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1078-0947
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Discrete & Continuous Dynamical Systems - A
August 2007 , Volume 17 , Issue 3
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2007, 17(3): 449-480
doi: 10.3934/dcds.2007.17.449
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Abstract:
In this paper we prove the existence of a solution in L∞loc$(\Omega)$ to the Euler-Lagrange equation for the variational problem
In this paper we prove the existence of a solution in L∞loc$(\Omega)$ to the Euler-Lagrange equation for the variational problem
inf$\bar u + W^{1,\infty}_0(\Omega)$$ \int_{\Omega} (\I_D(\nabla u) + g(u)) dx,\ \ \ $ (0.1)
with $D$ convex closed subset of $\R^n$ with non empty interior. We next show that if D* is strictly convex, then the Euler-Lagrange equation can be reduced to an ODE along characteristics, and we deduce that the solution to Euler-Lagrange is different from $0$ a.e. and satisfies a uniqueness property. Using these results, we prove a conjecture on the existence of variations on vector fields [6].
2007, 17(3): 481-500
doi: 10.3934/dcds.2007.17.481
+[Abstract](1515)
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Abstract:
We study the relations between the global dynamics of the 3D Leray-$\alpha $ model and the 3D Navier-Stokes system. We prove that time shifts of bounded sets of solutions of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and $ \alpha $ approaches zero. In particular, we show that the trajectory attractor of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system when $\alpha \rightarrow 0\+.$
We study the relations between the global dynamics of the 3D Leray-$\alpha $ model and the 3D Navier-Stokes system. We prove that time shifts of bounded sets of solutions of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and $ \alpha $ approaches zero. In particular, we show that the trajectory attractor of the Leray-$\alpha $ model converges to the trajectory attractor of the 3D Navier-Stokes system when $\alpha \rightarrow 0\+.$
2007, 17(3): 501-507
doi: 10.3934/dcds.2007.17.501
+[Abstract](1275)
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Abstract:
We give an example where for an open set of Lagrangians on the n-torus there is at least one cohomology class c with at least n different ergodic c-minimizing measures. One of the problems posed by Ricardo Mañé in his paper 'Generic properties and problems of minimizing measures of Lagrangian systems' (Nonlinearity, 1996) was the following:
Is it true that for generic Lagrangians every minimizing measure is uniquely ergodic?
A weaker statement is that for generic Lagrangians every cohomology class has exactly one minimizing measure, which of course will be ergodic. Our example shows that this can't be true and as a consequence one can hope to prove at most that for a generic Lagrangian, for every cohomology class there are at most n corresponding ergodic minimizing measures, where n is the dimension of the first cohomology group.
We give an example where for an open set of Lagrangians on the n-torus there is at least one cohomology class c with at least n different ergodic c-minimizing measures. One of the problems posed by Ricardo Mañé in his paper 'Generic properties and problems of minimizing measures of Lagrangian systems' (Nonlinearity, 1996) was the following:
Is it true that for generic Lagrangians every minimizing measure is uniquely ergodic?
A weaker statement is that for generic Lagrangians every cohomology class has exactly one minimizing measure, which of course will be ergodic. Our example shows that this can't be true and as a consequence one can hope to prove at most that for a generic Lagrangian, for every cohomology class there are at most n corresponding ergodic minimizing measures, where n is the dimension of the first cohomology group.
2007, 17(3): 509-528
doi: 10.3934/dcds.2007.17.509
+[Abstract](1281)
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Abstract:
In this paper, we study various dissipative mechanics associated with the Boussinesq systems which model two-dimensional small amplitude long wavelength water waves. We will show that the decay rate for the damped one-directional model equations, such as the KdV and BBM equations, holds for some of the damped Boussinesq systems.
In this paper, we study various dissipative mechanics associated with the Boussinesq systems which model two-dimensional small amplitude long wavelength water waves. We will show that the decay rate for the damped one-directional model equations, such as the KdV and BBM equations, holds for some of the damped Boussinesq systems.
2007, 17(3): 529-540
doi: 10.3934/dcds.2007.17.529
+[Abstract](1521)
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Abstract:
We consider the Lorenz system $\dot x = \s (y-x)$, $\dot y =rx -y-xz$ and $\dot z = -bz + xy$; and the Rössler system $\dot x = -(y+z)$, $\dot y = x +ay$ and $\dot z = b-cz + xz$. Here, we study the Hopf bifurcation which takes place at $q_{\pm}=(\pm\sqrt{br-b},\pm\sqrt{br-b},r-1),$ in the Lorenz case, and at $s_{\pm}=(\frac{c+\sqrt{c^2-4ab}}{2},-\frac{c+\sqrt{c^2-4ab}}{2a}, \frac{c\pm\sqrt{c^2-4ab}}{2a})$ in the Rössler case. As usual this Hopf bifurcation is in the sense that an one -parameter family in ε of limit cycles bifurcates from the singular point when ε=0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two $2$-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other $3$- or $n$-dimensional differential systems.
