Related papers: An ergodic study of Painleve VI

The third, fifth and sixth Painlev\'e equations are studied by means of the weighted projective spaces ${\mathbb C}P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton polyhedrons of the equations. Singular normal forms…

We consider an Abel polynomial differential equation. For two given points a and b, the "Poincare mapping" of the equation transforms the values of its solution at a into their values at b. In this article, we study global analytic…

Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching behavior. The Poincare return map…

We propose multidimensional versions of the Painlev\'e VI equation and its degenerations. These field theories are related to the isomonodromy problems of flat holomorphic infinite rank bundles over elliptic curves and take the form of…

For a class of flows on polytopes, including many examples from Evolutionary Game Theory, we describe a piecewise linear model which encapsulates the asymptotic dynamics along the heteroclinic network formed out of the polytope's vertexes…

Using continuation methods, we study the global solution structure of periodic solutions for a class of periodically forced equations, generalizing the case of relativistic pendulum. We obtain results on the existence and multiplicity of…

The solutions of the (nonlinear) Painleve VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F4 Weyl group action and many properties of the solutions will be…

We study the ergodic and statistical properties of a class of maps of the circle and of the interval of Lorenz type which present indifferent fixed points and points with unbounded derivative. These maps have been previously investigated in…

The isomonodromic deformations underlying the Painlev\'e transcendants are interpreted as nonautonomous Hamiltonian systems in the dual $\gR^*$ of a loop algebra $\tilde\grg$ in the classical $R$-matrix framework. It is shown how canonical…

We find four kinds of six-parameter family of coupled Painlev\'e VI systems in dimension four with affine Weyl group symmetry of types $B_6^{(1)}$, $D_6^{(1)}$ and $D_7^{(2)}$. Each system is the first example which gave higher-order…

Statistics of Poincar\' e recurrence for a class of circle maps, including sub-critical, critical, and super-critical cases, are studied. It is shown how the topological differences in the various types of the dynamics are manifested in the…

We employ an extension of ergodic theory to the random setting to investigate the existence of random periodic solutions of random dynamical systems. Given that a random dynamical system has a dissipative structure, we proved that a random…

We represent and analyze the general solution of the sixth Painleve transcendent in the Picard-Hitchin-Okamoto class in the Painleve form as the logarithmic derivative of the ratio of certain $\tau$-functions. These functions are…

In this paper we study the asymptotic behavior for large argument of a family of solutions of the Painlev\'e equation P$_{\rm VI} arising in the context of Random Matrix Theory [1]. We show this family of solutions are uniquely determined…

We study the sixth $q$-difference Painlev\'e equation ($q{\textrm{P}_{\textrm{VI}}}$) through its associated Riemann-Hilbert problem (RHP) and show that the RHP is always solvable for irreducible monodromy data. This enables us to identify…

In this article we prove that Lax pairs associated with $\hbar$-dependent six Painlev\'e equations satisfy the topological type property proposed by Berg\`ere, Borot and Eynard for any generic choice of the monodromy parameters.…

The onset of quantum ergodicity is often quantified by the average ratio of consecutive level spacings. The reference values for ergodic quantum systems have been obtained numerically from the spectra of large but finite-dimensional random…

The Gross Pitaevski map is a discrete time, split operator version of the Gross Pitaevski dynamics in the circle, for which exponential instability has been recently reported. Here it is studied as a classical dynamical system in its own…

We consider the inhomogeneous stochastic six vertex model with periodicity starting from step initial data. We prove that it converges almost surely to a deterministic limit shape. For the proof, we map the stochastic six vertex model to a…

We consider isomonodromic deformations of connections with a simple pole on the torus, motivated by the elliptic version of the sixth Painlev\'e equation. We establish an extended symmetry, complementing known results. The Calogero-Moser…