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Register a new userInertial Manifolds and Limit Cycles of Dynamical Systems in $mathbb R^n$
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We show that the presence of a two-dimensional inertial manifold for an ordinary differential equation in ${mathbb R}^{n}$ permits reducing the problem of determining asymptotically orbitally stable limit cycles to the Poincare--Bendixson theory. In the case $n=3$ we implement such a scenario for a model of a satellite rotation around a celestial body of small mass and for a biochemical model.
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We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of the equilibrium state of product form, we establish an almost-sure weighted equidistribution of cycles with respect to a natural stationary measure, as the periods of the cycles tend to infinity. This result is an analogue of Bowens theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms in random setup. We also prove averaging results over all samples, as well as another samplewise result. We apply our result to the random $beta$-expansion of real numbers, and obtain a new formula for the mean relative frequencies of digits in the series expansion.
The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauders fixed point theorem.
For the gravitational $n$-body problem, the simplest motions are provided by those rigid motions in which each body moves along a Keplerian orbit and the shape of the system is a constant (up to rotations and scalings) configuration featuring suitable properties. While in dimension $d leq 3$ the configuration must be central, in dimension $d geq 4$ new possibilities arise due to the complexity of the orthogonal group, and indeed there is a wider class of $S$-balanced configurations, containing central ones, which yield simple solutions of the $n$-body problem. Starting from recent results of the first and third authors, we study the existence of continua of bifurcations branching from a trivial branch of collinear $S$-balanced configurations and provide an estimate from below on the number of bifurcation instants. In the last part of the paper, by using the continuation method, we explicitly display the bifurcation branches in the case of the three body problem for different choices of the masses.
We introduce the notions of directional dynamical cubes and directional regionally proximal relation defined via these cubes for a minimal $mathbb{Z}^d$-system $(X,T_1,ldots,T_d)$. We study the structural properties of systems that satisfy the so called unique closing parallelepiped property and we characterize them in several ways. In the distal case, we build the maximal factor of a $mathbb{Z}^d$-system $(X,T_1,ldots,T_d)$ that satisfies this property by taking the quotient with respect to the directional regionally proximal relation. Finally, we completely describe distal $mathbb{Z}^d$-systems that enjoy the unique closing parallelepiped property and provide explicit examples.
We compare different notions of limit sets for the action of Kleinian groups on the $n-$dimensional projective space via the irreducible representation $varrho:PSL(2,mathbb{C})to PSL(n+1,mathbb{C}).$ In particular, we prove that if the Kleinian group is convex-cocompact, the Myrberg and the Kulkarni limit coincide.
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