We consider the Lorenz system $\dot x = \s (y-x)$, $\dot y =rx -y-xz$ and $\dot z = -bz + xy$; and the Rössler system $\dot x = -(y+z)$, $\dot y = x +ay$ and $\dot z = b-cz + xz$. Here, we study the Hopf bifurcation which takes place at $q_{\pm}=(\pm\sqrt{br-b},\pm\sqrt{br-b},r-1),$ in the Lorenz case, and at $s_{\pm}=(\frac{c+\sqrt{c^2-4ab}}{2},-\frac{c+\sqrt{c^2-4ab}}{2a}, \frac{c\pm\sqrt{c^2-4ab}}{2a})$ in the Rössler case. As usual this Hopf bifurcation is in the sense that an one -parameter family in ε of limit cycles bifurcates from the singular point when ε=0. Moreover, we can determine the kind of stability of these limit cycles. In fact, for both systems we can prove that all the bifurcated limit cycles in a neighborhood of the singular point are either a local attractor, or a local repeller, or they have two invariant manifolds, one stable and the other unstable, which locally are formed by two $2$-dimensional cylinders. These results are proved using averaging theory. The method of studying the Hopf bifurcation using the averaging theory is relatively general and can be applied to other $3$- or $n$-dimensional differential systems.
2007, 17(3): 541-552
doi: 10.3934/dcds.2007.17.541
+[Abstract](1073)
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Abstract:
In this paper we consider some systems of ordinary differential equations which are related to coagulation-fragmentation processes. In particular, we obtain explicit solutions $\{c_k(t)\}$ of such systems which involve certain coefficients obtained by solving a suitable algebraic recurrence relation. The coefficients are derived in two relevant cases: the high-functionality limit and the Flory-Stockmayer model. The solutions thus obtained are polydisperse (that is, $c_k(0)$ is different from zero for all $k \ge 1$) and may exhibit monotonically increasing or decreasing total mass. We also solve a monodisperse case (where $c_1(0)$ is different from zero but $c_k(0)$ is equal to zero for all $k \ge 2$) in the high-functionality limit. In contrast to the previous result, the corresponding solution is now shown to display a sol-gel transition when the total initial mass is larger than one, but not when such mass is less than or equal to one.
In this paper we consider some systems of ordinary differential equations which are related to coagulation-fragmentation processes. In particular, we obtain explicit solutions $\{c_k(t)\}$ of such systems which involve certain coefficients obtained by solving a suitable algebraic recurrence relation. The coefficients are derived in two relevant cases: the high-functionality limit and the Flory-Stockmayer model. The solutions thus obtained are polydisperse (that is, $c_k(0)$ is different from zero for all $k \ge 1$) and may exhibit monotonically increasing or decreasing total mass. We also solve a monodisperse case (where $c_1(0)$ is different from zero but $c_k(0)$ is equal to zero for all $k \ge 2$) in the high-functionality limit. In contrast to the previous result, the corresponding solution is now shown to display a sol-gel transition when the total initial mass is larger than one, but not when such mass is less than or equal to one.
2007, 17(3): 553-560
doi: 10.3934/dcds.2007.17.553
+[Abstract](1078)
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Abstract:
The sectional-hyperbolic sets constitute a class of partially hyperbolic sets introduced in [20] to describe robustly transitive singular dynamics on $n$-manifolds (e.g. the multidimensional Lorenz attractor [9]). Here we prove that a transitive sectional-hyperbolic set with singularities contains no local strong stable manifold through any of its points. Hence a transitive, isolated, sectional-hyperbolic set containing a local strong stable manifold is a hyperbolic saddle-type repeller. In addition, a proper transitive sectional-hyperbolic set on a compact $n$-manifold has empty interior and topological dimension $\leq n-1$. It follows that a singular-hyperbolic attractor with singularities [22] on a compact $3$-manifold has topological dimension $2$. Hence such an attractor is expanding, i.e., its topological dimension coincides with the dimension of its central subbundle. These results apply to the robustly transitive sets considered in [22], [17] and also to the Lorenz attractor in the Lorenz equation [25].
