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Discrete & Continuous Dynamical Systems - A

2001 , Volume 7 , Issue 3

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A case study in vanishing viscosity
Stefano Bianchini and Alberto Bressan
2001, 7(3): 449-476 doi: 10.3934/dcds.2001.7.449 +[Abstract](448) +[PDF](293.3KB)
Abstract:
We consider a special $2 \times 2$ viscous hyperbolic system of conservation laws of the form $u_t + A(u)u_{x} = \varepsilon u_{x x}$, where $A(u) = Df(u)$ is the Jacobian of a flux function $f$. For initialdata with smalltotalv ariation, we prove that the solutions satisfy a uniform BV bound, independent of $\varepsilon $. Letting $\varepsilon \to 0$, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system $u_t + f(u)_{x} = 0$. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.
Orbit complexity and data compression
Stefano Galatolo
2001, 7(3): 477-486 doi: 10.3934/dcds.2001.7.477 +[Abstract](586) +[PDF](189.1KB)
Abstract:
We consider data compression algorithms as a tool to get an approximate measure for the quantity of information contained in a string. By this it is possible to give a notion of orbit complexity for topological dynamical systems. In compact ergodic dynamical systems, entropy is almost everywhere equal to orbit complexity. The use of compression algorithms allows a direct estimation of the information content of the orbits.
Moment attractivity, stability and contractivity exponents of stochastic dynamical systems
Henri Schurz
2001, 7(3): 487-515 doi: 10.3934/dcds.2001.7.487 +[Abstract](536) +[PDF](279.6KB)
Abstract:
Nonlinear stochastic dynamical systems as ordinary stochastic differential equations and stochastic difference equations are in the center of this presentation in view of the asymptotic behavior of their moments. We study the exponential p-th mean growth behavior of their solutions as integration time tends to infinity. For this purpose, the concepts of attractivity, stability and contractivity exponents for moments are introduced as generalizations of well-known moment Lyapunov exponents of linear systems. Under appropriate monotonicity assumptions we gain uniform estimates of these exponents from above and below. Eventually, these concepts are generalized to describe the exponential growth behavior along certain Lyapunov-type functionals.
Infinite-dimensional complex dynamics: A quantum random walk
Brendan Weickert
2001, 7(3): 517-524 doi: 10.3934/dcds.2001.7.517 +[Abstract](486) +[PDF](181.6KB)
Abstract:
We describe a unitary operator $U(\alpha)$ on L2$(\mathbb T)$, depending on a real parameter $\alpha$, that is a quantization of a simple piecewise holomorphic dynamical system on the cylinder $\mathbf C^* \cong \mathbb T \times \mathbb R$. We give results describing the spectrum of $U(\alpha)$ in terms of the diophantine properties of $\alpha$, and use these results to compare the quantum to classical dynamics. In particular, we prove that for almost all $\alpha$, the quantum dynamics localizes, whereas the classical dynamics does not. We also give a condition implying that the quantum dynamics does not localize.
Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential
Reika Fukuizumi
2001, 7(3): 525-544 doi: 10.3934/dcds.2001.7.525 +[Abstract](624) +[PDF](251.3KB)
Abstract:
In this paper, we study the stability and the instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. We prove the existence of stable or unstable standing waves under certain conditions on the power of nonlinearity and the frequency of wave.
Equations with a $p$-Laplacian and an asymmetric nonlinear term
C. Fabry and Raul Manásevich
2001, 7(3): 545-557 doi: 10.3934/dcds.2001.7.545 +[Abstract](526) +[PDF](222.0KB)
Abstract:
For $p>1, $ and $\phi_p (s) = |s| ^{p-2} s,$ we consider the equation

$(\phi_p (x'))' + \alpha \phi_p (x^+ ) - \beta \phi_p (x^- ) = f(t,x),$

where $ x^{+}=\max\{x,0\}$; $x^{-} =\max\{-x,0\},$ in a situation of resonance or near resonance for the period $T,$ i.e. when $\alpha,\beta$ satisfy exactly or approximately the equation

$\frac{\pi_p }{\alpha^{1/p}} + \frac{\pi_p}{\beta^{1/p}} = \frac{T}{n},$

for some integer $n.$ We assume that $f$ is continuous, locally Lipschitzian in $x,$ $T$-periodic in $t,$ bounded on $\mathbf R^2,$ and having limits $f_{\pm}(t)$ for $x \to \pm \infty,$ the limits being uniform in $t.$ Denoting by $v $ a solution of the homogeneous equation

$(\phi_p (x'))' + \alpha \phi_p (x^+ ) - \beta \phi_p (x^- ) = 0,$

we study the existence of $T$-periodic solutions by means of the function

$ Z (\theta) = \int_{\{t\in I | v_{\theta }(t)>0\}} f_{+}(t)v(t + \theta) dt + \int_{\{t\in I | v_{\theta }(t)<0\}} f_-(t) v (t + \theta) dt,$

where $ I \stackrel{def}{=} [0,T].$ In particular, we prove the existence of $T$-periodic solutions at resonance when $Z$ has $2z$ zeros in the interval $[0,T/n),$ all zeros being simple, and $z$ being different from $1.$

Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity
Junping Shi and Ratnasingham Shivaji
2001, 7(3): 559-571 doi: 10.3934/dcds.2001.7.559 +[Abstract](551) +[PDF](163.8KB)
Abstract:
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Bifurcations of periodic solutions and chaos in Josephson system
Zhujun Jing, K.Y. Chan, Dashun Xu and Hongjun Cao
2001, 7(3): 573-592 doi: 10.3934/dcds.2001.7.573 +[Abstract](524) +[PDF](241.3KB)
Abstract:
The Josephson equation is investigated in detail: the existence and bifurcations for harmonic and subharmonic solutions under small perturbations are obtained by using second-order averaging method and subharmonic Melnikov function, and the criterion of existence for chaos is proved by Melnikov analysis; the bifurcation curves about n-subharmonic and heteroclinic orbits and the driving frequency $\omega$ effects to the forms of chaotic behaviors are given by numerical simulations.
The attractor for a nonlinear hyperbolic equation in the unbounded domain
S.V. Zelik
2001, 7(3): 593-641 doi: 10.3934/dcds.2001.7.593 +[Abstract](644) +[PDF](405.6KB)
Abstract:
We study the long-time behavior of solutions for damped nonlinear hyperbolic equations in the unbounded domains. It is proved that under the natural assumptions these equations possess the locally compact attractors which may have the infinite Hausdorff and fractal dimension. That is why we obtain the upper and lower bounds for the Kolmogorov's entropy of these attractors.
Moreover, we study the particular cases of these equations where the attractors occurred to be finite dimensional. For such particular cases we establish that the attractors consist of finite collections of finite dimensional unstable manifolds and every solution stabilizes to one of the finite number of equilibria points.
Nodal parametrisation of analytic attractors
Peter K. Friz, I. Kukavica and James C. Robinson
2001, 7(3): 643-657 doi: 10.3934/dcds.2001.7.643 +[Abstract](521) +[PDF](251.7KB)
Abstract:
Friz and Robinson showed that analytic global attractors consisting of periodic functions can be parametrised using the values of the solution at a finite number of points throughout the domain, a result applicable to the $2$d Navier-Stokes equations with periodic boundary conditions. In this paper we extend the argument to cover any attractor consisting of analytic functions; in particular we are now able to treat the $2$d Navier-Stokes equations with Dirichlet boundary conditions.

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