Tsucs2¶

Systems / ODEs / Coupled systems

The Three-Scroll Unified Chaotic System (TSUCS-2), a Dequan-Li-family flow unifying several three-scroll attractors. At the published parameters the orbit is a bounded, visually three-scroll-like 2-torus rather than a strange attractor: the measured spectrum is (0, 0, −1.50), so treat any "chaotic" quantity from the defaults with suspicion.

continuous · ODE3 dimensionsquasiperiodic

Interactive: drag to rotate

Definition¶

\[ \begin{aligned} \dot{x} &= - a x + a y + d x z \\ \dot{y} &= f y + k x - x z \\ \dot{z} &= c z - eps x^{2} + x y \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`40`	x–y coupling gain
`c`	`0.833`	z self-feedback
`d`	`0.5`	x·z term strength
`eps`	`0.65`	x² term strength
`f`	`20`	y self-feedback
`k`	`0`	y forcing gain

State variables: x, y, z

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - a + c + d z + f$
state-dependent

Equilibria

1 equilibria
0 stable · 1 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Tsucs2()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Pan, Zhou & Li (2013), Nonlinear Dyn. 73, 1965-1976
doi:10.1007/s11071-013-0922-8

BibTeX

@misc{tsucs2,
  title = {Tsucs2 system},
  note = {Pan, Zhou & Li (2013), Nonlinear Dyn. 73, 1965-1976},
  doi = {10.1007/s11071-013-0922-8}
}