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1078-0947
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1553-5231
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Discrete and Continuous Dynamical Systems
April 2000 , Volume 6 , Issue 2
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2000, 6(2): 255-274
doi: 10.3934/dcds.2000.6.255
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Abstract:
We improve Mather's proof on the existence of the connecting orbit around rotation number zero (Proposition 8.1 in [7]) in this paper. Our new proof not only assures the existences of the connecting orbit, but also gives a quantitative estimation on the diffusion time.
We improve Mather's proof on the existence of the connecting orbit around rotation number zero (Proposition 8.1 in [7]) in this paper. Our new proof not only assures the existences of the connecting orbit, but also gives a quantitative estimation on the diffusion time.
2000, 6(2): 275-292
doi: 10.3934/dcds.2000.6.275
+[Abstract](2173)
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Abstract:
In this paper, we will study a simple, piecewise linear model which mimics the transformation in a chaotic electrical circuit: R-L-Diode driven by a sinusoïdal voltage source. This map leads to a complicated chaotic structure, with infinitly many distinct, prime homoclinic points. We prove here that there are infinitly many distinct homoclinic points. Their dynamical classification is not completely understood. They are derived through a nonlinear ($-+$) map, built with piecewise linear pieces. To different two sequences, should correspond two distinct prime homoclinic points. We have derived, we believe, the basic phenomena leading to the complicated dynamics.
In this paper, we will study a simple, piecewise linear model which mimics the transformation in a chaotic electrical circuit: R-L-Diode driven by a sinusoïdal voltage source. This map leads to a complicated chaotic structure, with infinitly many distinct, prime homoclinic points. We prove here that there are infinitly many distinct homoclinic points. Their dynamical classification is not completely understood. They are derived through a nonlinear ($-+$) map, built with piecewise linear pieces. To different two sequences, should correspond two distinct prime homoclinic points. We have derived, we believe, the basic phenomena leading to the complicated dynamics.
2000, 6(2): 293-297
doi: 10.3934/dcds.2000.6.293
+[Abstract](2324)
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Abstract:
The equation of motion of a magnetically stabilized satellite in the plane of a circular polar orbit is studied through qualitative methods. Sufficient uniqueness conditions and bilateral bounds for odd periodic solutions are found. A solution with the largest amplitude is indicated and a criterion for its stability is obtained.
The equation of motion of a magnetically stabilized satellite in the plane of a circular polar orbit is studied through qualitative methods. Sufficient uniqueness conditions and bilateral bounds for odd periodic solutions are found. A solution with the largest amplitude is indicated and a criterion for its stability is obtained.
2000, 6(2): 299-304
doi: 10.3934/dcds.2000.6.299
+[Abstract](2714)
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Abstract:
After introducing a coding definition for the entropy of a piecewise monotone transformation with discontinuities, an algorithm is presented by which the entropy can be determined by the symbol sequences of the separation points.
After introducing a coding definition for the entropy of a piecewise monotone transformation with discontinuities, an algorithm is presented by which the entropy can be determined by the symbol sequences of the separation points.
2000, 6(2): 305-314
doi: 10.3934/dcds.2000.6.305
+[Abstract](2851)
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Abstract:
We study the finite speed of propagation of the Cauchy-Dirichlet problem for the porous media equation with absorption or convection terms in the strip $\mathfrak R_k^N\times (0,T)$, where $\mathfrak R_k^N=\mathfrak R^N\cap\{x_1, ..., x_k>0\}$, $1\leq k\leq N $ and we find new upper bounds of the free boundary.
Finally, we consider the case of higher order parabolic equations.
We study the finite speed of propagation of the Cauchy-Dirichlet problem for the porous media equation with absorption or convection terms in the strip $\mathfrak R_k^N\times (0,T)$, where $\mathfrak R_k^N=\mathfrak R^N\cap\{x_1, ..., x_k>0\}$, $1\leq k\leq N $ and we find new upper bounds of the free boundary.
Finally, we consider the case of higher order parabolic equations.
2000, 6(2): 315-328
doi: 10.3934/dcds.2000.6.315
+[Abstract](3132)
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Abstract:
This paper deals with existence and regularity of positive solutions of sublinear equations of the form $-\Delta u + b(x)u =\lambda f(u)$ in $\Omega$ where either $\Omega\in R^N$ is a bounded smooth domain in which case we consider the Dirichlet problem or $\Omega =R^N$, where we look for positive solutions, $b$ is not necessarily coercive or continuous and $f$ is a real function with sublinear growth which may have certain discontinuities. We explore the method of lower and upper solutions associated with some subdifferential calculus.
