Multiscale Dynamic

Linear ODE

• Matrix exponential flow.
• Jordan decomposition to Diagonal \(+\) Nilpotent.
  • Diagonal ( \(Re ( \lambda ) > 0\) or \(Re ( \lambda ) < 0\) ) exponential flow at most exponential (Unstable \(E^u\) or Stable \(E^s\) subspace respectively).
  • Nilpotent ( \(Re ( \lambda ) = 0\) exponential flow at most polynomial (Center subspace \(E^c\)).
  • Direct Sum decomposition \(\mathbb{R}^n = E^s \oplus E^c \oplus E^u\)

Invariant Manifold

Nonlinear system \(\dot{x} = f ( x )\), smooth vector field, has equilibrium point \(p\) if \(f ( p ) =0\).

• Matrix is hyperbolic if Direct Sum decomposition is diagonal: \(\mathbb{R}^n = E^s \oplus 0 \oplus E^u\).
• Hartman-Grobman: If equilibrium point is hyperbolic, Then in a neighborhood around p, the nonlinear system is topologically conjugate to its linearised system \(\dot{x} = Df ( x )\).
  • Stable-Unstable Manifold: Unstable \(W^u\) or Stable \(W^s\) manifold topologically conjugate to the linear Unstable \(E^u\) or Stable \(E^s\) subspace.
  • Center Manifold

Codimension 1 bifurcations of equilibrium

For \(x\) in \(\mathbb{R}\) , \(\mu\) in \(\mathbb{R}\), let \(f ( x , \mu )\) be a one parameter family of vector fields. Bifurcation theory characterize geometric transition in phase portrait with respect to \(\mu\). For hyperbolic equilibrium point \(p^*\) at \(\mu^*\),

• Implicit function theorem \(\implies\) the equilibrium perturbs smoothly to \(p^*(\mu)\) in some neighbourhood \(I\) of \(\mu^*\).
• Continuity of eigenvalues \(\implies\) the equilibrium \(p^*(\mu)\) is hyperbolic in the neighbourhood \(I\) of \(\mu^*\).

Generic way to lose hyperbolicity in a one parameter family:

• Real eigenvalue passes through zero (Saddle node bifurcation)
  • Then in a neighborhood around \(p\), the phase portrait is equivalent to its universal system \(\dot{x} = \mu - x^2\).
• Complex eigenvalue pair pass through zero (Hopf bifurcation).
• If the eigenvalue pass through zero with non-zero speed along x-direction, first Lyapunov coefficient not zero Then Hopf bifurcation.
• Hopf Bifurcation Scaling

Introduction: Van der Poly Oscillator

For \(a\) and \(\epsilon\) (small) positive constants, \(\begin{align} \dot{x} &= x - \frac{x^3}{3} + y \quad \textrm{fast}\\ \dot{y} &= \epsilon \cdot (a - x) \quad \textrm{slow} \end{align}\) There is a unique equilibrium point \(p := (a, \frac{a^3}{3})\) with Jacobian and eigenvalues \(\lambda = \frac{(1-a^2) \pm ((a^2-1)^2-4\cdot \epsilon)^{\frac{1}{2}}}{2}\)

• Along cubic: differentio-algebraic
• Away from cubic: parameter

R

• Grobman–Hartman theorem was first proved (Grobman, D. M. (1959). “О гомеоморфизме систем дифференциальных уравнений” [Homeomorphisms of systems of differential equations]. Doklady Akademii Nauk SSSR. 128: 880–881.) in 1959 by the Russian mathematician David Matveevich Grobman (born in 1922) from Moscow University, student of Nemytsckii. The next year, Philip Hartman (1915–2015) at John Hopkins University (USA) independently confirmed this result (Hartman, Philip (1960). “A lemma in the theory of structural stability of differential equations”. Proceedings of the American Mathematical Society, 11, (4): 610–620. doi:10.2307/2034720).
• Brown University, Vladimir_Dobrushkin, Methods of Applied Mathematics II course