In the dynamical systems theory, Thomas' cyclically symmetric attractor is a 3D strange attractor originally proposed by René Thomas. 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. The simple form has made it a popular example.
It is described by the differential equations
d
x
d
t
=
sin
(
y
)
−
b
x
d
y
d
t
=
sin
(
z
)
−
b
y
d
z
d
t
=
sin
(
x
)
−
b
z
where
b
is a constant.
b
corresponds to how dissipative the system is, and acts as a bifurcation parameter. For
b
>
1
the origin is the single stable equilibrium. At
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
b
≈
0.32899
, creating a stable limit cycle. The limit cycle the undergoes a period doubling cascade and becomes chaotic at
b
≈
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.
In the limit
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.
