Equations aux Différences dans R
Sandrine CHARLES - [email protected] Arnaud CHAUMOT [email protected] Christelle LOPES [email protected]
3 octobre 2016
Bifurcation diagram of the logistic mapXn+ 1 =rXn(1−Xn)
May RM. 1976. Simple mathematical models with very complicated dynamics.Nature:459–467.
0.0 0.2 0.4 0.6 0.8 1.0 2
4 6 8 10 12
x
(
t)
h = 0.1 h = 0.2
Solution exacte
−2 −1 0 1 2
−3
−2
−1 0 1 2 3
x
f
(
x)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
xn xn+1
0.0 x0 0.2 0.4 x * 0.8 1.0
λ = −1.5 Répulsif
λ = −1 Stable
λ = −0.5
Asymptotiquement stable
λ =0
Asymptotiquement stable
λ =0.5
Asymptotiquement stable λ =1
Asymptotiquement stable λ =1.5
Répulsif
−2 −1 0 1 2
−2
−1 0 1 2
xn xn+1=f(xn)
2 4 6 8 10
−3
−2
−1 0 1 2 3
n xn
−4 −3 −2 −1 0 1 2
−3
−2
−1 0 1 2
xn xn+1=f(xn)
2 4 6 8 10
−3
−2
−1 0 1 2 3
n xn
0 10 20 30 40 50
nt
λ =1.1 λ =1 λ =0.9
0 10 20 30 40 50 0.0
0.2 0.4 0.6 0.8 1.0
λ =1.5
xn
xE=0.333333
n Convergence directe
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
λ =2.7
xn
xE=0.62963
n Convergence avec oscillations
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
λ =3.1
xn
xE=0.677419
n Cycles de période 2
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
λ =3.51
xn
xE=0.7151
n Cycles de période 4
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
λ =3.6
xn
xE=0.722222
n Régime quasi−périodique
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
λ =3.892
xn
xE=0.743063
n Régime chaotique (1)
0 10 20 30 40 50
0.0 0.2 0.4 0.6 0.8 1.0
λ =3.892
xn
xE=0.743063
n Régime chaotique (2)
0 20 40 60 80
λ
1 e1 e2
N2* n'existe pas biologiquement N2* A.S. (convergence directe) N2* A.S. (oscillations)
N2* instable
Punaise Moustique
Scarabé
