Poincare Disk Research Articles

Global analysis on a continuous planar piecewise linear differential system with three zones

This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincare disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/83/abstr.html

Computing the Conformal Barycenter

The conformal barycenter of a point cloud on the sphere at infinity of the Poincare ball model of hyperbolic space is a hyperbolic analogue of the geometric median of a point cloud in Euclidean space. It was defined by Douady and Earle as part of a construction of a conformally natural way to extend homeomorphisms of the circle to homeomorphisms of the disk, and it plays a central role in Millson and Kapovich's model of the configuration space of cyclic linkages with fixed edgelengths. In this paper we consider the problem of computing the conformal barycenter. Abikoff and Ye have given an iterative algorithm for measures on $\mathbb{S}^1$ which is guaranteed to converge. We analyze Riemannian versions of Newton's method computed in the intrinsic geometry of the Poincare ball model. We give Newton-Kantorovich (NK) conditions under which we show that Newton's method with fixed step size is guaranteed to converge quadratically to the conformal barycenter for measures on any $\mathbb{S}^d$ (including infinite-dimensional spheres). For measures given by $n$ atoms on a finite dimensional sphere which obey the NK conditions, we give an explicit linear bound on the computation time required to approximate the conformal barycenter to fixed error. We prove that our NK conditions hold for all but exponentially few $n$ atom measures. For all measures with a unique conformal barycenter we show that a regularized Newton's method with line search will always converge (eventually superlinearly) to the conformal barycenter. Though we do not have hard time bounds for this algorithm, experiments show that it is extremely efficient in practice and in particular much faster than the Abikoff-Ye iteration.

Global solutions and blow-up for a Kirchhoff-type problem on a geodesic ball of the Poincare ball model

This article concerns a Kirchhoff-type parabolic problem on a geodesic ball of hyperbolic space. Firstly, we obtain conditions for finite time blow-up, and for the existence of global solutions for \(J(u_0)\leq d\), where \(J(u_0)\) denotes the initial energy and d denotes the depth of the potential well. Secondly, we estimate the upper and lower bounds of the blow-up time. In addition, we derive the growth rate of the blow-up solution and the decay rate of the global solution. Thirdly, we establish a new finite time blow-up condition which is independent of d and prove that the solution can blow up in finite time with arbitrary high initial energy, by using this blow-up condition. Finally, we present some equivalent conditions for the solution existing globally or blowing up in finite time.

Delaunay triangulations of generalized Bolza surfaces

The Bolza surface can be seen as the quotient of the hyperbolic plane, represented by the Poincare disk model, under the action of the group generated by the hyperbolic isometries identifying opposite sides of a regular octagon centered at the origin. We consider _generalized_ Bolza surfaces Mg, where the octagon is replaced by a regular 4g-gon, leading to a genus g surface. We propose an extension of Bowyer's algorithm to these surfaces. In particular, we compute the value of the systole of Mg. We also propose algorithms computing small sets of points on Mg that are used to initialize Bowyer's algorithm.

Phase portraits of a family of Kolmogorov systems depending on six parameters

We consider a general 3-dimensional Lotka-Volterra system with a rational first integral of degree two of the form \(H=x^i y^j z^k\). The restriction of this Lotka-Volterra system to each surface \(H(x,y,z)=h\) varying \(h\in \mathbb{R}\) provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form \(x^\ell y^m e^{st}\) they reduce to the Kolmogorov systems $$\displaylines{ \dot{x}=x\big(a_0- \mu (c_1 x + c_2 z^2 + c_3 z)\big),\cr \dot{z}=z\big(c_0+ c_1 x + c_2 z^2 + c_3 z\big). }$$ We classify the phase portraits in the Poincare disc of all these Kolmogorov systems which depend on six parameters. For more information see https://ejde.math.txstate.edu/Volumes/2021/35/abstr.html

Algebraic and topological classification of homogeneous quartic vector fields in the plane

We provide canonical forms for the homogeneous polynomials of degree five. Then we characterize all the phase portraits in the Poincare disk for all quartic homogeneous polynomial differential systems. More precisely, there are exactly 23 different topological phase portraits for the quartic homogeneous polynomial differential systems.

