Implementation "bifurcation.zip" woring on SageMath for Bifurcations:

• Main File "main.sage"
• Sub File Directory "sage.lib"
  • For Lie Derivative "lie_derivative.sage"
  • For Linear Algebra Problem "linear_problem.sage"
  • For Polynomial "polynomial.sage"

Hopf Bifurcations with Fixed Multiplicities (Experiment Environment: 2.9 GHz Quad-Core Intel Core i7)

• Two Dimensional System (dimension 2, multiplicity 2)
  • $a, b, \ldots, n$: Parameters \[ \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} -ry + ax^2 + bxy + cy^2 + gx^3 + hx^2y + ixy^2 + jy^3 \\ x + sy + dx^2 + exy + fy^2 + kx^3 + lx^2y + mxy^2 + ny^3 \end{pmatrix}. \]
  • Input, Output (Computation Time: 2.13 Seconds)
• FitzHugh-Nagumo System (dimension 2, multiplicity 7)
  • $a, b, c, I$: Parameters \[ \begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} x - \frac{x^3}{3} - y + I \\ cx - by + a \end{pmatrix}. \]
  • Input, Output (Computation Time: 1 Minute 59.16 Seconds)
• Lorenz System (dimension 3, multiplicity 2)
  • $a, b, r$: Parameters \[ \begin{pmatrix} \dot{x} \\ \dot{y} \\ \dot{z} \end{pmatrix} = \begin{pmatrix} a y - ax \\ rx - y -xz \\ bz + xy \end{pmatrix}. \]
  • Parameter Point such that $(a, b, r, 0, 0, 0)$ is a Hopf Point : \[ a^2b + ab^2 - a^2r + a^2 + 2ab + b^2 - ar + a + b = 0 \wedge -abr + ab \not= 0 \wedge \\ \exists (a_1, b_1, c_1) \in \mathbb{R}^3 (2aba_1 + b^2a_1 - 2ara_1 + a^2b_1 + 2abb_1 - a^2r_1 + 2aa_1 + 2ba_1 - ra_1 + 2ab_1 + 2bb_1 - ar_1 + a_1 + b_1 > 0). \]
  • Since $a^2b + ab^2 - a^2r + a^2 + 2ab + b^2 - ar + a + b = (ab-ar+b^2+a+b)(a+1)$, note that \[ a^2b + ab^2 - a^2r + a^2 + 2ab + b^2 - ar + a + b = 0 \\ \Leftrightarrow ab-ar+b^2+a+b = 0 \vee a+1 = 0. \]
  • Case $ab-ar+b^2+a+b = 0$: Input, Output (Computation Time: 24 Minute 3.86 Seconds)
  • Case $a+1 = 0$: Input, Output (Computation Time: 1 Minute 6.21 Seconds)