Multi-parameter planar polynomial dynamical systems

In this paper, we study multi-parameter planar dynamical systems and carry out the global bifurcation analysis of such systems. To control the global bifurcations of limit cycle in these systems, it is necessary to know the properties and combine the effects of all their field rotation parameters. It can be done by means of the development of our bifurcational geometric method based on the Wintner-Perko termination principle and application of canonical
systems with field rotation parameters. Using this method, we solve, e.g., Hilbert's Sixteenth Problem on the maximum number of limit cycles and their distribution for the general Li\'{e}nard
polynomial system and a Holling-type quartic dynamical system. We also conduct some numerical experiments to illustrate the obtained results.

Pre-Calabi-Yau algebras as noncommutative Poisson structures

We show how the double Poisson algebra introduced in \cite{VdB} appear as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $Ac part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structure. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson bracket on representation spaces $({\rm Rep}_n A)^{Gl_n}$ for any associative algebra $A$.

Introduction to Higher Cubical Operads. Second Part: The Functor of Fundamental Cubical Weak $\infty$-Groupoids for Spaces

Some Aspects of Dynamical Topology: Dynamical Compactness and Slovak Spaces

The area of dynamical systems where one investigates
dynamical properties that can be described in topological terms is "Topological Dynamics".
Investigating the topological
properties of spaces and maps that can be described in dynamical
terms is in a sense the opposite idea. This area is recently called as "Dynamical
Topology". For (discrete) dynamical systems given by compact metric spaces and
continuous (surjective) self-maps we (mostly) survey some results on two
new notions: "Slovak Space" and "Dynamical Compactness". Slovak space
is a dynamical analogue of the rigid space: a nontrivial compact
metric space whose homeomorphism group is cyclic and generated by a
minimal homeomorphism.
Dynamical compactness is a new concept of chaotic dynamics. The
omega-limit set of a point is a basic notion in theory of dynamical
systems and means the collection of states which "attract" this point
while going forward in time. It is always nonempty when the phase
space is compact. By changing the time we introduced the notion of the
omega-limit set of a point with respect to a Furstenberg family. A
dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness:
all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property.

From Euler's play with infinite series to the anomalous magnetic moment