Paweł Góra

Invariant densities for generalized β -maps

We find an explicit formula for the invariant density of a generalized β -map. This allows us to also find an explicit formula for the invariant density of Chebyshev map and discuss the monotonicity of the asymptotic average for such maps. Our results are based on a generalization of works of Parry ( Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416; 15 (1964), 95–105).

Family of piecewise expanding maps having singular measure as a limit of ACIMs

Keller [Stochastic stability in some chaotic dynamical systems. Monatsh. Math.94(4) (1982), 313–333] introduced families of W-shaped maps that can have a great variety of behaviors. As a family approaches a limit W map, he observed behavior that was either described by a probability density function (PDF) or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (ACIM) can result only from the existence of small invariant neighborhoods of the fixed critical point of the limit map. In this paper, we show that the conjecture is not true. We construct a very simple family of W-maps with ACIMs supported on the whole interval, whose limiting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [P. Gora. Invariant densities for piecewise linear maps of interval. Ergod. Th. & Dynam. Sys. 29(5) (2009), 1549–1583].

STOCHASTIC PERTURBATIONS OF POSITION DEPENDENT RANDOM MAPS

A random map is a dynamical system consisting of a collection of maps which are selected randomly by means of fixed probabilities at each iteration. In this note, we consider absolutely continuous invariant measures of random maps with position dependent probabilities and prove that they are stable under small stochastic perturbations. This result depends on a new lemma which handles arbitrarily small extra partition elements that may arise from the perturbation of the random map. For perturbations satisfying additional conditions, we give precise estimates of the error in the invariant density.

DERIVING CHAOTIC DYNAMICAL SYSTEMS FROM ENERGY FUNCTIONALS

Simple one-dimensional chaotic dynamical systems are derived by optimizing energy functionals. The Euler–Lagrange equation yields a nonlinear second-order differential equation whose solution yields a 2–1 map which admits an absolutely continuous invariant measure. The solutions of the differential equation are studied.

ON THE SIGNIFICANCE OF THE TENT MAP

Discrete time dynamical systems generated by the iteration of nonlinear maps provide simple and interesting examples of chaotic systems. But what is the physical principle behind the emergence of these maps? In this note we present an approach to this problem by considering a class, Y, of 2–1 chaotic maps on [0, 1] that are symmetric and have symmetric invariant densities. We prove that such maps are conjugate to the tent map. In Y we search for maps that minimizes a functional that depends on y ∈ Y and fy, the probability density function invariant under y. We define a simple functional whose extremal value is achieved by the tent map.

OPTIMAL CONTROL OF CHAOTIC SYSTEMS

The problem of controlling a chaotic system is treated from a long term statistical basis. Unlike the OGY targeting method that exploits individual unstable orbits, this approach is concerned with targeting the density function of an invariant probability measure. Given a point transformation T, possessing a density function f, we choose , a different probability density function, to be the target. Using optimization methods, we construct a point transformation , close to T, whose invariant probability density function is , or close to .

A FRAMEWORK FOR INTERNEURAL DYNAMICS IN A CEREBRAL CORTEX CENTER

We model the innervation dynamics of interneurons in a cerebral cortex center A between the time of initial sensory input and acquisition of a sustained steady state. The model assumes that interneurons in A are heavily interconnected allowing synchronization. This invites modeling the dynamics by means of a discrete time map. The model takes into account the influence of excitatory and inhibitory cells and reflects the architecture of synapses along the axons. The acquisition of a sustained chaotic state is characterized by means of a natural invariant probability measure. The time to attain this probability measure can be estimated.

CHAOS OF DYNAMICAL SYSTEMS ON GENERAL TIME DOMAINS

We consider dynamical systems on time domains that alternate between continuous time intervals and discrete time intervals. The dynamics on the continuous portions may represent species growth when there is population overlap and are governed by differential or partial differential equations. The dynamics across the discrete time intervals are governed by a chaotic map and may represent population growth which is seasonal. We study the long term dynamics of this combined system. We study various conditions on the continuous time dynamics and discrete time dynamics that produce chaos and alternatively nonchaos for the combined system. When the discrete system alone is chaotic we provide a condition on the continuous dynamical component such that the combined system behaves chaotically. We also provide a condition that ensures that if the discrete time system has an absolutely continuous invariant measure so will the combined system. An example based on the logistic continuous time and logistic discrete time component is worked out.

INVARIANT DENSITIES FOR POSITION-DEPENDENT RANDOM MAPS ON THE REAL LINE: EXISTENCE, APPROXIMATION AND ERROR BOUNDS

A random map is a discrete-time dynamical system in which a transformation is randomly selected from a collection of transformations according to a probability function and applied to the process. In this note, we study random maps with position-dependent probabilities on ℝ. This means that the random map under consideration consists of transformations which are piecewise monotonic with countable number of branches from ℝ into itself and a probability function which is position dependent. We prove existence of absolutely continuous invariant probability measures and construct a method for approximating their densities. Explicit quantitative bound on the approximation error is given.

CONDITIONALLY INVARIANT MEASURES FOR POSITION DEPENDENT RANDOM MAPS OF AN INTERVAL WITH HOLES

We study position dependent random maps on the unit interval with holes where the possible laws of motion are piecewise monotonic transformations. The main result of this note is proving the existence of absolutely continuous conditionally invariant measures.