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 {\displaystyle {\frac {dx}{dt}}=\sin(y)-bx}
d y d t = sin ( z ) − b y {\displaystyle {\frac {dy}{dt}}=\sin(z)-by}
d z d t = sin ( x ) − b z {\displaystyle {\frac {dz}{dt}}=\sin(x)-bz}
where b {\displaystyle b} is a constant.
b {\displaystyle b} corresponds to how dissipative the system is, and acts as a bifurcation parameter. For b > 1 {\displaystyle b>1} the origin is the single stable equilibrium. At b = 1 {\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 b ≈ 0.32899 {\displaystyle b\approx 0.32899} , creating a stable limit cycle. The limit cycle then undergoes a period doubling cascade and becomes chaotic at b ≈ 0.208186 {\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. In the limit b = 0 {\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.
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