Regular and Chaotic Dynamics

, Volume 23, Issue 6, pp 654–664 | Cite as

Moser’s Quadratic, Symplectic Map

  • Arnd BäckerEmail author
  • James D. Meiss


In 1994, Jürgen Moser generalized Hénon’s area-preserving quadratic map to obtain a normal form for the family of four-dimensional, quadratic, symplectic maps. This map has at most four isolated fixed points. We show that the bounded dynamics of Moser’s six parameter family is organized by a codimension-three bifurcation, which we call a quadfurcation, that can create all four fixed points from none.

The bounded dynamics is typically associated with Cantor families of invariant tori around fixed points that are doubly elliptic. For Moser’s map there can be two such fixed points: this structure is not what one would expect from dynamics near the cross product of a pair of uncoupled Hénon maps, where there is at most one doubly elliptic point. We visualize the dynamics by escape time plots on 2d planes through the phase space and by 3d slices through the tori.


Hénon map symplectic maps saddle-center bifurcation Krein bifurcation invariant tori 

MSC2010 numbers

37J40 70H08 34C28 37C05 


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Supplementary material

11819_2018_6525_MOESM1_ESM.mp4 (4.4 mb)
Fig. 4: 3D phase space slice for Moser’s quadratic symplectic map: quadfurcation 2EE + 2EH
11819_2018_6525_MOESM2_ESM.mp4 (5.3 mb)
Fig. 6: 3D phase space slice for Moser’s quadratic symplectic map: quadfurcation EE + HH + 2EH


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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Technische Universität Dresden, Institut für Theoretische Physik and Center for DynamicsDresdenGermany
  2. 2.Max-Planck-Institut für Physik komplexer SystemeDresdenGermany
  3. 3.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

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