Several Classes of Plain Dynamic Systems Qualitative Investigation

Dynamic systems in applications are useful as mathematical models of those processes and phenomena, where statistical events, or fluctuations, may be disregarded. Dynamic systems may be divided into the two main categories – the systems with continuous time (the flows) and systems with discrete time (the cascades). During the investigations of flows normal autonomous systems of ordinary differential equations are used. The present work is devoted to the original rigorous research of some important family of dynamic systems having reciprocal polynomial right parts, which are the forms of theirs phase variables. The whole wide family under consideration is being split into numeric subfamilies belonging to different hierarchical levels, and is subjected to the first and second Poincare transformations, or mappings. As a result, the full qualitative pattern of trajectories is constructed – using the Poincare sphere – in the Poincare disk. A series of new special investigation methods developed, useful for further investigations of similar dynamic systems’ classes.

THE HYPERBOLIC MENELAUS THEOREM IN THE POINCARE DISC MODEL OF HYPERBOLIC GEOMETRY

In this note, we present the hyperbolic Menelaus theorem in the Poincare disc of hyperbolic geometry.

Differential Geometry and Binary Operations

We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.

Traveling wave solutions for degenerate nonlinear parabolic equations

We consider the traveling wave solutions of the degenerate nonlinear parabolic equation $$u_{t}=u^{p}(u_{xx}+u)$$ which arises in the model of heat combustion, solar flares in astrophysics, plane curve evolution problems and the resistive diffusion of a force-free magnetic field in a plasma confined between two walls. We also deal with the equation $$v_{\tau }=v^{p}(v_{xx}+v-v^{-p+1})$$ related with it. We first give a result on the whole dynamics on the phase space $${\mathbb {R}}^{2}$$ with including infinity about two-dimensional ordinary differential equation that introduced the traveling wave coordinates: $$\xi = x-ct$$ by applying the Poincare compactification and dynamical system approach. Second, we focus on the connecting orbits on it and give a result on the existence of the weak traveling wave solutions with quenching for $$c>0$$ and $$p\in 2{\mathbb {N}}$$ . Moreover, we give the detailed information about the asymptotic behavior of $$u(\xi )$$ , $$u'(\xi )$$ , $$v(\xi )$$ and $$v'(\xi )$$ for $$p\in 2{\mathbb {N}}$$ . In the case that $$p \in 2 {\mathbb {N}}+1$$ , it is too complicated to determine the dynamics near the singularities on the Poincare disk, however, we classify the connecting orbits and corresponding traveling wave solutions and obtain their asymptotic behavior.

Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles

<p style='text-indent:20px;'>We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the <inline-formula><tex-math id="M1">\begin{document}$ x- $\end{document}</tex-math></inline-formula>axis having a nilpotent saddle at the origin.

Structurally Unstable Quadratic Vector Fields of Codimension Two: Families Possessing Either a Cusp Point or Two Finite Saddle-Nodes

The goal of this paper is to contribute to the classification of the phase portraits of planar quadratic differential systems according to their structural stability. Artes et al. (Mem Am Math Soc 134:639, 1998) proved that there exist 44 structurally stable topologically distinct phase portraits in the Poincare disc modulo limit cycles in this family, and Artes et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) showed the existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits of codimension-one quadratic systems, modulo limit cycles. In this work we begin the classification of planar quadratic systems of codimension two in the structural stability. Combining the sets of codimension-one quadratic vector fields one to each other, we obtain ten new sets. Here we consider set AA obtained by the coalescence of two finite singular points, yielding either a triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible topological phase portraits of set AA and prove their realization. We got 34 new topologically distinct phase portraits in the Poincare disc modulo limit cycles. Moreover, in this paper we correct a mistake made by the authors in the book of Artes et al. (Structurally unstable quadratic vector fields of codimension one, Springer, Berlin, 2018) and we reduce to 203 the number of topologically distinct phase portrait of codimension one modulo limit cycles.

Bifurcation Diagram and Global Phase Portraits of a Family of Quadratic Vector Fields in Class I

We study a family of quadratic vector fields in Class I $$\dot{x}=y, \dot{y}=-x-\alpha y+\mu x^2-y^2$$ , where $$(\alpha , \mu ) \in \mathbb {R}^2$$ . To study the equilibria at infinity on the Poincare disk of this system completely, we follow the method of generalized normal sectors of Tang and Zhang (Nonlinearity 17:1407–1426, 2004) and give further two new criterions, which allows us to obtain not only the qualitative properties of the equilibria but also asymptotic expressions of these orbits connecting the equilibria at infinity of this system. Further, the complete bifurcation diagram including saddle connection bifurcation curves of this system is given. Moreover, by qualitative properties of the equilibria, the nonexistence of limit cycle and rotated properties about $$\alpha $$ and $$\mu $$ , all global phase portraits on the Poincare disk of this system are also obtained and the number is 19.

