Nonlinear system: ( y ˙ 1 y ˙ 2 ) = ( 2 y 1 y 1 2 ) Giving the jacobian of transformation being: F ˙ = ( 2 0 2 y 1 0 ) = ( 0 0 2 0 ) ( F 1 F 2 ) + ( 2 0 ) Which gives me the eigenvalues λ = 0 2 = 0 , 0 meaning a degenerate node since we have only the eigenvector: F ( 1 ) = ( 0 1 ) Non-linear system with all trajectories converging on the line x = 0, rather than ( 2 , 0 )?

Question

Nonlinear system:      (                                                                      y                ˙                                      1                                                                                          y                ˙                                      2                                )    =      (                            2                      y            1                                                            y            1            2                                )  Giving the jacobian of transformation being:            F      ˙        =      (                            2                          0                                      2                      y            1                                    0                      )    =      (                            0                          0                                      2                          0                      )        (                                        F            1                                                            F            2                                )    +      (                            2                                      0                      )  Which gives me the eigenvalues   λ  =      0    2    =  0  ,  0 meaning a degenerate node since we have only the eigenvector:      F          (      1      )        =      (                            0                                      1                      )  Non-linear system with all trajectories converging on the line   x  =  0, rather than   (  2  ,  0  )?

iletsa2ym · Accepted Answer

(                                                                      y                ˙                                      1                                                                                          y                ˙                                      2                                )    =      (                            2                      y            1                                                            y            1            2                                )  We want the critical points of this system, which we can find by taking:      (                            0                                      0                      )    =      (                            2                      y            1                                                            y            1            2                                )    (  0  ,  α  ) are the spots of the critical points, thus there are infinite critical points at   x  =  0

Rosemary Chase · Accepted Answer

You have made a mistake in your Jacobian.Since we are taking   F  =            y      ˙      , we need to be very clear that this means:      (                                        F            1                                                            F            2                                )    =      (                            2                      y            1                                                            y            1            2                                )  And hence:            F      ˙        =      (                                                                      F                ˙                                      1                                                                                          F                ˙                                      2                                )    =      (                            2                          0                                      2                      y            1                                    0                      )    =      (                            0                          0                                      1                          0                      )        (                                        F            1                                                            F            2                                )    +      (                            2                                      0                      )

Nonlinear system: (y_1y_2)=(2y_1 y^2_1) Giving the jacobian of transformation being: F=(2 0, 2y_1 0)=(0 0, 2 0)(F_1 F_2)+(2 0)

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