Table 3 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 2.

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

0.4

0.4

0.4

0.400001

0.399997

0.399992

1.23E-06

3.21E-06

7.73E-06

0.1

0.399573

0.357514

0.431772

0.399579

0.357509

0.431791

6.28E-06

4.62E-06

1.95E-05

0.2

0.398342

0.317882

0.463138

0.398361

0.317881

0.463137

1.94E-05

8.59E-07

1.04E-06

0.3

0.396377

0.280865

0.494175

0.396391

0.280858

0.494162

1.44E-05

6.76E-06

1.27E-05

0.4

0.393742

0.246248

0.524965

0.393741

0.246238

0.52498

7.23E-07

1.03E-05

1.49E-05

0.5

0.390496

0.213839

0.555592

0.390488

0.213834

0.555624

8.42E-06

4.6E-06

3.25E-05

0.6

0.386693

0.183464

0.586142

0.386692

0.183464

0.586152

7.04E-07

7.27E-08

1.03E-05

0.7

3.82E-01

1.55E-01

6.17E-01

3.82E-01

1.55E-01

6.17E-01

1.42E-05

5.36E-06

2.27E-05

0.8

0.377611

0.128206

0.647359

0.37763

0.128193

0.647341

1.86E-05

1.26E-05

1.76E-05

0.9

0.37242

0.103052

0.678202

0.372424

0.103044

0.678231

4.05E-06

7.25E-06

2.92E-05

1

0.366849

0.079388

0.70932

0.366847

0.079386

0.709349

2.14E-06

2.52E-06

2.89E-05