If I have the following differential equations: { x s ˙ = γ y s + x s ( R c 2 − x s 2 − y s 2 ) y s ˙ = − γ x s + y s ( R c 2 − x s 2 − y s 2 ) that represent the coordinates x and y of a robot following a limit-cycle trajectory, How to use Euler method to find the approximations of ( x s )?

Question

If I have the following differential equations:      {                                                                      x                s                            ˙                                =          γ                      y            s                    +                      x            s                    (                      R            c            2                    −                      x            s            2                    −                      y            s            2                    )                                                                                y                s                            ˙                                =          −          γ                      x            s                    +                      y            s                    (                      R            c            2                    −                      x            s            2                    −                      y            s            2                    )                        that represent the coordinates x and y of a robot following a limit-cycle trajectory, How to use Euler method to find the approximations of (      x    s  )?

margenar0g · Accepted Answer

Why Euler&#039;s method and approximations? This is an exactly solvable system. Simply switch to polar coordinates                    x                            =        r                cos        ⁡                  θ                                    y                            =        r                sin        ⁡                  θ                    Then the system turns into                                          d            r                                d            t                                              =        r                                      (                          R        −        r                              )                                                (                          R        +        r                              )                                                                    d            θ                                d            t                                              =        −        γ            Then   θ  (  t  )  =      θ    0    −  γ    t and for the limit cycle you have   r  (  t  )  ≡  R, so the periodic solution in Cartesian coordinates is                    x        (        t        )                            =        R                cos        ⁡        (                              θ            0                    −          γ                    t                )                            y        (        t        )                            =        R                sin        ⁡        (                              θ            0                    −          γ                    t                )

If I have the following differential equations: {dot x_s=yy_s+x_s(R^2_c−x^2_s−y^2_s), dot y_s=−yx_s+y_s(R^2_c−x^2_s−y^2_s)} that represent the coordinates x and y of a robot following a limit-cycle trajectory, How to use Euler method to find the approximations of (x_s)?

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