Shrimp and Canonical Quartic Mapping: Basic Infos & papers

Prof. Edward Lorenz and shrimps


In his last paper, Prof. Edward N. Lorenz rediscovered "shrimps" in discrete-time systems (maps)
and, in an endnote, graciously aknowledged the priority of our works being published since 1993:
Structure of the parameter space of the Hénon map, J.A.C. Gallas, PRL 70,   2714 (1993)
Prof. Lorenz cites 4 of our earlier (1993-99) papers:
Edward N. Lorenz, Compound windows of the Hénon map, Physica D  237,   1689-1704 (2008)

Shrimps in discrete-time systems (maps) have been studied since the 1990s: Check references here

A current hot topic (cover of several journals !!) is the recent discovery of shrimps
abundantly and curiously organized in continuous-time systems:
i.e. in flows, models ruled by sets of differential equations (not maps)






Review of Basic Facts for Maps and Flows:

• HRES Color figs:   Structure of the parameter space of a ring cavity,
  LRES B&W figs:   Structure of the parameter space of a ring cavity,
  J.A.C. Gallas
  Applied Physics B 60, S203-S213 (1995), Special supplement, Festschrift Herbert Walther.




First results: Papers about Mappings (discrete time systems):

• Structure of the parameter space of the Hénon map,
  J.A.C. Gallas
  Physical Review Letters 70, 2714 (1993).

• Mandelbrot-like sets in dynamical systems with no critical points
  Antonio Endler  and  Jason A.C. Gallas,
  Comptes Rendus Academie des Sciences, Mathematique (Paris)   342,  681-684 (2006)

• Dissecting shrimps: results for some one-dimensional physical systems,
  J.A.C. Gallas
  Physica A 202, 196 (1994).

• Accumulation points in nonlinear parameter-lattices,
  Marcus Werner Beims and J.A.C. Gallas
  Physica A 238, 225-244 (1997).




Enticing news: Papers about Flows (continuous time systems):

• Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator
  Cristian Bonatto   Jason A.C. Gallas  and  Yoshisuke Ueda,
  Physical Review E  77,   026217 (2008)

• Accumulations boundaries: Codimension-two accumulation of accumulations in phase diagrams of
  semiconductor lasers, electric circuits, atmospheric, and chemical oscillators
  Cristian Bonatto  and  Jason A.C. Gallas,
  Philosophical Transactions of the Royal Society of London, Series A   366, 505-517 (2008)

• Accumulations horizons and period adding cascades in optically injected semiconductor lasers
  Cristian Bonatto  and  Jason A.C. Gallas,
  Physical Review E  75,   055204(R) (2007), Rapid Communications

• Self-similarities in the frequency-amplitude space of a loss-modulated CO2 laser  =  physics 0505213
  Cristian Bonatto, Jean Claude Garreau  and  Jason A.C. Gallas,
  Physical Review Letters  95,  143905 (2005)

• HRES Color figs:   Structure of the parameter space of a ring cavity,
  LRES B&W figs:   Structure of the parameter space of a ring cavity ,
  J.A.C. Gallas
  Applied Physics B 60, S203-S213 (1995), Special supplement, Festschrift Herbert Walther.

• Lyapunov exponents for a Duffing oscillator,
  Andrea Regina Zeni and J.A.C. Gallas
  Physica D 89, 71-82 (1995).




Papers about the Origins of Periodicity in Physical Models:

• On the origin of periodicity in dynamical systems
  J.A.C. Gallas,
  Physica A, vol. 283, pages 17-23 (2000).

• Nonlinear dependencies between sets of periodic orbits
  J.A.C. Gallas,
  Europhysics Letters, vol. 47, pages 649-655 (1999).

• Infinite hierarchies of nonlinearly dependent periodic orbits
  J.A.C. Gallas,
  Physical Review E, 63, 016216 (2001)

• Arithmetical signatures of the dynamics of the Hénon map
  A. Endler and J.A.C. Gallas,
  Physical Review E, 65,  036231 (2002).

• O novo brilho das equações Abelianas   [The new glow of the Abelian equations],
  Em comemoração do Bicentenário do Nascimento de Abel
  J.A.C. Gallas,
  Boletim da Sociedade Portuguesa de Matemática, 47, 1-15 (2002).

• Tradução do artigo de Niels Henrik ABEL (Comemoração do Bicentenário do Nascimento):
  Mémoire sur une classe particulière d'équations résolubles algébriquement,
  Journal für die reine und angewandte Mathematik, 4, 131-156(1829).
  J.A.C. Gallas,
  Boletim da Sociedade Portuguesa de Matemática, 47, 17-48(2002).