locally Lipschitz continuous integrated semigroup on an abstract L space $X$ and $C: D(B) \to X$ perturbs $B$ positively, then $A = B + C$ is again the generator of an increasing l.L.c. integrated semigroup. In this paper we study the growth bound and the compactness properties of the $C_0$ semigroup $S_\circ$ that is generated by the part of $A$ in $X_\circ = \overline {D(B)}$. We derive conditions in terms of the resolvent outputs $ F(\lambda) = C (\lambda - B)^{-1} $ for the semigroup $S_\circ$ to be eventually compact or essentially compact and to exhibit asynchronous exponential growth. We apply our results to age-structured population models with additional structures. We consider an age-structured model with spatial diffusion and an age-size-structured model.
If $B$ is the generator of an increasing locally Lipschitz continuous integrated semigroup on an abstract L space $X$ and $C: D(B) \to X$ perturbs $B$ positively, then $A = B + C$ is again the generator of an increasing l.L.c. integrated semigroup. In this paper we study the growth bound and the compactness properties of the $C_0$ semigroup $S_\circ$ that is generated by the part of $A$ in $X_\circ = \overline {D(B)}$. We derive conditions in terms of the resolvent outputs $ F(\lambda) = C (\lambda - B)^{-1} $ for the semigroup $S_\circ$ to be eventually compact or essentially compact and to exhibit asynchronous exponential growth. We apply our results to age-structured population models with additional structures. We consider an age-structured model with spatial diffusion and an age-size-structured model.
1998, 4(4): 765-782
doi: 10.3934/dcds.1998.4.765
+[Abstract](1052)
+[PDF](268.9KB)
Abstract:
In dimension two, there are no paths from an Anosov diffeomorphism reaching the boundary of the stability components while attaining a first quadratic tangency associated to a periodic point. Therefore we analyse the possibility to construct an arc ending with a first cubic homoclinic tangency. For several reasons that will be explained in the sequel, we will restrict to area preserving diffeomorphisms.
In dimension two, there are no paths from an Anosov diffeomorphism reaching the boundary of the stability components while attaining a first quadratic tangency associated to a periodic point. Therefore we analyse the possibility to construct an arc ending with a first cubic homoclinic tangency. For several reasons that will be explained in the sequel, we will restrict to area preserving diffeomorphisms.
1998, 4(4): 783-793
doi: 10.3934/dcds.1998.4.783
+[Abstract](1183)
+[PDF](184.3KB)
Abstract:
In this paper, the long time behavior of solution for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities is considered. First the global weak attractor for this equation in $E_1$ is constructed. And then by exact analysis of energy equations, it is showed that the global weak attractor is actually the global strong attractor in $E_1$.
In this paper, the long time behavior of solution for the dissipative Hamiltonian amplitude equation governing modulated wave instabilities is considered. First the global weak attractor for this equation in $E_1$ is constructed. And then by exact analysis of energy equations, it is showed that the global weak attractor is actually the global strong attractor in $E_1$.
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