The sectional-hyperbolic sets constitute a class of partially hyperbolic sets introduced in [20] to describe robustly transitive singular dynamics on $n$-manifolds (e.g. the multidimensional Lorenz attractor [9]). Here we prove that a transitive sectional-hyperbolic set with singularities contains no local strong stable manifold through any of its points. Hence a transitive, isolated, sectional-hyperbolic set containing a local strong stable manifold is a hyperbolic saddle-type repeller. In addition, a proper transitive sectional-hyperbolic set on a compact $n$-manifold has empty interior and topological dimension $\leq n-1$. It follows that a singular-hyperbolic attractor with singularities [22] on a compact $3$-manifold has topological dimension $2$. Hence such an attractor is expanding, i.e., its topological dimension coincides with the dimension of its central subbundle. These results apply to the robustly transitive sets considered in [22], [17] and also to the Lorenz attractor in the Lorenz equation [25].
2007, 17(3): 561-587
doi: 10.3934/dcds.2007.17.561
+[Abstract](1151)
+[PDF](316.7KB)
Abstract:
We study chain transitivity and Morse decompositions of discrete and continuous-time semiflows on fiber bundles with emphasis on (generalized) flag bundles. In this case an algebraic description of the chain transitive sets is given. Our approach consists in embedding the semiflow in a semigroup of continuous maps to take advantage of the good properties of the semigroup actions on the flag manifolds.
We study chain transitivity and Morse decompositions of discrete and continuous-time semiflows on fiber bundles with emphasis on (generalized) flag bundles. In this case an algebraic description of the chain transitive sets is given. Our approach consists in embedding the semiflow in a semigroup of continuous maps to take advantage of the good properties of the semigroup actions on the flag manifolds.
2007, 17(3): 589-627
doi: 10.3934/dcds.2007.17.589
+[Abstract](892)
+[PDF](507.3KB)
Abstract:
We present a model illustrating heterodimensional cycles (i.e., cycles associated to saddles having different indices) as a mechanism leading to the collision of hyperbolic homoclinic classes (of points of different indices) and thereafter to the persistence of (heterodimensional) cycles. The collisions are associated to secondary (saddle-node) bifurcations appearing in the unfolding of the initial cycle.
We present a model illustrating heterodimensional cycles (i.e., cycles associated to saddles having different indices) as a mechanism leading to the collision of hyperbolic homoclinic classes (of points of different indices) and thereafter to the persistence of (heterodimensional) cycles. The collisions are associated to secondary (saddle-node) bifurcations appearing in the unfolding of the initial cycle.
2007, 17(3): 629-652
doi: 10.3934/dcds.2007.17.629
+[Abstract](1206)
+[PDF](629.6KB)
Abstract:
We present a topological method of obtaining the existence of infinite number of symmetric periodic orbits for systems with reversing symmetry. The method is based on covering relations. We apply the method to a four-dimensional reversible map.
We present a topological method of obtaining the existence of infinite number of symmetric periodic orbits for systems with reversing symmetry. The method is based on covering relations. We apply the method to a four-dimensional reversible map.
2007, 17(3): 653-670
doi: 10.3934/dcds.2007.17.653
+[Abstract](1069)
+[PDF](242.9KB)
Abstract:
Properties of the sequence ind$(\infty,\id-f^{m}),$ $m=1,2,\ldots$ where $f^{m}$ is the $m$-th iterate of the mapping $ f:R^{d}\to R^{d}$, and ind denotes the Kronecker index are investigated. The case when $f$ has the asymptotic derivative $A$ at infinity and some eigenvalues of $A$ are roots of unity is of primary interest.
Properties of the sequence ind$(\infty,\id-f^{m}),$ $m=1,2,\ldots$ where $f^{m}$ is the $m$-th iterate of the mapping $ f:R^{d}\to R^{d}$, and ind denotes the Kronecker index are investigated. The case when $f$ has the asymptotic derivative $A$ at infinity and some eigenvalues of $A$ are roots of unity is of primary interest.
2007, 17(3): 671-689
doi: 10.3934/dcds.2007.17.671
+[Abstract](1198)
+[PDF](567.6KB)
Abstract:
Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almost-invariant sets, however, often little can be said rigorously about such calculations. We attempt to explain some of the numerically observed behaviour by constructing a hyperbolic map with a Perron-Frobenius operator whose eigendecomposition is representative of numerical calculations for hyperbolic systems. We explicitly construct an eigenfunction associated with an isolated eigenvalue and prove that a special form of Ulam's method well approximates the isolated spectrum and eigenfunctions of this map.
Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almost-invariant sets, however, often little can be said rigorously about such calculations. We attempt to explain some of the numerically observed behaviour by constructing a hyperbolic map with a Perron-Frobenius operator whose eigendecomposition is representative of numerical calculations for hyperbolic systems. We explicitly construct an eigenfunction associated with an isolated eigenvalue and prove that a special form of Ulam's method well approximates the isolated spectrum and eigenfunctions of this map.
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