This paper deals with existence and regularity of positive solutions of sublinear equations of the form $-\Delta u + b(x)u =\lambda f(u)$ in $\Omega$ where either $\Omega\in R^N$ is a bounded smooth domain in which case we consider the Dirichlet problem or $\Omega =R^N$, where we look for positive solutions, $b$ is not necessarily coercive or continuous and $f$ is a real function with sublinear growth which may have certain discontinuities. We explore the method of lower and upper solutions associated with some subdifferential calculus.
2000, 6(2): 329-350
doi: 10.3934/dcds.2000.6.329
+[Abstract](3136)
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Abstract:
We consider the notion of shift tangent vector introduced in [7] for real valued BV functions and introduced in [9] for vector valued BV functions. These tangent vectors act on a function $u\in L^1$ shifting horizontally the points of its graph at different rates, generating in such a way a continuous path in $L^1$. The main result of [7] is that if the semigroup $\mathcal S$ generated by a scalar strictly convex conservation law is shift differentiable, i.e. paths generated by shift tangent vectors at $u_0$ are mapped in paths generated by shift tangent vectors at $\mathcal S_t u_0$ for almost every $t\geq 0$. This leads to the introduction of a sort of differential, the "shift differential", of the map $u_0 \to \mathcal S_t u_0$.
In this paper, using a simple decomposition of $u\in $BV in terms of its derivative, we extend the results of [9] and we give a unified definition of shift tangent vector, valid both in the scalar and vector case. This extension allows us to study the shift differentiability of the flow generated by a hyperbolic system of conservation laws.
We consider the notion of shift tangent vector introduced in [7] for real valued BV functions and introduced in [9] for vector valued BV functions. These tangent vectors act on a function $u\in L^1$ shifting horizontally the points of its graph at different rates, generating in such a way a continuous path in $L^1$. The main result of [7] is that if the semigroup $\mathcal S$ generated by a scalar strictly convex conservation law is shift differentiable, i.e. paths generated by shift tangent vectors at $u_0$ are mapped in paths generated by shift tangent vectors at $\mathcal S_t u_0$ for almost every $t\geq 0$. This leads to the introduction of a sort of differential, the "shift differential", of the map $u_0 \to \mathcal S_t u_0$.
In this paper, using a simple decomposition of $u\in $BV in terms of its derivative, we extend the results of [9] and we give a unified definition of shift tangent vector, valid both in the scalar and vector case. This extension allows us to study the shift differentiability of the flow generated by a hyperbolic system of conservation laws.
2000, 6(2): 351-360
doi: 10.3934/dcds.2000.6.351
+[Abstract](2551)
+[PDF](201.8KB)
Abstract:
Let $\sigma:\Sigma\to\Sigma$ be a topologically mixing shift of finite type. If $G$ is a group, and $\beta:\Sigma\to G$ is a continuous function, denote by $\sigma_\beta$ the skew-product of $\sigma$ by $\beta$. If $G$ is $\mathbb R^n$, we show examples of continuous multiparameters families of functions $\beta$ for which the skew-products $\sigma_\beta$ are topologically transitive for sets of parameters of full measure. If $G$ is a connected semisimple matrix Lie group, we show examples of functions $\beta$ for which the skew-products $\sigma_beta$ are topologically transitive.
Let $\sigma:\Sigma\to\Sigma$ be a topologically mixing shift of finite type. If $G$ is a group, and $\beta:\Sigma\to G$ is a continuous function, denote by $\sigma_\beta$ the skew-product of $\sigma$ by $\beta$. If $G$ is $\mathbb R^n$, we show examples of continuous multiparameters families of functions $\beta$ for which the skew-products $\sigma_\beta$ are topologically transitive for sets of parameters of full measure. If $G$ is a connected semisimple matrix Lie group, we show examples of functions $\beta$ for which the skew-products $\sigma_beta$ are topologically transitive.
2000, 6(2): 361-380
doi: 10.3934/dcds.2000.6.361
+[Abstract](3406)
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Abstract:
In this paper we study the boundary value problem for the Hamilton-Jacobi-Isaacs equation of pursuit-evasion differential games with state constraints. We prove existence of a continuous viscosity solution and a comparison theorem that we apply to establish uniqueness of such a solution and its uniform approximation by solutions of discretized equations.
