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Dynamics of the Modified n-Degree Lorenz System

Sk. Sarif Hassan
Sk. Sarif Hassan
, 
Moole Parameswar Reddy
Moole Parameswar Reddy
 und 
Ranjeet Kumar Rout
Ranjeet Kumar Rout
   | 22. Aug. 2019
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Online veröffentlicht: 22. Aug. 2019
Seitenbereich: 315 - 330
Eingereicht: 24. März 2019
Akzeptiert: 24. Mai 2019
DOI: https://doi.org/10.2478/AMNS.2019.2.00028
Schlüsselwörter
© 2019 Sk. Sarif Hassan et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 Public License.

Fig. 1

First, second and third rows from top, x-stacked planes, y-stacked planes and z-stacked planes respectively are shown.
First, second and third rows from top, x-stacked planes, y-stacked planes and z-stacked planes respectively are shown.

Fig. 2

Parameter Subspace of ℝ3 such that the origin is locally asymptotically stable.
Parameter Subspace of ℝ3 such that the origin is locally asymptotically stable.

Fig. 3

Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (0,0,0) where the initial values are taken from the neighbourhood of the origin.
Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (0,0,0) where the initial values are taken from the neighbourhood of the origin.

Fig. 4

Unstable trajectories which are away from the origin in the system D56,15,16[6].
Unstable trajectories which are away from the origin in the system D56,15,16[6].

Fig. 5

Unstable trajectories (limit cycle) which are away from the origin in the system D26,93,36[5].
Unstable trajectories (limit cycle) which are away from the origin in the system D26,93,36[5].

Fig. 6

Unstable trajectories (limit cycle) which are away from the origin in the system D9,7,54[16].
Unstable trajectories (limit cycle) which are away from the origin in the system D9,7,54[16].

Fig. 7

Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (−8.84531,−0.315904,17) where the initial values are taken from the neighbourhood of the origin.
Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (−8.84531,−0.315904,17) where the initial values are taken from the neighbourhood of the origin.

Fig. 8

Unstable trajectories which are away from the equilibrium point (3.65333, 0.405925, 41) in the system D52,41,9[15].
Unstable trajectories which are away from the equilibrium point (3.65333, 0.405925, 41) in the system D52,41,9[15].

Fig. 9

Parameter Plot
Parameter Plot

Fig. 10

Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (8.84531, 0.315904, 17) where the initial values are taken from the neighbourhood of the origin.
Left: Locally asymptotically stable trajectories and in Right: the 3D plot of the trajectories of the equilibrium point (8.84531, 0.315904, 17) where the initial values are taken from the neighbourhood of the origin.

Fig. 11

Unstable trajectories which are away from the equilibrium point (3.76576, 0.418418, 41) in the system D52,41,9[13].
Unstable trajectories which are away from the equilibrium point (3.76576, 0.418418, 41) in the system D52,41,9[13].

Fig. 12

Unstable trajectories which are away from the all the equilibrium points of the system D52,41,9[n].
Unstable trajectories which are away from the all the equilibrium points of the system D52,41,9[n].

Fig. 13

Trajectories (Up) and three dimensional plots (Down) respectively for the system D51,3,4[n] where the degree of the system n varying from 1 to 15.
Trajectories (Up) and three dimensional plots (Down) respectively for the system D51,3,4[n] where the degree of the system n varying from 1 to 15.

Fig. 14

Trajectories are plotted for the system D76,36,31[n] where the degree of the system n varying from 1 to 15.
Trajectories are plotted for the system D76,36,31[n] where the degree of the system n varying from 1 to 15.

Fig. 15

Trajectories and 3D plots are plotted for the system D3,12,24[n] where the degree of the system n varying from 1 to 10.
Trajectories and 3D plots are plotted for the system D3,12,24[n] where the degree of the system n varying from 1 to 10.

Transition of the dynamical behavior of the system D51,3,4[n] according as the degree of the system n varying from 1 to 15.

Sl. No. System D51,3,4[n] Character of the System
1 D76,36,31[1] Periodic with high period.
2 D76,36,31[2] Chaotic with Positive Lyapunov exponent 0.6324
3 D76,36,31[3] Chaotic with Positive Lyapunov exponent 0.7634
4 D76,36,31[4] Chaotic with Positive Lyapunov exponent 0.4589
5 D76,36,31[5] Chaotic with Positive Lyapunov exponent 0.3412
6 D76,36,31[6] Chaotic with Positive Lyapunov exponent 0.2875
7 D76,36,31[7] Chaotic with Positive Lyapunov exponent 0.2337
8 D76,36,31[8] Chaotic with Positive Lyapunov exponent 0.5643
9 D76,36,31[9] Chaotic with Positive Lyapunov exponent 0.2981
10 D76,36,31[10] Chaotic with Positive Lyapunov exponent 0.6873
11 D76,36,31[11] Chaotic with Positive Lyapunov exponent 0.5438
12 D76,36,31[12] Chaotic with Positive Lyapunov exponent 0.9865
13 D76,36,31[13] Chaotic with Positive Lyapunov exponent 0.7541
14 D76,36,31[14] Chaotic with Positive Lyapunov exponent 0.6987
15 D76,36,31[15] Chaotic with Positive Lyapunov exponent 0.3086

The system D51,3,4[n] behavior according as the degree of the system n varying from 1 to 15.

Sl. No. System D51,3,4[n] Character of the System
1 D51,3,4[1] converges to the equilibrium point (6.9282, 1.7321, 3)
2 D51,3,4[2] converges to the equilibrium point (3.7225, 0.9306, 3)
3 D51,3,4[3] converges to the equilibrium point (3.0262, 0.7565, 3)
4 D51,3,4[4] converges to the equilibrium point (2.7285, 0.6821, 3)
5 D51,3,4[5] Periodic with high period.
6 D51,3,4[6] Periodic with high period.
7 D51,3,4[7] Periodic with high period.
8 D51,3,4[8] Periodic with high period.
9 D51,3,4[9] Periodic with high period.
10 D51,3,4[10] Periodic with high period.
11 D51,3,4[11] Periodic with high period.
12 D51,3,4[12] Periodic with high period.
13 D51,3,4[13] Periodic with high period.
14 D51,3,4[14] Periodic with high period.
15 D51,3,4[15] Periodic with high period.
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