The Related Extension and Application of the Ši'lnikov Theorem

Table 3

The systems with three and more equilibrium points.

System	Typical parameter values	The equivalent points	The eigenvalues
Lorenz	𝑎=10,𝑏=8/3,𝑐=28	(0,0,0); (±6√2,±6√2,27)	−22.8277,−2.6667,11.8277; −13.8546,0.094±1.3585𝑖
Rucklidge	𝑎=−2,𝑏=−6.7	(0,0,0); (0,±2.5884,6.7)	−3.7749, −1, 1.7749; −3.5154,0.2577±1.9353𝑖
Chen	𝑎=0.3	(0,0,0); (±3√7,±3√7,21)	−30.8359,−1,1.7749; −18.4288,4.214±14.8846𝑖
SM	𝑎=0.85,𝑏=0.5	(0,0,0); (±√2/2,0,1)	−1.5116,−0.5,0.6616; 1.5079,0.079±0.8105𝑖
Lü	𝑎=36,𝑏=3,𝑐=20	(0, 0, 0); (±2√15,±2√15,20)	−36,20,−3; −22.6516,1.8258±13.6867𝑖
Modified Lorenz	𝑎=10,𝑏=8/3,𝑐=28	(0, 0, 0); (±3√2/5,±3√2/5,1.35)	−22.8277,11.8277,−2.6667; −13.8546,0.094±10.1943𝑖
The united system	𝛼=0.5	(0, 0, 0); (±2√17,±2√17,24)	−28.1696, 19.1696, −2.8333; −16.9593,2.563±13.1857𝑖
Zhou(b)	𝑎1=−4.1,𝑎2=1.2,𝑎3=13.45; 𝑐1=2.76,𝑐2=0.6,𝑐3=13.13; 𝑑=1.6	(0.9211,−1.9226,0.1092); (−0.7106,0.8407,−0.1416); (−0.112,−9.6873,−0.8984); (−0.0985,−11.1136,−1.0216)	−0.9776,4.427±3.8675𝑖; 15.0739,−5.7162,0.1768; 0.0041,1.6068±18.1354𝑖; −0.004,1.1829±19.6666𝑖
Zhou(d)	𝑎1=0.5,𝑎2=0.1,; 𝑏1=−12,𝑏2=−0.062,; 𝑏3=−6.79	(0,0,0); (±8.9676,∓1.8808,−2.484); (∓9.0944,±1.8056,3.327)	−12,−6.79,0.5; −18.6583,0.2611±2.9646𝑖; −18.8001,0.1519±3.4702𝑖
liu	𝑎=5,𝑏=−10; 𝑐=−3.4,𝑑1=−1; 𝑑2=𝑑3=1	(0,0,0); (√34,±√17,±5√2); (−√34,±√17,∓5√2)	−10, −3.4, 5; −12.6496,2.1248±7.0172𝑖
CL	𝑎=5,𝑏=10,𝑐=3.8	(0,0,0); (±√114,±√57,√50); (±√114,∓√57,−√50)	−10, −3.8, 5; −13.177,2.1885±7.2723𝑖
NL	𝑎=0.4,𝑏=0.175	(0,0,0); (∓0.0705,±0.2737,−0.1103); (±0.5352,±0.0689,0.2107)	0.175,−0.4±𝑖; −0.7997,0.0874±0.8754𝑖; −0.803,0.089±1.2118𝑖