Periodic windows distribution resulting from homoclinic bifurcations in the two-parameter space R. O. Medrano-T. 1,2 and I. L. Caldas 2 1 Departamento de Ciˆ encias Exatas e da Terra, Universidade Federal de S˜ ao Paulo, Diadema, S˜ ao Paulo, Brasil and 2 Instituto de F´ ısica, Universidade de S˜ao Paulo, S˜ao Paulo, Brasil Periodic solution parameters, in chaotic dynamical systems, form periodic windows with charac- teristic distribution in two-parameter spaces. Recently, general properties of this organization have been reported, but a theoretical explanation for that remains unknown. Here, for the first time we associate the distribution of these periodic windows with scaling laws based in fundamental dynamic properties. For the R¨ ossler system, we present a new scenery of periodic windows composed by mul- tiple spirals, continuously connected, converging to different points along of a homoclinic bifurcation set. We show that the bi-dimensional distribution of these periodic windows unexpectedly follows scales given by the one-parameter homoclinic theory. Our result is a strong evidence that, close to homoclinic bifurcations, periodic windows are aligned in the two-parameter space. PACS numbers: 05.45.-a, 02.30.Oz, 05.45.Pq Keywords: Homoclinic systems, Periodic windows, Bifurcation I. INTRODUCTION For several smooth nonlinear maps and differential equations, stable periodic orbits and their dependence on the system control parameters are well known. The existence of these orbits can be properly visualized on a bi-dimensional parameter space, where generally we find periodic windows, i.e., continuous sets of parameters, em- bedded into chaotic regions, for which periodical orbits exist [1, 2]. There is an intricate periodic window, quite general in dynamical systems, whose local nature was explained in [1, 3] but the global features, despite the great number of studies [4–16], remain not well under- stood. For local and global features we mean typically qualities of an isolated and a multiple periodic windows, respectively. Recently, it has been found that these peri- odic windows, baptized shrimp [2], are continuously con- nected along spirals emerging from in a homoclinic bi- furcation point [17, 18] (set of parameters for which a bi-asymptotic curve, the homoclinic orbit, converges to a saddle-focus equilibrium point). These spiral structures were verified experimentally in [19, 20] and are also ob- served in [4, 21, 22]. However, in these researches the distribution of shrimps in the parameter space have not yet been clearly associated to any fundamental dynamical property. To accomplish this, we investigate the relation between the shrimps and the homoclinic curves in the parameter space [23, 24][30]. For the R¨ossler system, we present a new and remarkable two-parameter space scenery where from each shrimp emerge infinity spirals with focus in dis- crete points along of a homoclinic bifurcation curve (con- tinuous parameter sets for which homoclinic orbits exist). Each spiral is composed by a shrimp family, i.e., infinite shrimps continuously connected in a spiral sequence. We show that, even the shrimps are a codimension-two phe- nomena (two parameters are necessary to obtain it), they are accumulating at the spiral focus following scaling laws predicted by the one-parameter space homoclinic theory [25]. We also show that the reported period adding cas- cades observed in shrimps accumulations [11, 12] is a con- sequence of the spiral periodic windows approach to the homoclinic bifurcation point. Recently was published a work [20] about scaling laws in a electronic homoclinic system where the authors as- sociate scales of tangent bifurcation with the shrimp dis- tribution. Here we discuss this issue in details and show that, in the two-parameter space, the scales measured in shrimps correspond to the distance between crosses of superstable periodic curves. Furthermore, we call the at- tention to the necessity of a rigorous prove of these scales in shrimp distributions which was done in [26]. In section II we present the spiral scenery of periodic windows, in III we show the scaling laws concerning its distribution and periodicity, and in section IV we present the conclusions. II. SHRIMP DISTRIBUTIONS We consider the homoclinic R¨ossler system given by ˙ x = -y - z ˙ y = x + ay ˙ z = bx - cz + xz, (1) where we fix b =0.3 and analyze the parameter space c ′ × a ′ , where a = a ′ sin(θ)+ c ′ cos(θ)+ a 0 and c = a ′ cos(θ) - c ′ sin(θ)+ c 0 with a 0 =0.3301, c 0 =4.9305, and θ = 88.8 ◦ . For the region investigated (Fig. 1) the origin phase space (P 0 ) is a saddle-focus and the eigenvalues of Eqs. (1) Jacobian matrix evaluated at P 0 are λ 1,2 = ρ ± iω and λ 3 = λ, where λ, ρ, and ω are R ∗ . Periodic and chaotic asymptotic solutions of Eqs. (1) are determined numerically by evaluating the largest nonzero Lyapunov exponent l. In Fig. 1, from black arXiv:1012.2241v1 [nlin.CD] 10 Dec 2010