A class of two-dimensional rational maps with self-excited and hidden attractors

This paper studies a new class of two-dimensional rational maps exhibiting self-excited and hidden attractors. The mathematical model of these maps is firstly formulated by introducing a rational term. The analysis of existence and stability of the fixed points in these maps suggests that there are four types of fixed points, i.e., no fixed point, one single fixed point, two fixed points and a line of fixed points. To investigate the complex dynamics of these rational maps with different types of fixed points, numerical analysis tools, such as time histories, phase portraits, basins of attraction, Lyapunov exponent spectrum, Lyapunov(Kaplan–Yorke) dimension and bifurcation diagrams, are employed. Our extensive numerical simulations identify both self-excited and hidden attractors, which were rarely reported in the literature. Therefore, the multi-stability of these maps, especially the hidden one, is further explored in the present work.

Rational maps as Schwarzian primitives

We examine when a meromorphic quadratic differential φ with prescribed poles is the Schwarzian derivative of a rational map. We give a necessary and sufficient condition: In the Laurent series of φ around each pole c, the most singular term should take the form(1- d2)/(2(z- c)2), where d is an integer, and then a certain determinant in the next d coefficients should vanish. This condition can be optimized by neglecting some information on one of the poles(i.e., by only requiring it to be a double pole). The case d = 2 was treated by Eremenko(2012). We show that a geometric interpretation of our condition is that the complex projective structure induced by φ outside the poles has a trivial holonomy group. This statement was suggested to us by Thurston in a private communication. Our work is related to the problem of finding a rational map f with a prescribed set of critical points, since the critical points of f are precisely the poles of its Schwarzian derivative.Finally, we study the pole-dependency of these Schwarzian derivatives. We show that, in the cubic case with simple critical points, an analytic dependency fails precisely when the poles are displaced at the vertices of a regular ideal tetrahedron of the hyperbolic 3-ball.

Rees-Shishikura's Theorem for Geometrically Finite Rational Maps

In this paper, we generalize Rees-Shishikura＇s theorem to the class of geometrically finite rational maps.

The Dynamics and Control of the Fractional Forms of Some Rational Chaotic Maps

This paper proposes three fractional discrete chaotic systems based on the Rulkov,Chang,and Zeraoulia–Sprott rational maps.The dynamics of the proposed maps are investigated by means of phase plots and bifurcations diagrams.Adaptive stabilization schemes are proposed for each of the three maps and the convergence of the states is established by using the Lyapunov method.Furthermore,a combination synchronization scheme is proposed whereby a combination of the fractional Rulkov and Chang maps is synchronized to the fractional Zeraoulia-Sprott map.Numerical results are used to confirm the findings of the paper.

Results About Parabolic-Like Mappings

In this paper we present the most important definitions and results of the theory of parabolic-like mappings, and we will give an example. The proofs of the results can be found in [2,4] and [3].

AN EDGE CRACK PROBLEM IN A SEMI-INFINITE PLANE SUBJECTED TO CONCENTRATED FORCES

An oblique edge crack problem in a semi-infinite plane is discussed. Re concentrated forces are applied on the edge crack face, or on the line boundary of the cracked semi-infinite plane. The rational mapping function approach is suggested to solve the boundary value problem and a solution in a closed form is obtained. Finally, several numerical examples with the calculated results are given.

Vector meson effects on multi-Skyrmion states from the rational map ansatz

The roles of the lightest vector mesons ρ and ω in the multi-Skyrmion states are studied using the hidden local symmetry approach up to the next-to-leading order,including the homogeneous Wess-Zumino terms.The low-energy constants in the effective field theory are determined using the Sakai-Sugimoto model and the flat-space five-dimensional Yang-Mills action.With only two inputs,m_(ρ) and f_(π),it is possible to determine all low-energy constants without ambiguity.The vector meson effects can be investigated by sequentially integrating vector mesons,and their geometry can be elucidated by comparing the results using the low-energy constants estimated from the Sakai-Sugimoto model and the flat-space five-dimensional Yang-Mills action.We found that theρmeson reduces the masses of the multi-Skyrmion states and increases the overlaps of their constituents,whereas theωmeson repulses the constituents of the multi-Skyrmion states and increases their masses.Therefore,these vector mesons are crucial in the Skyrme model approach to nuclei.We also found that the warping factor,an essential element in the holographic model of QCD,affects the properties of the multi-Skyrmion states and cannot be ignored.

Hausdorf Dimension of Quadratic Rational Julia Sets

Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two.