In this paper we study the boundary value problem for the Hamilton-Jacobi-Isaacs equation of pursuit-evasion differential games with state constraints. We prove existence of a continuous viscosity solution and a comparison theorem that we apply to establish uniqueness of such a solution and its uniform approximation by solutions of discretized equations.
2000, 6(2): 381-392
doi: 10.3934/dcds.2000.6.381
+[Abstract](3142)
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Abstract:
Let $F:(M,\omega) \mapsto (M,\omega)$ be a smooth symplectic diffeomorphism with a fixed point a and a heteroclinic orbit in the sense of been in the intersection of the central stable and the central unstable manifolds of the fixed point. It is studied the case when the tangent space of a point in the heteroclinic orbit is the direct sum of three subspaces. The first one is the characteristic bundle of the central stable manifold of $\mathbf a$ the second one is the characteristic bundle of the central unstable manifold of $\mathbf a$, and the third one is tangent to the intersection of the central stable and unstable manifolds.
In this situation, the homoclinic map $\Lambda$ is a smooth and symplectic diffeomorphism of open subsets of the central manifold of $\mathbf a$.
Moreover, if an invariant circle intersects the domain of definition of $\Lambda$ and its image intersects other circle, there are orbits that wander from one circle to the other. This phenomenon is similar to the Arnold diffusion.
The Melnikov Method gives sufficient conditions for the existence of homoclinic maps, and non identity homoclinic maps in a perturbation of a Hamiltonian system.
Let $F:(M,\omega) \mapsto (M,\omega)$ be a smooth symplectic diffeomorphism with a fixed point a and a heteroclinic orbit in the sense of been in the intersection of the central stable and the central unstable manifolds of the fixed point. It is studied the case when the tangent space of a point in the heteroclinic orbit is the direct sum of three subspaces. The first one is the characteristic bundle of the central stable manifold of $\mathbf a$ the second one is the characteristic bundle of the central unstable manifold of $\mathbf a$, and the third one is tangent to the intersection of the central stable and unstable manifolds.
In this situation, the homoclinic map $\Lambda$ is a smooth and symplectic diffeomorphism of open subsets of the central manifold of $\mathbf a$.
Moreover, if an invariant circle intersects the domain of definition of $\Lambda$ and its image intersects other circle, there are orbits that wander from one circle to the other. This phenomenon is similar to the Arnold diffusion.
The Melnikov Method gives sufficient conditions for the existence of homoclinic maps, and non identity homoclinic maps in a perturbation of a Hamiltonian system.
2000, 6(2): 393-418
doi: 10.3934/dcds.2000.6.393
+[Abstract](2351)
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Abstract:
In this paper, we establish the existence of inertial sets for a class of wave equations in which the coefficient of the second order time derivative is $\varepsilon$. We show that the fractal dimension of these inertial sets does not depend on $\varepsilon$ for $\varepsilon$ small enough. We then compare the asymptotic behavior of the problem (as $\varepsilon\to 0$) through a continuity like property of the inertial sets. The autonomous case and nonautonomous case are studied.
In this paper, we establish the existence of inertial sets for a class of wave equations in which the coefficient of the second order time derivative is $\varepsilon$. We show that the fractal dimension of these inertial sets does not depend on $\varepsilon$ for $\varepsilon$ small enough. We then compare the asymptotic behavior of the problem (as $\varepsilon\to 0$) through a continuity like property of the inertial sets. The autonomous case and nonautonomous case are studied.
2000, 6(2): 419-430
doi: 10.3934/dcds.2000.6.419
+[Abstract](3407)
+[PDF](695.3KB)
Abstract:
We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. This paper is a continuation of our program (see [CY1,CY2]) in studying the interaction of nonlinear waves in the Riemann problem. The central point in this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. In this paper we focus mainly on the interaction of contact discontinuities, which consists of two genuinely different cases. For each case, the structure of the Riemann solution is analyzed by using the method of characteristics, and the corresponding numerical solution is illustrated via contour plots by using the upwind averaging scheme that is second-order in the smooth region of the solution developed in [CY1]. For one case, the four contact discontinuities role up and generate a vortex, and the density monotonically decreases to zero at the center of the vortex along the stream curves. For the other, two shock waves are formed and, in the subsonic region between two shock waves, a new kind of nonlinear hyperbolic waves (called smoothed Delta-shock waves) is observed.
