Table 4 Comparative analysis with absolute error of \(x\left(t\right), y\left(t\right)\text{and} z(t)\) between reference and LENN-FA-AOA solution for scenario 3.

From: Legendre based neural networks integrated with heuristic algorithms for the analysis of Lorenz chaotic model: an intelligent and comparative study

\(t\)

\(x(t)\)

\(y(t)\)

\(z(t)\)

\(\widehat{x}(t)\)

\(\widehat{y}(t)\)

\(\widehat{z}(t)\)

\(A{E}_{x(t)}\)

\(A{E}_{y(t)}\)

\(A{E}_{z(t)}\)

0

0.3

0.3

0.3

0.299994

0.299992

0.3

6.32E-06

8.31E-06

4.63E-07

0.1

0.299609

0.273955

0.324204

0.29961

0.273924

0.324208

1.3E-06

3.1E-05

3.25E-06

0.2

0.298486

0.249687

0.348856

0.298487

0.249635

0.348859

1.1E-06

5.17E-05

3.13E-06

0.3

0.296703

0.227014

0.374017

0.296694

0.226993

0.374003

9.17E-06

2.05E-05

1.32E-05

0.4

0.294325

0.205775

0.399751

0.294314

0.205781

0.399737

1.11E-05

6.19E-06

1.39E-05

0.5

0.291408

0.185829

0.426128

0.291407

0.185826

0.426134

1.8E-06

2.59E-06

6.48E-06

0.6

0.288006

0.167047

0.453219

0.288012

0.167021

0.453231

5.54E-06

2.66E-05

1.19E-05

0.7

0.284166

0.149318

0.481097

0.284166

0.149287

0.481081

2.11E-07

3.15E-05

1.63E-05

0.8

0.279929

0.132543

0.509841

0.279916

0.132529

0.509802

1.28E-05

1.36E-05

3.83E-05

0.9

0.275335

0.116633

0.539529

0.275317

0.116628

0.539526

1.75E-05

4.47E-06

3.25E-06

1

0.270418

0.101509

0.570244

0.270411

0.101495

0.570263

6.89E-06

1.39E-05

1.84E-05