Hyperbolic Components Research Articles

From hyperbolic to parabolic parameters along internal rays

For the quadratic family f c ( z ) = z 2 + c f_{c}(z) = z^2+c with c c in a hyperbolic component of the Mandelbrot set, it is known that every point in the Julia set moves holomorphically. In this paper we give a uniform derivative estimate of such a motion when the parameter c c converges to a parabolic parameter c ^ {\hat {c}} radially; in other words, it stays within a bounded Poincaré distance from the internal ray that lands on c ^ {\hat {c}} . We also show that the motion of each point in the Julia set is uniformly one-sided Hölder continuous at c ^ {\hat {c}} with exponent depending only on the petal number. This paper is a parabolic counterpart of the authors’ paper “From Cantor to semi-hyperbolic parameters along external rays” [Trans. Amer. Math. Soc. 372 (2019), pp. 7959–7992].

Meromorphic functions with a polar asymptotic value

This paper is part of a general program in complex dynamics to understand parameter spaces of transcendental maps with finitely many singular values. The simplest families of such functions have two asymptotic values and no critical values. These families, up to affine conjugation, depend on two complex parameters. Understanding their parameter spaces is key to understanding families with more asymptotic values, just as understanding quadratic polynomials was for rational maps more generally. The first such families studied were the one-dimensional slices of the exponential family, exp ⁡ ( z ) + a \exp (z) + a , and the tangent family λ tan ⁡ z \lambda \tan z . The exponential case exhibited phenomena not seen for rational maps: Cantor bouquets in both the dynamic and parameter spaces, and no bounded hyperbolic components. The tangent case, with its two finite asymptotic values ± λ i \pm \lambda i , is closer to the rational case, a kind of infinite degree version of the latter. In this paper, we consider a general family that interpolates between exp ⁡ ( z ) + a \exp (z) + a and λ tan ⁡ z \lambda \tan z . Our new family has two asymptotic values and a one-dimensional slice for which one of the asymptotic values is constrained to be a pole, the “polar asymptotic value” of the title. We show how the dynamic and parameter planes for this slice exhibit behavior that is a surprisingly delicate interplay between that of the exp ⁡ ( z ) + a \exp (z) + a and λ tan ⁡ z \lambda \tan z families.

Intersections between boundaries of bounded hyperbolic components of cubic polynomials

Intersections between boundaries of bounded hyperbolic components of cubic polynomials

Constructions of b-semitoric systems

In this article, we introduce b-semitoric systems as a generalization of semitoric systems, specifically tailored for b-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A b-semitoric system is a four-dimensional b-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global S1-action and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of b-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps.

Asymptotic behavior of the linearized compressible barotropic Navier‐Stokes system with a time varying delay term in the boundary or internal feedback

In this paper, we consider the linearized compressible barotropic Navier‐Stokes system in a bounded interval with a time‐varying delay term acting in the Dirichlet boundary or internal feedback of the hyperbolic component. Assuming some suitable conditions on the time‐dependent delay term and the coefficients of feedback (delayed or not), we study the exponential stability of the concerned hyperbolic‐parabolic system. Due to the presence of the time‐varying delay term, the corresponding spatial operator is also time dependent. Using classical semigroup theory with Kato's variable norm approach, we first show the existence and uniqueness of the Navier‐Stokes system with time delay, acting in the boundary or interior. Next, we prove the two stabilization results by means of interior delay feedback and boundary delay. In both cases, we establish the exponential stability results by introducing some suitable functional energy and using the Lyapunov function approach.

On geometrically finite degenerations I: boundaries of main hyperbolic components

We develop a theory of quasi post-critically finite degenerations of Blaschke products. This gives us tools to study the boundaries of hyperbolic components of rational maps in higher-dimensional moduli spaces. We use it to obtain a combinatorial classification of geometrically finite polynomials on the boundary of the main hyperbolic component \mathcal{H}_d , i.e., the hyperbolic component in the space of monic and centered polynomials that contains z^d . We also show that the closure \overline{\mathcal{H}_d} is not a topological manifold with boundary for d\geq 4 by constructing self-bumps on its boundary.

Modeling Solar Orbiter dust detection rates in the inner heliosphere as a Poisson process

