THE ROSSLER ATTRACTOR

<span class='text_page_counter'>(1)</span>THE ROSSLER ATTRACTOR The Tran. The Rossler system.  x' = − y − z  '  y = x + ay z' = b + z x − c ( ) . where a=1/4; b=1;. c ∈ [0, 7]. 1) Find all equilibrium.  x' = − y − z z = − y  ' −c ± c 2 − 4ab  ⇒  x = −ay ⇒y=  y = x + ay 2a z' = b + z x − c b + cy + ay 2 = 0 ( )   1 where a = ; b = 1; c ∈ [ 0, 7 ] 4 ⇒ y = −2c ± 2 c 2 − 1 + c < 1: No equilibrium. + c = 1:. 1 1  y = −2c ± 2 c 2 − 1 ⇒ y = −2 ; x = ; z = 2 so P1 =  ; − 2 ; 2  2 2 .

<span class='text_page_counter'>(2)</span> + c > 1:. y = −2c ± 2 c 2 − 1; x =. 1 c∓ 2. (. ). c 2 − 1 ; z = 2c ± 2 c 2 − 1. 1  ⇒ P2 =  c − c 2 − 1 ; − 2c + 2 c 2 − 1 ; 2c − 2 c 2 − 1  2  1  ⇒ P3 =  c + c 2 − 1 ; − 2c − 2 c 2 − 1 ; 2c + 2 c 2 − 1  ; c ∈ (1, 7] 2 . ( (. ) ). 2) Describe the bifurcation that occurs at c=1 Since a = 1/4; b = 1; c = 1.  x' = − y − z  1  ' y = x + y 4  '  z = 1 + z ( x − 1) Get the matrix from the equation system. 1  0 −1 −1  λ    1 1 0  ⇒ det ( λ I − A ) = det  −1 λ − A = 1 4 4     z 0 x − 1 − z 0    3 1 3  1  = λ3 −  − x  λ2 +  x + + z  λ +1− x − z = 0 4 4 4  4  + c = 1:. 1 x = ; y = −2 ; z = 2 2.   0   λ − x + 1 1.

<span class='text_page_counter'>(3)</span> The codes in Mathematica 7.0. ⇒ λ ( 8λ 2 + 2λ + 23) = 0 The eigenvalues of the linearized dynamics are found ⇒ λ = 0; λ =. 1 1 −1 − i 183 ; λ = −1 + i 183 8 8. (. ). (. ). There are two complex eigenvalues with negative real parts, so the equilibrium is an unstable.. The Graph in 3D space where c =1 The codes in Mathematica 7.0 f={X,Y,Z}/.NDSolve[{ X'[t]
-(Y[t]+Z[t]), Y'[t]
X[t]+a Y[t], Z'[t]
b+X[t]Z[t]-c Z[t], X[0]
1,Y[0]
1,Z[0]
1}/.{a→0.25,b→1,c→1}, {X,Y,Z},{t,0,50}][[1]] ParametricPlot3D[Evaluate[Append[#[t]&/@f,Red]],{t,0,50},BoxRatios→{1 ,1,1},PlotRange→All,PlotPoints→1500,AxesLabel→TraditionalForm/@l].

<span class='text_page_counter'>(4)</span> 1  ⇒ P2 =  c − c 2 − 1 ; − 2c + 2 c 2 − 1 ; 2c − 2 c 2 − 1  2 . (. ).     λ 1 1   1   det ( λ I − A ) = det −1 0 λ−   4   1  −2c + 2 c 2 − 1 λ − c − c 2 − 1 + 1 0  2 . (. f (λ ) =. The codes in Mathematica 7.0. ).

<span class='text_page_counter'>(5)</span> 1 ⇒ P3 =  c + 2. (.  c 2 − 1 ; − 2c − 2 c 2 − 1 ; 2c + 2 c 2 − 1  . ).   λ 1  1 det ( λ I − A ) = det  −1 λ−  4   −2c − 2 c 2 − 1 0 .   1   0   1 λ − c + c 2 − 1 + 1 2 . ). (. f (λ ) =.   λ 1  1 det ( λ I − A ) = det  −1 λ−  4   −2c − 2 c 2 − 1 0 .     0   1 λ − c + c 2 − 1 + 1 2 . The codes in Mathematica 7.0. 1. (. ).

<span class='text_page_counter'>(6)</span> 3) Investige numerically the behavior of this system as c increases. What bifurcations do you observe? WHERE C =0 TO 7 f={X,Y,Z}/.NDSolve[{ X'[t]
-(Y[t]+Z[t]), Y'[t]
X[t]+a Y[t], Z'[t]
b+X[t]Z[t]-c Z[t], X[0]
1,Y[0]
1,Z[0]
1}/.{a→0.25,b→1,c→3}, {X,Y,Z},{t,0,50}][[1]] ParametricPlot3D[Evaluate[Append[#[t]&/@f,Red]],{t,0,50},BoxRatios→{1 ,1,1},PlotRange→All,PlotPoints→1500,AxesLabel→TraditionalForm/@l].

<span class='text_page_counter'>(7)</span> 4) In Figure 14.12 we have plotted a single solution for c= 5.5. Computer other solutions for this parameter value, and display the results from other viewpoint R3. What conjectures do you make about the behavior of this system? + c = 5.5:. 1 11 ∓ 117 ; z = 11 ± 117 4 1   ⇒ P2 =  −11 + 117 ; 11 − 117 ;11 + 117  4   y = −11 ± 117; x =. (. ). (. ). 1   ⇒ P3 =  −11 − 117 ; 11 + 117 ;11 − 117  4  . (. ). f={X,Y,Z}/.NDSolve[{ X'[t]
-(Y[t]+Z[t]), Y'[t]
X[t]+a Y[t], Z'[t]
b+X[t]Z[t]-c Z[t], X[0]
1,Y[0]
1,Z[0]
1}/.{a→0.25,b→1,c→5.5}, {X,Y,Z},{t,0,50}][[1]] ParametricPlot3D[Evaluate[Append[#[t]&/@f,Red]],{t,0,50},BoxRatios→{1 ,1,1},PlotRange→All,PlotPoints→1500,AxesLabel→TraditionalForm/@l].

<span class='text_page_counter'>(8)</span> Figure 14.12 The Rossler attractor. + Display the results from other viewpoint R3.

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