1,406 research outputs found
Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
In this tutorial, we discuss self-excited and hidden attractors for systems
of differential equations. We considered the example of a Lorenz-like system
derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to
demonstrate the analysis of self-excited and hidden attractors and their
characteristics. We applied the fishing principle to demonstrate the existence
of a homoclinic orbit, proved the dissipativity and completeness of the system,
and found absorbing and positively invariant sets. We have shown that this
system has a self-excited attractor and a hidden attractor for certain
parameters. The upper estimates of the Lyapunov dimension of self-excited and
hidden attractors were obtained analytically.Comment: submitted to EP
Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase
This paper study the well--posedness of the entropy formulation given by
Plotnikov in [{Differential Equations}, 30 (1994), pp. 614--622] for
forward-backward parabolic problem obtained as singular limit of a proper
pseudoparabolic approximation. It was proved in [C. Mascia, A. Terracina, and
A. Tesei, {Arch.\ Ration.\ Mech.\ Anal.}, 194 (2009), pp. 887--925] that such
formulation gives uniqueness when the solution takes values in the stable
phases. Here we consider the situation in which unstable phase is taken in
account, proving that, in general, uniqueness does not hold
Some open problems in low dimensional dynamical systems
The aim of this paper is to share with the mathematical community a list of
33 problems that I have found along the years during my research. I believe
that it is worth to think about them and, hopefully, it will be possible either
to solve some of the problems or to make some substantial progress. Many of
them are about planar differential equations but there are also questions about
other mathematical aspects: Abel differential equations, difference equations,
global asymptotic stability, geometrical questions, problems involving
polynomials or some recreational problems with a dynamical component
Survey of decentralized control methods
An overview is presented of the types of problems that are being considered by control theorists in the area of dynamic large scale systems with emphasis on decentralized control strategies. Approaches that deal directly with decentralized decision making for large scale systems are discussed. It is shown that future advances in decentralized system theory are intimately connected with advances in the stochastic control problem with nonclassical information pattern. The basic assumptions and mathematical tools associated with the latter are summarized, and recommendations concerning future research are presented
Action minimizing fronts in general FPU-type chains
We study atomic chains with nonlinear nearest neighbour interactions and
prove the existence of fronts (heteroclinic travelling waves with constant
asymptotic states). Generalizing recent results of Herrmann and Rademacher we
allow for non-convex interaction potentials and find fronts with non-monotone
profile. These fronts minimize an action integral and can only exists if the
asymptotic states fulfil the macroscopic constraints and if the interaction
potential satisfies a geometric graph condition. Finally, we illustrate our
findings by numerical simulations.Comment: 19 pages, several figure
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