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Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics

Marina Esteban

Marina Esteban

Departamento de Matemática Aplicada II, Universidad de Sevilla, Escuela Técnica Superior de IngenieríaSevilla, Spain
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Esteban, Marina
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Enrique Ponce

Enrique Ponce

Departamento de Matemática Aplicada II, Universidad de Sevilla, Escuela Técnica Superior de IngenieríaSevilla, Spain
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Ponce, Enrique
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Francisco Torres

Francisco Torres

Departamento de Matemática Aplicada II, Universidad de Sevilla, Escuela Técnica Superior de IngenieríaSevilla, Spain
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Torres, Francisco
  
04 nov 2017
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Publicado en línea: 04 nov 2017
Páginas: 449 - 464
Recibido: 06 abr 2017
Aceptado: 04 nov 2017
DOI: https://doi.org/10.21042/AMNS.2017.2.00036
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© 2017 Marina Esteban, Enrique Ponce, Francisco Torres, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

The transition mechanism between systems (SU) and (SL). We use blue colour for solutions of the upper system and red colour for solutions of the lower one. We can see also the falls at the line Σ+ and the rises at the line Σ−.
The transition mechanism between systems (SU) and (SL). We use blue colour for solutions of the upper system and red colour for solutions of the lower one. We can see also the falls at the line Σ+ and the rises at the line Σ−.

Fig. 2

The ‘graph’ of a normalized hysteresis function. The hysteresis value H(x) is unambiguous for x < −1 and x > 1. However, for −1 ⩽ x ⩽ 1 the output depends on the past, as explained in the text.
The ‘graph’ of a normalized hysteresis function. The hysteresis value H(x) is unambiguous for x < −1 and x > 1. However, for −1 ⩽ x ⩽ 1 the output depends on the past, as explained in the text.

Fig. 3

The saddle point (xE, yE) and its invariant manifolds for the upper system (SU). Other distinguished values are emphasized.
The saddle point (xE, yE) and its invariant manifolds for the upper system (SU). Other distinguished values are emphasized.

Fig. 4

Bifurcation set in the parameter plane (γ, yE) for xE < −1. We emphasized the number of symmetric periodic orbits in each region.
Bifurcation set in the parameter plane (γ, yE) for xE < −1. We emphasized the number of symmetric periodic orbits in each region.

Fig. 5

The transition map U for different values of the parameter yE and γ ∈ (−1, 0). The black points in this figure represent the point u*=(u−*,u+*) $u^*=(u^*_-,u^*_+)$ for each case.
The transition map U for different values of the parameter yE and γ ∈ (−1, 0). The black points in this figure represent the point u*=(u−*,u+*) $u^*=(u^*_-,u^*_+)$ for each case.

Fig. 6

The transition map U for different values of yE. Again, the terminal black points in this figure represent the point u*=(u−*,u+*) $u^*=(u^*_-,u^*_+)$ for each case.
The transition map U for different values of yE. Again, the terminal black points in this figure represent the point u*=(u−*,u+*) $u^*=(u^*_-,u^*_+)$ for each case.

Fig. 7

The two symmetric periodic orbits existing for (xE, yE) = (−2, −1) and γ = 0.8. One of them takes the three zones and is unstable. The other one takes only the central zone and is stable. The blue lines (resp. red lines) correspond to valid solutions for the SU−system (resp. SL−system).
The two symmetric periodic orbits existing for (xE, yE) = (−2, −1) and γ = 0.8. One of them takes the three zones and is unstable. The other one takes only the central zone and is stable. The blue lines (resp. red lines) correspond to valid solutions for the SU−system (resp. SL−system).

Fig. 8

Typical graph for φ(Z) and surfaceX=φ(Z).
Typical graph for φ(Z) and surfaceX=φ(Z).
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