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Register a new userApproximate Description of the Mandelbrot Set. Thermodynamic Analogy
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Analogy between an approximate version of Feigenbaum renormalization group analysis in complex domain and the phase transition theory of Yang-Lee (based on consideration of formally complexified thermodynamic values) is discussed. It is shown that the Julia sets of the renormalization transformation correspond to the approximation of Mandelbrot set of the original map. New aspects of analogy between the theory of dynamical systems and the phase transition theory are uncovered.
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According to the method, suggested in our previous work (nlin/0509012) and based on the consideration of the specially coupled systems, the possibility of physical realization of the phenomena of complex analytic dynamics (such as Mandelbrot and Julia sets) is discussed. It is shown, that unlike the case of discrete maps or differential systems with periodic driving, investigated in mentioned work, there are some difficulties in attempts to obtain the Mandelbrot set for the coupled autonomous continuous systems. A system of coupled autonomous R{o}ssler oscillators is considered as an example.
We suggest an approach to constructing physical systems with dynamical characteristics of the complex analytic iterative maps. The idea follows from a simple notion that the complex quadratic map by a variable change may be transformed into a set of two identical real one-dimensional quadratic maps with a particular coupling. Hence, dynamical behavior of similar nature may occur in coupled dissipative nonlinear systems, which relate to the Feigenbaum universality class. To substantiate the feasibility of this concept, we consider an electronic system, which exhibits dynamical phenomena intrinsic to complex analytic maps. Experimental results are presented, providing the Mandelbrot set in the parameter plane of this physical system.
Let f be a degree d polynomial defined over the nonarchimedean field C_p, normalized so f is monic and f(0)=0. We say f is post-critically bounded, or PCB, if all of its critical points have bounded orbit under iteration of f. It is known that if p is greater than or equal to d and f is PCB, then all critical points of f have p-adic absolute value less than or equal to 1. We give a similar result for primes between d/2 and d. We also explore a one-parameter family of cubic polynomials over the 2-adic numbers to illustrate that the p-adic Mandelbrot set can be quite complicated when p is less than d, in contrast with the simple and well-understood p > d case.
We prove that there exists a homeomorphism $chi$ between the connectedness locus $mathcal{M}_{Gamma}$ for the family $mathcal{F}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $mathcal{M}_1$. The homeomorphism $chi$ is dynamical ($mathcal{F}_a$ is a mating between $PSL(2,mathbb{Z})$ and $P_{chi(a)}$), it is conformal on the interior of $mathcal{M}_{Gamma}$, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces. Following the recent proof by Petersen and Roesch that $mathcal{M}_1$ is homeomorphic to the classical Mandelbrot set $mathcal{M}$, we deduce that $mathcal{M}_{Gamma}$ is homeomorphic to $mathcal{M}$.
We consider analytical formulae that describe the chaotic regions around the main periodic orbit $(x=y=0)$ of the H{e}non map. Following our previous paper (Efthymiopoulos, Contopoulos, Katsanikas $2014$) we introduce new variables $(xi, eta)$ in which the product $xieta=c$ (constant) gives hyperbolic invariant curves. These hyperbolae are mapped by a canonical transformation $Phi$ to the plane $(x,y)$, giving Moser invariant curves. We find that the series $Phi$ are convergent up to a maximum value of $c=c_{max}$. We give estimates of the errors due to the finite truncation of the series and discuss how these errors affect the applicability of analytical computations. For values of the basic parameter $kappa$ of the H{e}non map smaller than a critical value, there is an island of stability, around a stable periodic orbit $S$, containing KAM invariant curves. The Moser curves for $c leq 0.32$ are completely outside the last KAM curve around $S$, the curves with $0.32<c<0.41$ intersect the last KAM curve and the curves with $0.41leq c< c_{max} simeq 0.49$ are completely inside the last KAM curve. All orbits in the chaotic region around the periodic orbit $(x=y=0)$, although they seem random, belong to Moser invariant curves, which, therefore define a structure of chaos. Orbits starting close and outside the last KAM curve remain close to it for a stickiness time that is estimated analytically using the series $Phi$. We finally calculate the periodic orbits that accumulate close to the homoclinic points, i.e. the points of intersection of the asymptotic curves from $x=y=0$, exploiting a method based on the self-intersections of the invariant Moser curves. We find that all the computed periodic orbits are generated from the stable orbit $S$ for smaller values of the H{e}non parameter $kappa$, i.e. they are all regular periodic orbits.
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