Linear and Nonlinear Arabesques

A study of closed chains of negative 2-element circuits

Chris Antonopoulos*, Vasileios Basios, Jacques Demongeot, Pasquale Nardone, Rene Thomas

*Corresponding author for this work

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

In this paper we consider a family of dynamical systems that we call "arabesques", defined as closed chains of 2-element negative circuits. An n-dimensional arabesque system has n 2element circuits, but in addition, it displays by construction, two n-element circuits which are both positive versus one positive and one negative, depending on the parity (even or odd) of the dimension n. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they cannot possess any attractor. First, we analyze a linear variant of them which we call "arabesque 0" or for short "A0". For increasing dimensions, the trajectories are increasingly complex open tori. Next, we inserted a single cubic nonlinearity that does not affect the signs of its circuits (that we call "arabesque 1" or for short "A1"). These systems have three steady states, whatever be the dimension, in agreement with the order of the nonlinearity. All three are unstable, as there cannot be any attractor in their state-space. The 3D variant (that we call for short "A1_3D") has been analyzed in some detail and found to display a complex mixed set of quasi-periodic and chaotic trajectories. Inserting n cubic nonlinearities (one per equation) in the same way as above, we generate systems "A2_nD". A2_3D behaves essentially as A1_3D, in agreement with the fact that the signs of the circuits remain identical. A2_4D, as well as other arabesque systems with even dimension, has two positive n-circuits and nine steady states. Finally, we investigate and compare the complex dynamics of this family of systems in terms of their symmetries.

Original languageEnglish
Article number1330033
Number of pages20
JournalInternational Journal of Bifurcation and Chaos
Volume23
Issue number9
DOIs
Publication statusPublished - Sep 2013

Keywords

  • Jacobian circuits and frontiers
  • conservative systems
  • arabesque systems
  • Lyapunov exponents
  • smaller alignment index (SALI)
  • Hamiltonian-systems
  • phase-space
  • dynamics
  • chaos
  • networks
  • feedback
  • partition
  • galaxies
  • diagrams
  • orbits

Cite this

Linear and Nonlinear Arabesques : A study of closed chains of negative 2-element circuits. / Antonopoulos, Chris; Basios, Vasileios; Demongeot, Jacques; Nardone, Pasquale; Thomas, Rene.

In: International Journal of Bifurcation and Chaos, Vol. 23, No. 9, 1330033, 09.2013.

Research output: Contribution to journalArticle

Antonopoulos, Chris ; Basios, Vasileios ; Demongeot, Jacques ; Nardone, Pasquale ; Thomas, Rene. / Linear and Nonlinear Arabesques : A study of closed chains of negative 2-element circuits. In: International Journal of Bifurcation and Chaos. 2013 ; Vol. 23, No. 9.
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N2 - In this paper we consider a family of dynamical systems that we call "arabesques", defined as closed chains of 2-element negative circuits. An n-dimensional arabesque system has n 2element circuits, but in addition, it displays by construction, two n-element circuits which are both positive versus one positive and one negative, depending on the parity (even or odd) of the dimension n. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they cannot possess any attractor. First, we analyze a linear variant of them which we call "arabesque 0" or for short "A0". For increasing dimensions, the trajectories are increasingly complex open tori. Next, we inserted a single cubic nonlinearity that does not affect the signs of its circuits (that we call "arabesque 1" or for short "A1"). These systems have three steady states, whatever be the dimension, in agreement with the order of the nonlinearity. All three are unstable, as there cannot be any attractor in their state-space. The 3D variant (that we call for short "A1_3D") has been analyzed in some detail and found to display a complex mixed set of quasi-periodic and chaotic trajectories. Inserting n cubic nonlinearities (one per equation) in the same way as above, we generate systems "A2_nD". A2_3D behaves essentially as A1_3D, in agreement with the fact that the signs of the circuits remain identical. A2_4D, as well as other arabesque systems with even dimension, has two positive n-circuits and nine steady states. Finally, we investigate and compare the complex dynamics of this family of systems in terms of their symmetries.

AB - In this paper we consider a family of dynamical systems that we call "arabesques", defined as closed chains of 2-element negative circuits. An n-dimensional arabesque system has n 2element circuits, but in addition, it displays by construction, two n-element circuits which are both positive versus one positive and one negative, depending on the parity (even or odd) of the dimension n. In view of the absence of diagonal terms in their Jacobian matrices, all these dynamical systems are conservative and consequently, they cannot possess any attractor. First, we analyze a linear variant of them which we call "arabesque 0" or for short "A0". For increasing dimensions, the trajectories are increasingly complex open tori. Next, we inserted a single cubic nonlinearity that does not affect the signs of its circuits (that we call "arabesque 1" or for short "A1"). These systems have three steady states, whatever be the dimension, in agreement with the order of the nonlinearity. All three are unstable, as there cannot be any attractor in their state-space. The 3D variant (that we call for short "A1_3D") has been analyzed in some detail and found to display a complex mixed set of quasi-periodic and chaotic trajectories. Inserting n cubic nonlinearities (one per equation) in the same way as above, we generate systems "A2_nD". A2_3D behaves essentially as A1_3D, in agreement with the fact that the signs of the circuits remain identical. A2_4D, as well as other arabesque systems with even dimension, has two positive n-circuits and nine steady states. Finally, we investigate and compare the complex dynamics of this family of systems in terms of their symmetries.

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KW - conservative systems

KW - arabesque systems

KW - Lyapunov exponents

KW - smaller alignment index (SALI)

KW - Hamiltonian-systems

KW - phase-space

KW - dynamics

KW - chaos

KW - networks

KW - feedback

KW - partition

KW - galaxies

KW - diagrams

KW - orbits

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DO - 10.1142/S0218127413300334

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VL - 23

JO - International Journal of Bifurcation and Chaos

JF - International Journal of Bifurcation and Chaos

SN - 0218-1274

IS - 9

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ER -