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On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic

Mohamed El-Borhamy

Mohamed El-Borhamy

Department of Engineering Mathematics and Physics, Faculty of Engineering, University of TantaEgypt
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El-Borhamy, Mohamed
 und 
Nahla Mosalam

Nahla Mosalam

Department of Engineering Mathematics and Physics, Faculty of Engineering, University of TantaEgypt
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Mosalam, Nahla
  
30. März 2020
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Online veröffentlicht: 30. März 2020
Seitenbereich: 93 - 108
Eingereicht: 06. Aug. 2019
Akzeptiert: 09. Nov. 2019
DOI: https://doi.org/10.2478/amns.2020.1.00010
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© 2020 Mohamed El-Borhamy et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Motion of Lagrange’s gyroscope represented by a cylindrical body mounted on a vibrating base with periodic excitation(d(t) = do sinωt).
Motion of Lagrange’s gyroscope represented by a cylindrical body mounted on a vibrating base with periodic excitation(d(t) = do sinωt).

Fig. 2

Existence of stable limit cycle with α = 10, β = 1, c = 0.1, e = 0.05, Γ = 1 and ω = 3.
Existence of stable limit cycle with α = 10, β = 1, c = 0.1, e = 0.05, Γ = 1 and ω = 3.

Fig. 3

Solution and phase plane in case of a stable motion at α = 10, β = 1, c = 0.5, e = 0.2, Γ = 1 and ω = 3.
Solution and phase plane in case of a stable motion at α = 10, β = 1, c = 0.5, e = 0.2, Γ = 1 and ω = 3.

Fig. 4

Bifurcation diagrams in case of c = 0.5, ω = 1, e = 0 (left) and e = 0.1 (right).
Bifurcation diagrams in case of c = 0.5, ω = 1, e = 0 (left) and e = 0.1 (right).

Fig. 5

Chaos diagrams in case of c = 0.5, ω = 1, e = 0 at Γ=0.522,1,25 and 50 from to left to right.
Chaos diagrams in case of c = 0.5, ω = 1, e = 0 at Γ=0.522,1,25 and 50 from to left to right.

Fig. 6

Chaos diagrams in case of c = 0.5, ω = 1, e = 0.1 at Γ=0.529,1,25 and 50 from to left to right.
Chaos diagrams in case of c = 0.5, ω = 1, e = 0.1 at Γ=0.529,1,25 and 50 from to left to right.
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