We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. This paper is a continuation of our program (see [CY1,CY2]) in studying the interaction of nonlinear waves in the Riemann problem. The central point in this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. In this paper we focus mainly on the interaction of contact discontinuities, which consists of two genuinely different cases. For each case, the structure of the Riemann solution is analyzed by using the method of characteristics, and the corresponding numerical solution is illustrated via contour plots by using the upwind averaging scheme that is second-order in the smooth region of the solution developed in [CY1]. For one case, the four contact discontinuities role up and generate a vortex, and the density monotonically decreases to zero at the center of the vortex along the stream curves. For the other, two shock waves are formed and, in the subsonic region between two shock waves, a new kind of nonlinear hyperbolic waves (called smoothed Delta-shock waves) is observed.
2000, 6(2): 431-450
doi: 10.3934/dcds.2000.6.431
+[Abstract](3685)
+[PDF](280.9KB)
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In this paper, optimal control problems for semilinear parabolic equations with distributed and boundary controls are considered. Pointwise constraints on the control and on the state are given. Main emphasis is laid on the discussion of second order sufficient optimality conditions. Sufficiency for local optimality is verified under different assumptions imposed on the dimension of the domain and on the smoothness of the given data.
In this paper, optimal control problems for semilinear parabolic equations with distributed and boundary controls are considered. Pointwise constraints on the control and on the state are given. Main emphasis is laid on the discussion of second order sufficient optimality conditions. Sufficiency for local optimality is verified under different assumptions imposed on the dimension of the domain and on the smoothness of the given data.
2000, 6(2): 451-458
doi: 10.3934/dcds.2000.6.451
+[Abstract](2786)
+[PDF](179.2KB)
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In this paper, by using a trace theorem in the theory of functions of bounded variation, we prove the existence of absolutely continuous invariant measures for a class of piecewise expanding mappings of general bounded domains in any dimension.
In this paper, by using a trace theorem in the theory of functions of bounded variation, we prove the existence of absolutely continuous invariant measures for a class of piecewise expanding mappings of general bounded domains in any dimension.
2000, 6(2): 459-473
doi: 10.3934/dcds.2000.6.459
+[Abstract](2316)
+[PDF](200.7KB)
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We give a definition of bound set for a very general boundary value problem that generalizes those already known in literature. We then find sufficient conditions for the intersection of the sublevelsets of a family of scalar functions to be a bound set for the Floquet boundary value problem. Indeed, we distinguish the two cases of locally Lipschitz continuous and only continuous scalar functions.
We give a definition of bound set for a very general boundary value problem that generalizes those already known in literature. We then find sufficient conditions for the intersection of the sublevelsets of a family of scalar functions to be a bound set for the Floquet boundary value problem. Indeed, we distinguish the two cases of locally Lipschitz continuous and only continuous scalar functions.
2000, 6(2): 475-482
doi: 10.3934/dcds.2000.6.475
+[Abstract](2422)
+[PDF](182.1KB)
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We prove the multiplicity of periodic solutions to second order ordinary differential equations in $\mathbb R^2$ with nonlinearities crossing the two first eigenvalues of the differential operator.
We prove the multiplicity of periodic solutions to second order ordinary differential equations in $\mathbb R^2$ with nonlinearities crossing the two first eigenvalues of the differential operator.
2000, 6(2): 483-499
doi: 10.3934/dcds.2000.6.483
+[Abstract](3185)
+[PDF](243.3KB)
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We study the global well-posedness of the Cauchy problem for the KP II equation. We prove the global well-posedness in the inhomogeneous-homogeneous anisotropic Sobolev spaces $H_{x,y}^{-1/78+\epsilon,0}\cap H_{x,y}^{-17/144,0}$. Though we require the use of the homogeneous Sobolev space of negative index, we obtain the global well-posedness below $L^2$.
We study the global well-posedness of the Cauchy problem for the KP II equation. We prove the global well-posedness in the inhomogeneous-homogeneous anisotropic Sobolev spaces $H_{x,y}^{-1/78+\epsilon,0}\cap H_{x,y}^{-17/144,0}$. Though we require the use of the homogeneous Sobolev space of negative index, we obtain the global well-posedness below $L^2$.
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