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Discrete & Continuous Dynamical Systems - S

April 2013 , Volume 6 , Issue 4

Issue dedicated to Petr P. Zabreiko on the occasion of his 70th birthday

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On mathematical contributions of Petr Petrovich Zabreĭko
Zalman Balanov, I. Gaishun, V. Gorohovik, Wieslaw Krawcewicz and A. Lebedev
2013, 6(4): 837-860 doi: 10.3934/dcdss.2013.6.837 +[Abstract](128) +[PDF](895.9KB)
Chaos in forced impact systems
Flaviano Battelli and Michal Fe?kan
2013, 6(4): 861-890 doi: 10.3934/dcdss.2013.6.861 +[Abstract](146) +[PDF](575.7KB)
We follow a functional analytic approach to study the problem of chaotic behaviour in time-perturbed impact systems whose unperturbed part has a piecewise continuous impact homoclinic solution that transversally enters the discontinuity manifold. We show that if a certain Melnikov function has a simple zero at some point, then the system has impact solutions that behave chaotically. Applications of this result to quasi periodic systems are also given.
Intertwining semiclassical solutions to a Schrödinger-Newton system
Silvia Cingolani, Mnica Clapp and Simone Secchi
2013, 6(4): 891-908 doi: 10.3934/dcdss.2013.6.891 +[Abstract](96) +[PDF](468.0KB)
We study the problem\[\begin{cases}\left( -\varepsilon\mathrm{i}\nabla+A(x)\right) ^{2}u+V(x)u=\varepsilon^{-2}\left( \frac{1}{|x|}\ast|u|^{2}\right) u,\\u\in L^{2}(\mathbb{R}^{3},\mathbb{C}),    \varepsilon\nablau+\mathrm{i}Au\in L^{2}(\mathbb{R}^{3},\mathbb{C}^{3}),\end{cases}\]where $A\colon\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ is an exterior magneticpotential, $V\colon\mathbb{R}^{3}\rightarrow\mathbb{R}$ is an exteriorelectric potential, and $\varepsilon$ is a small positive number. If $A=0$ and$\varepsilon=\hbar$ is Planck's constant this problem is equivalent to theSchr?dinger-Newton equations proposed by Penrose in [23] todescribe his view that quantum state reduction occurs due to somegravitational effect. We assume that $A$ and $V$ are compatible with theaction of a group $G$ of linear isometries of $\mathbb{R}^{3}$. Then, for anygiven homomorphism $\tau:G\rightarrow\mathbb{S}^{1}$ into the unit complexnumbers, we show that there is a combined effect of the symmetries and thepotential $V$ on the number of semiclassical solutions $u:\mathbb{R}^{3}\rightarrow\mathbb{C}$ which satisfy $u(gx)=\tau(g)u(x)$ for all $g\in G$,$x\in\mathbb{R}^{3}$. We also study the concentration behavior of thesesolutions as $\varepsilon 0.$
Fatigue accumulation in an oscillating plate
Michela Eleuteri, Jana Kopfov and Pavel Krej?
2013, 6(4): 909-923 doi: 10.3934/dcdss.2013.6.909 +[Abstract](162) +[PDF](389.0KB)
A thermodynamic model for fatigue accumulation in an oscillating elastoplastic Kirchhoffplate based on the hypothesis that the fatigue accumulation rate is proportional tothe dissipation rate, is derived for the case that both the elastic and the plasticmaterial characteristics change with increasing fatigue. We prove the existence ofa unique solution in the whole time interval before a singularity (material failure) occursunder the simplifying hypothesis that the temperature history is a priori given.
Dynamics of the the dihedral four-body problem
Davide L. Ferrario and Alessandro Portaluri
2013, 6(4): 925-974 doi: 10.3934/dcdss.2013.6.925 +[Abstract](113) +[PDF](1336.0KB)
Consider four point particles with equal masses in the euclideanspace,subject to the following symmetry constraint: at each instant theyare symmetric with respect to the dihedral group $D_2$,that is the groupgenerated by two rotations of angle $\pi$ around twoorthogonal axes.Under ahomogeneous potential of degree $-\alpha$ for $0<\alpha<2$,this is a subproblem of the four-body problem,inwhich all orbits have zero angular momentum and the configurationspace is three-dimensional.In this paper westudy the flow in McGehee coordinates on the collision manifold,anddiscuss the qualitative behavior of orbits which reach or come close to a total collision.
Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators
Carlos Garca-Azpeitia and Jorge Ize
2013, 6(4): 975-983 doi: 10.3934/dcdss.2013.6.975 +[Abstract](124) +[PDF](295.0KB)
This paper gives an analysis of the periodic solutions of a ring of $n$oscillators coupled to their neighbors. We prove the bifurcation of branchesof such solutions from a relative equilibrium, and we study theirsymmetries. We give complete results for a cubic Schr?dinger potential andfor a saturable potential and for intervals of the amplitude of theequilibrium. The tools for the analysis are the orthogonal degree andrepresentation of groups. The bifurcation of relative equilibria was givenin a previous paper.
Equivariant Conley index versus degree for equivariant gradient maps
Anna Go??biewska and S?awomir Rybicki
2013, 6(4): 985-997 doi: 10.3934/dcdss.2013.6.985 +[Abstract](115) +[PDF](220.2KB)
In this article we study the relationship between the degree forinvariant strongly indefinite functionals and the equivariantConley index. We prove that, under certain assumptions, achange of the equivariant Conley indices is equivalent to thechange of the degrees for equivariant gradient maps. Moreover, weformulate easy to verify sufficient conditions for theexistence of a global bifurcation of critical orbits of invariantstrongly indefinite functionals.