Phase portraits of Bernoulli quadratic polynomial differential systems

In this article we study a new class of quadratic polynomial differential systems. We classify all global phase portraits in the Poincare disk of Bernoulli quadratic polynomial differential systems in R^2.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/48/abstr.html

Darboux integrability and dynamics of the Basener–Ross population model

We deal with the Basener and Ross model for the evolution of human population in Easter island. We study the Darboux integrability of this model and characterize all its global dynamics in the Poincare disc, obtaining 15 different topological phase portraits.

Formalization of the Poincaré Disc Model of Hyperbolic Geometry

We describe formalization of the Poincare disc model of hyperbolic geometry within the Isabelle/HOL proof assistant. The model is defined within the complex projective line $$\mathbb {C}{}P^1$$ and is shown to satisfy Tarski’s axioms except for Euclid’s axiom—it is shown to satisfy it’s negation, and, moreover, to satisfy the existence of limiting parallels axiom.

The Hyperbolic Geometry of Financial Networks

Based on data from the European banking stress tests of 2014, 2016 and the transparency exercise of 2018 we demonstrate for the first time that the latent geometry of financial networks can be well-represented by geometry of negative curvature, i.e., by hyperbolic geometry. This allows us to connect the network structure to the popularity-vs-similarity model of Papdopoulos et al., which is based on the Poincare disc model of hyperbolic geometry. We show that the latent dimensions of `popularity' and `similarity' in this model are strongly associated to systemic importance and to geographic subdivisions of the banking system. In a longitudinal analysis over the time span from 2014 to 2018 we find that the systemic importance of individual banks has remained rather stable, while the peripheral community structure exhibits more (but still moderate) variability.

Investigation of Phase Portraits Belonging to Polynomial Dynamic Systems in a Poincare Disk

The paper describes especially developed research methods and results for some original study of a broad dynamic systems’ family, which is characterized with the reciprocal polynomials in the right parts of differential equations which compose a system. The goal of this work is to obtain and analyze all the existing (and different in the topological meaning) phase portraits in a Poincare disk, as well as to indicate the criteria of each topological type of phase portraits appearance. Poincare methods of sequential transformations and displays have been successfully used along with a whole set of new research techniques developed for the purposes of this study. Over than 250 of different portraits were obtained and depicted eventually for different subsystems belong to the dynamic systems’ family under consideration. All portraits were described up to each the so-called invariant cell of the phase portrait, a boundary of it, and features of a phase flow in it, including it’s a source and a sink. Results of dynamic systems’ behavior research are useful for a wide spectrum of applications such as the mathematical modeling of physical processes, vital problems of civil engineering, for example, in the consideration of seismic stability of buildings etc., computing and producing systems, biological phenomena, and of sociological events.

Angular inflation in multi-field α-attractors

We explore the dynamics of multi-field models of inflation in which the field-space metric is a hyperbolic manifold of constant curvature. Such models are known as α-attractors and their single-field regimes have been extensively studied in the context of inflation and supergravity. We find a variety of multi-field inflationary trajectories in different regions of parameter space, which is spanned by the mass parameters and the hyperbolic curvature. Amongst these is a novel dynamical attractor along the boundary of the Poincare disc which we dub “angular inflation”. We calculate the evolution of adiabatic and isocurvature fluctuations during this regime and show that, while isocurvature modes decay during this phase, the duration of the angular inflation period can shift the single-field predictions of α-attractors. For highly curved field-space manifolds, this can lead to predictions that lie outside the current observational bounds.

Global phase portraits of the key pitchfork bifurcation

This paper deals with the following quadratic polynomial differential system$\frac{dx}{dt}=y^{2}-y-x,$ $\frac{dy}{dt}=x^{2}$ $-\,~\mu~x-y,$with parameter $\mu\in\mathbb{R}$, which is the key example for studying the pitchfork bifurcation of a singular point. We classify the global phase portraits in the Poincare disc of this system when $\mu$ varies.