Seminário de Sistemas Dinâmicos e Estocásticos

Departamento de Matemática - IMECC – UNICAMP



Próximos seminários


16 de setembro de 2016 às 10hs.


Palestrante: Alexandre Miranda Alves (Universidade Federal de Viçosa).

Título: Conditions to the existence of center-focus in planar systems and
center for Abel equations.

Resumo


Abel equations of the form $x (t) = f (t)x^3(t) + g(t)x^2(t)$, $t \in [−a, a]$, where a > 0 is a constant, $f$ and $g$ are continuous functions, are of interest because of their close relation to planar vector fields. If $f$ and $g$ are odd functions we prove that the Abel equation has a center at the origin. We also consider a class of polynomial differential equations $x=-y + P_n(x, y)$ and $\dot y= x + Q_n(x, y)$, where $P_n$ and $Q_n$ are homogeneous polynomials of degree n. Using the results obtained for Abel’s equation we obtain a new subclass of systems having a center and another subclass having a focus at the origin.

16 de setembro de 2016 às 14hs.


Palestrante: Danilo Antonio Caprio(UNESP-IBILCE).

Título: Dynamics of stochastic Fibonacci adding machines.

Resumo


In this work I will define the stochastic adding machine associated to the Fibonacci base $(F_n)_{n\geq 0}$ (where $F_0=1$, $F_1=2$ and $F_n=F_{n-1}+F_{n-2}$, for all $n\geq 2$) and to the probabilities sequence $(p_i)_{i\geq 1}$. I will consider the transition operator $S$ and I will prove that the Markov chain is transient if and only if $\prod_{i=1}^{\infty}p_i>0$. Otherwise, if $\sum_{i=1}^{+\infty}p_i=+\infty$, then the Markov chains is null recurrent and if $\sum_{i=2}^{+\infty}p_iF_{2(i-1)}<+\infty$, then
the Markov chain is recurrent positive.

I will compute the point spectrum and prove that it is connected to the fibered Julia sets for a class of endomorphisms in $\mathbb{C}^2$. Precisely $\sigma_{pt}(S)\subset E \subset \sigma(S)$ where $E=\{z\in\mathbb{C}:(g_n\circ\ldots\circ g_0(z,z))_{n\geq 1}\textrm{ is bounded}\}$ and $g_n:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ are maps defined by $g_0(x,y)=\left(\frac{x-(1-p_1)}{p_1},\frac{y-(1-p_1)}{p_1}\right)$ and $g_n(x,y)=\left(\frac{1}{r_n}xy-\left(\frac{1}{r_n}-1\right),x\right)$ for all $n\geq 1$, where $r_n=p_{\left[\frac{n+1}{2}\right]+1}$. Moreover, if $\liminf_{i\to+\infty}p_i>0$ then $E$ is compact and $\mathbb{C}\setminus E$ is connected.

Finally, I will prove that $\sigma_{pt}(S)\cap\mathbb{R}= E\cap \mathbb{R}$.

Bibliography

[1] A class of adding machines and Julia sets, Discrete and Continuous Dynamical Systems-A, 2016 (arXiv:1508.05062).

[2] P.R. Killeen, T.J. Taylor, A stochastic adding machine and complex dynamics, Nonlinearity 13 (2000) 1889-1903.

[3] A. Messaoudi, O. Sester, G. Valle, Spectrum of stochastic adding machines and fibered Julia sets, Stochastics and Dynamics, 13(3), 26p, 2013.



Local: sala 321 Imecc





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