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Dynamics of Kahan-Hirota-Kimura maps with rational invariant fibrations

Víctor Mañosa, Chara Pantazi

Abstract

We present a simple method to study the dynamics of planar Kahan-Hirota-Kimura (KHK) maps preserving rational fibrations. Using this approach, we show that integrable KHK maps may exhibit complex dynamics, even when obtained from vector fields with trivial behavior. As an application, we study the KHK map associated with a quadratic planar vector field with an isochronous center. This map preserves the original first integral and admits the vector field as a Lie symmetry. Moreover, for a dense set of values of the integration step, it is globally periodic and exhibits all possible periods except 2. We also provide evidence of non-integrability for KHK maps associated with other quadratic vector fields possessing isochronous centers. To overcome this issue, we introduce the notion of pseudo-KHK maps, as alternative integrable discretizations for vector fields with isochronous centers. These maps are constructed to preserve the first integrals of the original vector field and to ensure that the vector field itself is a Lie symmetry of the map. The construction can be extended to isochronous centers of degree greater than two.

Dynamics of Kahan-Hirota-Kimura maps with rational invariant fibrations

Abstract

We present a simple method to study the dynamics of planar Kahan-Hirota-Kimura (KHK) maps preserving rational fibrations. Using this approach, we show that integrable KHK maps may exhibit complex dynamics, even when obtained from vector fields with trivial behavior. As an application, we study the KHK map associated with a quadratic planar vector field with an isochronous center. This map preserves the original first integral and admits the vector field as a Lie symmetry. Moreover, for a dense set of values of the integration step, it is globally periodic and exhibits all possible periods except 2. We also provide evidence of non-integrability for KHK maps associated with other quadratic vector fields possessing isochronous centers. To overcome this issue, we introduce the notion of pseudo-KHK maps, as alternative integrable discretizations for vector fields with isochronous centers. These maps are constructed to preserve the first integrals of the original vector field and to ensure that the vector field itself is a Lie symmetry of the map. The construction can be extended to isochronous centers of degree greater than two.
Paper Structure (16 sections, 13 theorems, 83 equations, 5 figures, 2 tables)

This paper contains 16 sections, 13 theorems, 83 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Vector fields with quadratic first integrals
  4. KHK-map of a particular Petrera-Suris vector field with quadratic integral
  5. Proof of Propositions \ref{['p:dinamica-exa']}, \ref{['p:periodes-exa']} and \ref{['p:liesymmesuraexa']}
  6. KHK maps associate with isochronous quadratic planar vector fields
  7. Detailed analysis of the KHK map associated to the quadratic vector field $S_1$
  8. Dynamics of the KHK map $\Phi_{1,\epsilon}$
  9. Proof of Propositions \ref{['p:dinamicaS1']} and \ref{['p:periodesS1']}
  10. Evidences of non integrability of the KHK maps associated with the quadratic fields $S_2$, $S_3$ and $S_4$
  11. Pseudo-KHK maps associated with Isochronous vector fields
  12. Definition and main properties
  13. Pseudo-KHK maps of quadratic isochronous vector fields
  14. Pseudo-KHK maps of quadratic isochronous from the perspective of the methodology in Section \ref{['s:method']}
  15. Pseudo-KHK maps for non quadratic isochronous vector fields
  16. ...and 1 more sections

Key Result

Proposition 1

Consider the map $M(t)=(at+b)/(ct+d)$, where $a,b,c,d\in{\mathbb R}$, with $c\neq 0$, defined for $t\in \widehat{{\mathbb R}}= \mathbb{R} \cup \{\infty\}$. Set $\Delta=(d-a)^2+4bc$ and $\xi=(a+d+\sqrt{\Delta})/(a+d-\sqrt{\Delta})$.

Figures (5)

  • Figure 1: Scheme of the energy level curves $C_h$ surrounding the center $O_1=(0,0)$ for $h\geq 0$ (in blue), and the center $O_2=(0,-1)$ for $h\leq -1$ (in brown); The separatrix of the central basins $y=-1/2$ (in red).
  • Figure 2: Some orbits of the KHK map $\Phi_{2,\epsilon}$ associated with the isochronous vector field $S_2$ for $\epsilon=0.1$ showing some of the typical characteristics of a non-integrable perturbed twist maps (left). Detail of one island surrounding an elliptic orbit (right).
  • Figure 3: Some orbits of the KHK map $\Phi_{2,\epsilon}$ associated with the isochronous vector field $S_2$ for $\epsilon=1$.
  • Figure 4: Some orbits of the KHK map $\Phi_{3,\epsilon}$ associated with the isochronous vector field $S_3$ for $\epsilon=0.5$.
  • Figure 5: Detail of the orbits of the KHK map $\Phi_{3,\epsilon}$ associated with the isochronous vector field $S_3$ for $\epsilon=0.5$.

Theorems & Definitions (25)

  • Proposition 1
  • Lemma 2
  • proof
  • Proposition 3
  • Remark 4
  • Corollary 5
  • Proposition 6
  • Proposition 7
  • proof : Proof of Proposition \ref{['p:dinamica-exa']}
  • proof : Proof of Proposition \ref{['p:periodes-exa']}
  • ...and 15 more