Context.Solar Orbiter provides dust detection capability in the inner heliosphere, but estimating physical properties of detected dust from the collected data is far from straightforward.Aims.First, a physical model for dust collection considering a Poisson process is formulated. Second, it is shown that dust on hyperbolic orbits is responsible for the majority of dust detections with Solar Orbiter’s Radio and Plasma Waves (RPW). Third, the model for dust counts is fitted to Solar Orbiter RPW data and parameters of the dust are inferred, namely radial velocity, hyperbolic meteoroids predominance, and the solar radiation pressure to gravity ratio as well as the uncertainties of these.Methods.Nonparametric model fitting was used to get the difference between the inbound and outbound detection rate and dust radial velocity was thus estimated. A hierarchical Bayesian model was formulated and applied to available Solar Orbiter RPW data. The model uses the methodology of integrated nested Laplace approximation, estimating parameters of dust and their ncertainties.Results.Solar Orbiter RPW dust observations can be modeled as a Poisson process in a Bayesian framework and observations up to this date are consistent with the hyperbolic dust model with an additional background component. Analysis suggests a radial velocity of the hyperbolic component around (63 ± 7) km s−1with the predominance of hyperbolic dust being about (78 ± 4)%. The results are consistent with hyperbolic meteoroids originating between 0.02 AU and 0.1 AU and showing substantial deceleration, which implies effective solar radiation pressure to a gravity ratio ≳ 0.5. The flux of the hyperbolic component at 1 AU is found to be (1.1 ± 0.2) × 10−4m−2s−1and the flux of the background component at 1 AU is found to be (5.4 ± 1.5) × 10−5m−2s−1.

Factoring Gleason polynomials modulo 2

Among the connected components of the interior of the Mandelbrot set are those that are hyperbolic. These components consist of parameters c∈ℂ for which the critical point z 0 =0 of f c :z↦z 2 +c is attracted to an attracting periodic cycle. Every hyperbolic component contains a unique center; that is, a parameter c for which the critical point z 0 is periodic. For a given n≥1, the Gleason polynomial for period n is the monic polynomial G n ∈ℤ[c] whose roots are exactly the centers of the hyperbolic components of period n. It is unknown if G n factors over ℤ. In this article, we factor G n modulo 2. We prove the following remarkable fact: the number of irreducible factors of G n modulo 2 is equal to the number of real roots of G n .

Research on springback compensation for multi-point forming of corrosion-resistant aluminum alloy hyperbolic component

Springback is the reverse elastic deformation produced by the unloading process, which is a common phenomenon in the sheet metal stamping and forming process and affects the final forming accuracy of the part. Aiming at the problem of springback error in the multi-point forming process of hyperbolic component for ships, the springback phenomenon of hyperbolic component was studied by combining theoretical analysis, numerical simulation and forming test. The mechanical property parameters and stress-strain curves were obtained by tensile tests using a new corrosion-resistant aluminum alloy as the sheet material. The springback compensation method is introduced, the springback compensation formula based on surface curvature is derived, and the compensated mold surface is further corrected and fitted using the cubic B-sample method. The multi-point punch tooling was generated and modeled by the multi-point 3D modeling software MPFCAD and finite element software to simulate the multi-point forming and unloading springback process of the hyperbolic member. Multi-point forming tests were conducted on a multi-point forming press for corrosion-resistant aluminum alloy hyperbolic component, and the formed parts were checked for accuracy and error analysis by 3D scanning and GOM Inspect software. The results show that the springback compensation method used in this paper can effectively reduce the springback error in the multi-point forming process of corrosion-resistant aluminum alloy hyperbolic component, and the effect is very good.

A Riemannian metric on hyperbolic components

A Riemannian metric on hyperbolic components

Boundary controllability and stabilizability of a coupled first-order hyperbolic-elliptic system

In this paper, we study the controllability of a coupled first-order hyperbolic-elliptic system in the interval $ (0, 1) $ by a Dirichlet boundary control acting at the left endpoint of the hyperbolic component only. Using the multiplier approach and compactness-uniqueness argument, we establish the exact controllability for the hyperbolic component of the model, at any time $ T>1 $. We explore the method of moments to conclude the exact controllability at the critical time $ T = 1 $. For the case of small time, that is for $ T<1 $, we show that the system is not null controllable. Further, using a Gramian-based approach introduced by Urquiza, we prove the exponential stabilization of the corresponding closed-loop system with arbitrary prescribed decay rate by means of boundary feedback control law.

Hyperbolic Genetic Mechanism of Human Evolution

The concept of “hyperbolic medicine” covers several aspects. Its main characteristic is that hyperbolic spatiotemporal curves found in nature generate hyperbolic mechanisms that influence human physiology. A bibliographic review of scientific works related to genetic aspects of human evolution and the structure of the genetic code has been carried out. Then, theoretically, the hyperbolic patterns that occur in human physiology and the genetic code have been studied, as a possible hyperbolic mechanism of human evolution. Images in nature are space-time hyperbolas and exist independently of the longitude and latitude of the Earth where they are observed. The lines of force of electromagnetic fields and many curves of processes that occur in human physiology are also hyperbolic curves. Electromagnetic fields have effects on human physiology through hyperbolic curves. The genetic code is shaped like a helix because when a current passes through it, a transverse magnetic field is generated, and its hyperbolic lines of force follow a counterclockwise helical path. The different types of evolution occur according to different mechanisms. The mutation is an evolution mechanism that is related to a stable and heritable alteration of DNA. Nature is hyperbolic, and this conditions human physiology and genetics to make it hyperbolic. The alteration of the genetic code is a mechanism of human evolution. This evolution may be conditioned by a change in the hyperbolic helical structure of the genetic code. The primary mechanism of evolution can be found in an alteration of the hyperbolic component of the genetic code, linked to nature.

