# Registered Data

## [02015] Theory and applications of random/non-autonomous dynamical systems part II

**Session Date & Time**: 1D (Aug.21, 15:30-17:10)**Type**: Proposal of Minisymposium**Abstract**: Dynamical systems evolving in the existence of noise, is called random dynamical systems. The basic properties, such as stability, bifurcation, and statistical properties, of such random or non-autonomous dynamical systems have not been well studied in mathematics and physics. Recently, cooperative research on random dynamical systems has been developed in the fields in statistical and nonlinear physics, dynamical system theory, ergodic theory, and stochastic process theory. In this mini-symposium, we consider theory and applications of random dynamical systems. In Part II, we discuss Newton methods and random dynamical systems.**Organizer(s)**: Hiroki Sumi, Yuzuru Sato, Kouji Yano, Takuma Akimoto**Classification**:__37H05__,__90C52__,__30D05__,__65K10__,__37F10__**Speakers Info**:**Hiroki Sumi**(Kyoto University)- Takayuki Watanabe (Kyoto University)
- Mark Comerford (University of Rhode Island)
- Tuyen Trung Truong (University of Oslo)

**Talks in Minisymposium**:**[02705] Complex two-dimensional random relaxed Newton's methods****Author(s)**:**Hiroki Sumi**(Kyoto University)

**Abstract**: We develop the theory of random dynamical systems of meromorphic maps on the complex two-dimensional projective space and we consider complex two-dimensional random relaxed Newton's methods to find backward images of the origin (0,0) in C^{2} under complex two dimensional regular polynomial maps on C^{2}. We will see the randomness or noise in the systems bring us very nice things, noise-induced order and collapsing of nasty attractors, which cannot hold in deterministic relaxed Newton's methods.

**[02999] New Q-Newton's method and Backtracking line search****Author(s)**:**Tuyen Trung Truong**(University of Oslo)

**Abstract**: New Q-Newton's method is a variant of Newton's method which preserves the fast rate of convergence while having the additional good property of being able to avoid saddle points. It works by adding a term into the Hessian, and change the sign of negative eigenvalues. Backtracking line search boosts the convergence guarantee. This talk will describe the algorithm, good experimental performance (including finding roots of meromorphic functions), and good theoretical guarantees.

**[03870] Worked out examples of Random Relaxed Newton's Methods****Author(s)**:**Takayuki Watanabe**(Chubu University)

**Abstract**: We give some numerical results on Random Relaxed Newton's Methods which were proposed by Sumi to compute an approximate root of a given polynomial. He proved that this randomized algorithm almost surely works well if large noise is inserted. In this talk, we demonstrate by numerical experiments that even small noise can make the randomized algorithm successful, and discuss a mathematical conjecture.

**[05306] A Universal Fatou Component****Author(s)**:**Mark David Comerford**(University of Rhode Island)

**Abstract**: We show that, for the non-autonomous iteration of polynomials with suitably bounded degrees and coefficients, it is possible to obtain the whole of the classical schlicht family of normalized univalent functions on the unit disc as limit functions on a single Fatou component for a single bounded sequence of quadratic polynomials.