Unbounded sequences of cycles in general autonomous equations with periodic nonlinearities
Alexander M. Krasnoselskii
2013, 6(4): 999-1016 doi: 10.3934/dcdss.2013.6.999 +[Abstract](130) +[PDF](458.4KB)
Autonomous higher order differential equations with scalarnonlinearities, periodic with respect to the main phasevariable under appropriate generic conditions, have an infinitesequence of isolated cycles with amplitudes growing to infinityand periods converging to some specific value $T_{0}$.
On a structure of the fixed point set of homogeneous maps
Yakov Krasnov, Alexander Kononovich and Grigory Osharovich
2013, 6(4): 1017-1027 doi: 10.3934/dcdss.2013.6.1017 +[Abstract](134) +[PDF](358.3KB)
A spectral and inverse spectral problem for homogeneous polynomial maps is discussed.The $m$-independence of vectors based on the symmetric tensor powers performs as a main toolto study the structure of the spectrum. Possible restrictions on this structureare described in terms of syzygies provided by the Euler-Jacobi formula.Applications to projective dynamics are discussed.
Pointwise estimates for solutions of singular quasi-linear parabolic equations
Vitali Liskevich and Igor I. Skrypnik
2013, 6(4): 1029-1042 doi: 10.3934/dcdss.2013.6.1029 +[Abstract](111) +[PDF](378.0KB)
For a class of singular divergence type quasi-linear parabolicequations with a Radon measure on the right hand side we derivepointwise estimates for solutions via the nonlinear Wolffpotentials.
Iterative methods for approximating fixed points of Bregman nonexpansive operators
Victoria Martín-Márquez, Simeon Reich and Shoham Sabach
2013, 6(4): 1043-1063 doi: 10.3934/dcdss.2013.6.1043 +[Abstract](84) +[PDF](426.1KB)
Diverse notions of nonexpansive type operators have been extended to the more general framework of Bregman distances in reflexive Banach spaces. We study these classes of operators, mainly with respect to the existence and approximation of their (asymptotic) fixed points. In particular, the asymptotic behavior of Picard and Mann type iterations is discussed for quasi-Bregman nonexpansive operators. We also present parallel algorithms for approximating common fixed points of a finite family of Bregman strongly nonexpansive operators by means of a block operator which preserves the Bregman strong nonexpansivity. All the results hold, in particular, for the smaller class of Bregman firmly nonexpansive operators, a class which contains the generalized resolvents of monotone mappings with respect to the Bregman distance.
Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential
Jean Mawhin
2013, 6(4): 1065-1076 doi: 10.3934/dcdss.2013.6.1065 +[Abstract](139) +[PDF](372.9KB)
T-periodic solutions of systems of difference equations of the form\begin{eqnarray*}\Delta \phi[\Delta q(n-1)] = \nabla_q F[n,q(n)] + h(n) \quad (n \in \mathbb{Z})\end{eqnarray*}where $\phi = \nabla \Phi$, with $\Phi$ strictly convex, is a homeomorphism of $\mathbb{R}^N$ onto the ball $B_a \subset \mathbb{R}^N$, or a homeomorphism of the ball $B_{a} \subset \mathbb{R}^N$ onto $\mathbb{R}^N$, are considered when $F(n,u)$ is periodic in the $u_j$. The approach is variational.
Topology and homoclinic trajectories of discrete dynamical systems
Jacobo Pejsachowicz and Robert Skiba
2013, 6(4): 1077-1094 doi: 10.3934/dcdss.2013.6.1077 +[Abstract](121) +[PDF](442.3KB)
We show that nontrivial homoclinic trajectories ofa family of discrete, nonautonomous, asymptotically hyperbolic systems parametrized by a circlebifurcate from a stationary solution if the asymptotic stable bundles $E^s(+\infty)$ and$E^s(-\infty)$ of the linearization at the stationary branch are twisted in different ways.
Effect of positive feedback on Devil's staircase input-output relationship
Alexei Pokrovskii and Dmitrii Rachinskii
2013, 6(4): 1095-1112 doi: 10.3934/dcdss.2013.6.1095 +[Abstract](115) +[PDF](606.3KB)
We consider emerging hysteresis behaviour in a closed loop systemthat includes a nonlinear link $f$ of the Devil's staircase (Cantorfunction) type and a positive feedback. This type of closed loopsarises naturally in analysis of networks where local ``negative''coupling of network elements is combined with ``positive'' couplingat the level of the mean-field interaction (in the limit case whenthe impact of each individual vertex is infinitesimal, while thenumber of vertices is growing). For the Cantor function $f$, takenas a model, and for a monotonically increasing input, we present thecorresponding output of the system explicitly, showing that theoutput is piecewise constant and has a finite number of equal jumps.We then discuss hysteresis loops of the system for genericnon-monotone inputs. The results are presented in the context of differential equations describingnonlinear control systems with almost immediate linear feedback, i.e., in the limit where the time of propagation of the signalthrough the feedback loop tends to zero.

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