Determining the Lyapunov Spectrum of Continuous-Time 1D and 2D Multiscroll Chaotic Oscillators via the Solution of 𝑚-PWL Variational Equations

Table 1

Numerical simulation results of the proposed algorithm.

Case	Chua’s oscillator (2-scrolls)	1D 6-scroll chaotic oscillator			2D 4-scroll chaotic oscillator
	Chua-FS-ELS	Normalized design	ELS design	FS-ELS design	2D-FS-ELS design
Circuit elements	𝐶1=450 pF,; 𝐶2=1.5 nF,; 𝐿=1 mH,; 𝑅=1620 Ω,; 𝑔1=1/1358,; 𝑔2=1/2464,; HP1 = 0.114 V,; HP2 = 0.4 V.	𝑅𝑝=7  kΩ,; 𝑅𝑥,𝑅𝑦,𝑅𝑧=10  kΩ,; 𝑅𝑖𝑥=10  kΩ,; 𝐶=142.85 𝜇F,; 𝑉sat=9 V,; 𝑅𝑐=10 kΩ,; 𝑅𝑓=9 kΩ,; 𝑅𝑖=1 kΩ.	𝑅𝑝=7 kΩ,; 𝑅𝑥,𝑅𝑦,𝑅𝑧=10 kΩ,; 𝑅𝑖𝑥=10 kΩ,; 𝐶=142.85 μF,; 𝑉sat=9 V,; 𝑅𝑐=360 kΩ,; 𝑅𝑓=3.6 MΩ,; 𝑅𝑖=1 kΩ.	𝑅𝑝=7 kΩ,; 𝑅𝑥,𝑅𝑦,𝑅𝑦=10 kΩ,; 𝑅𝑖𝑥=10 kΩ,; 𝐶=2.2 nF,; 𝑉sat=9 V,; 𝑅𝑐=360 kΩ,; 𝑅𝑓=3.6 MΩ,; 𝑅𝑖=1 kΩ.	𝑅𝑝=7 kΩ,; 𝑅𝑥,𝑅𝑦,𝑅𝑧=10 kΩ,; 𝑅𝑖𝑥=𝑅=10 kΩ,; 𝐶=1.4 nF,; 𝑉sat=9 V,; 𝑅𝑐=360 kΩ,; 𝑅𝑓=3.6 MΩ,; 𝑅𝑖1=1 kΩ.
Initial conditions	𝑉𝐶1(0)=0.01 V,; 𝑉𝐶2(0)=0 V,; 𝐼𝐿(0)=0 A,; Γ(𝟎)=𝐈.	𝑉𝑥(0)=0.5 V,; 𝑉𝑦(0)=0 V,; 𝑉𝑧(0)=0 V,; Γ(𝟎)=𝐈.	𝑉𝑥(0)=0.00125 V,; 𝑉𝑦(0)=0 V,; 𝑉𝑧(0)=0 V,; Γ(𝟎)=𝐈.	𝑉𝑥(0)=0.00125 V,; 𝑉𝑦(0)=0 V,; 𝑉𝑧(0)=0 V,; Γ(𝟎)=𝐈.	𝑉𝑥(0)=0.00125 V,; 𝑉𝑦(0)=0.000625 V,; 𝑉𝑧(0)=0 V,; Γ(𝟎)=𝐈.
Frequency bandwidth	𝝎BW=1/𝑅𝐶1 411𝑒3 rad/s	𝝎BW=1/𝑅𝑝𝐶; 1 rad/s	𝝎BW=1/𝑅𝑝𝐶; 1 rad/s	𝝎BW=1/𝑅𝑝𝐶  65𝑒3 rad/s	𝝎BW=1/𝑅𝑝𝐶  102𝑒3 rad/s
Voltage dynamic range	±0.3 V	±45 V	±250 mV	±250 mV	±250 mV
Absolute minimum eigenvalue	5.3926𝑒5	1.4571	3.8333	2.4916𝑒5	4.0314𝑒5
Renormalization factor (RNF)	0.000001	0.6862	0.2608	0.000004	0.000002
Discretization time (h)	0.4 𝜇s	100 ms	70 ms	1.5 𝜇s	1 𝜇s
Orthogonalization time (4𝝅/𝝎BW)	30.5 𝜇s	12.6 s	12.6 s	193.3 𝜇s	123 𝜇s
LE1	0.2307	0.1487	0.1553	0.1426	0.1749
LE2	0.0031	0.0029	0.0030	0.0046	0.0101
LE3	−5.3050	−0.6867	−0.8511	−0.8705	−0.7598
Error	1.4𝑒−10	5𝑒−10	8𝑒−10	2𝑒−9	1𝑒−8