Escape components of McMullen maps

AbstractWe consider the McMullen maps $f_{\unicode{x3bb} }(z)=z^{n}+\unicode{x3bb} z^{-n}$ with $\unicode{x3bb} \in \mathbb {C}^{*}$ and $n \geq 3$ . We prove that the closures of escape hyperbolic components are pairwise disjoint and the boundaries of all bounded escape components (the McMullen domain and Sierpiński holes) are quasi-circles with Hausdorff dimension strictly between $1$ and $2$ .

Trees, length spectra for rational maps via barycentric extensions, and Berkovich spaces

In this paper, we study the dynamics of degenerating sequences of rational maps on Riemann sphere Cˆ using R-trees. As an analogue of isometric group actions on R-trees for Kleinian groups, we give two constructions for limiting dynamics on R-trees: one geometric and one algebraic. The geometric construction uses the limit of rescalings of barycentric extensions of rational maps, while the algebraic construction uses the Berkovich space of complexified Robinson’s field. We show that the two approaches are equivalent. As an application, we use it to give a classification of hyperbolic components of rational maps that admit degeneracies with bounded multipliers.

Perturbations of graphs for Newton maps I: bounded hyperbolic components

AbstractWe consider graphs consisting of finitely many internal rays for degenerating Newton maps and state a convergence result. As an application, we prove that a hyperbolic component in the moduli space of quartic Newton maps is bounded if and only if every element has degree $2$ on the immediate basin of each root. This provides the first complete description of bounded hyperbolic components in a complex two-dimensional moduli space.

Computation of Troesch Problem by a Modified Newton’s Approach

We introduce an improved Newton scheme to solve the nonlinear Troesch problem; which is a nonlinear parameter-sensitive differential equation applied to model plasma confinement in a column.. A predictor-corrector scheme based on a modified backward Euler method was adopted for the Newton’s update. The hyperbolic sine component of the governing equation was converted to its logarithmic analog in order to handle high gradients and discontinuities. This was also found to be useful when the initial position values are not very close to the projected equilibrium. The proposed algorithm is straightforward and offers a relatively high degree of accuracy for a remarkably wide range of values of the sensitivity parameter. The trade-off is a slightly more computation time than the classical Newton’s approach.

A Novel Fractional Analytical Technique for the Time-space Fractional Equations Appearing in Oil Pollution

Oil spills in the seas and oceans cause pollution and have many destructive environmental effects. The diffusion (parabolic) equations are the most reasonable option to model the propagation of this leakage and contamination. These equations allow statistics regarding the amount of oil that has outreached the ocean outlet, to be used as initial and boundary conditions for a mathematical model of oil diffusion and alteration in seas. As it involves the hyperbolic (advection/wave) component of the equation, the most reasonable choices are diffusion and Allen–Cahn (AC) equations, which are difficult to solve numerically. Equations of diffusion and Allen-Cahn were solved with different degrees of fractional derivatives (α= 0.25, α=0.5, α=0.75 and α=0.75), and the oil pollution concentration was obtained at a specific time and place. This study adopts the homotopy perturbation method (HPM) for nonlinear Allen–Cahn (AC) equation and time fractional diffusion equation to express oil pollution in the water. Fractional derivatives are portrayed in the sense of Caputo. Two presented examples illustrate the applicability and validity of the proposed method. Pollution concentrations in flow field over an interval of time and space for different degrees of fractional derivation are shown. At lower fraction derivative degrees, the pollution concentration behavior is nonlinear, and as the degree of fraction derivation increases to one, the nonlinear behavior of the pollution concentration decreases. The results produced by the suggested technique compared to the exact solutions shows that it is efficient and convenient; it is also reduces computational time.

Bounded hyperbolic components of bicritical rational maps

We prove that the hyperbolic components of bicritical rational maps having two distinct attracting cycles each of period at least two are bounded in the moduli space of bicritical rational maps. Our arguments rely on arithmetic methods.

Subhyperbolic rational maps on boundaries of hyperbolic components

<p style='text-indent:20px;'>In this paper, we prove that every quasiconformal deformation of a subhyperbolic rational map on the boundary of a hyperbolic component <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{H} $\end{document}</tex-math></inline-formula> still lies on <inline-formula><tex-math id="M2">\begin{document}$ \partial \mathcal{H} $\end{document}</tex-math></inline-formula>. As an application, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components.</p>

Significant Curves of the Mandelbrot Set

The paper provides a description of some interesant curves contained in the Mandelbrot set. These curves are the boundaries of the areas called „bulbs“ which are described approximately only in present. In this paper, some of them are described analyticaly – curves of so called first period, the boundary of the main hyperbolic component, internal and external bounds and also some curves of the second period.