Thomas' cyclically symmetric attractor

Thomas' cyclically symmetric attractor.

In the dynamical systems theory, Thomas' cyclically symmetric attractor is a 3D strange attractor originally proposed by René Thomas.^[1] It has a simple form which is cyclically symmetric in the x, y, and z variables and can be viewed as the trajectory of a frictionally dampened particle moving in a 3D lattice of forces.^[2] The simple form has made it a popular example.

It is described by the differential equations

${\displaystyle {\frac {dx}{dt}}=\sin(y)-bx}$

${\displaystyle {\frac {dy}{dt}}=\sin(z)-by}$

${\displaystyle {\frac {dz}{dt}}=\sin(x)-bz}$

where ${\displaystyle b}$ is a constant.

${\displaystyle b}$ corresponds to how dissipative the system is, and acts as a bifurcation parameter. For ${\displaystyle b>1}$ the origin is the single stable equilibrium. At ${\displaystyle b=1}$ it undergoes a pitchfork bifurcation, splitting into two attractive fixed points. As the parameter is decreased further they undergo a Hopf bifurcation at ${\displaystyle b\approx 0.32899}$, creating a stable limit cycle. The limit cycle then undergoes a period doubling cascade and becomes chaotic at ${\displaystyle b\approx 0.208186}$. Beyond this the attractor expands, undergoing a series of crises (up to six separate attractors can coexist for certain values). The fractal dimension of the attractor increases towards 3.^[2]

In the limit ${\displaystyle b=0}$ the system lacks dissipation and the trajectory ergodically wanders the entire space (with an exception for 1.67%, where it drifts parallel to one of the coordinate axes: this corresponds to quasiperiodic torii). The dynamics has been described as deterministic fractional Brownian motion, and exhibits anomalous diffusion.^[2][3]

References

1. Thomas, René (1999). "Deterministic chaos seen in terms of feedback circuits: Analysis, synthesis, 'labyrinth chaos'". Int. J. Bifurc. Chaos. 9 (10): 1889–1905. Bibcode:1999IJBC....9.1889T. doi:10.1142/S0218127499001383.
2. Sprott, J. C.; Chlouverakis, Konstantinos E. (2007). "Labyrinth Chaos". Int. J. Bifurc. Chaos. 17 (6): 2097. Bibcode:2007IJBC...17.2097S. doi:10.1142/S0218127407018245.
3. Rowlands, G.; Sprott, J. C. (2008). "A simple diffusion model showing anomalous scaling". Physics of Plasmas. 15 (8): 082308. Bibcode:2008PhPl...15h2308R. doi:10.1063/1.2969429.