3.5 The flow and dependence on initial data
Write \(\phi (t; t_0, x_0)\) for the maximal solution with \(x(t_0) = x_0\). For an autonomous system \(x' = f(x)\), write \(\phi _t(x_0) = \phi (t; 0, x_0)\), the flow: \(\phi _t\) maps the state at time \(0\) to the state at time \(t\).
Theorem 3.5.1. \(\phi \bigl (t; s, \phi (s; t_0, x_0)\bigr ) = \phi (t; t_0, x_0)\) whenever both sides are defined. For an autonomous system, \(\phi _0 = \mathrm {id}\) and \(\phi _t \circ \phi _s = \phi _{t+s}\).
Proof. Both sides of the first identity are solutions of \(x' = f(t, x)\), as functions of \(t\), and they agree at \(t = s\); by Lemma 3.1.1 they agree wherever both are defined. For an autonomous system, \(t \mapsto \phi _{t+s}(x_0)\) is a solution (the equation does not involve \(t\)) with value \(\phi _s(x_0)\) at \(t = 0\), so it equals \(\phi _t(\phi _s(x_0))\). □
The flow is the central object of the geometric theory: a state determines its future and its past uniquely, and \(\{\phi _t\}\) is a one-parameter group of maps of the state space. The final question is how smooth \(\phi _t\) is.
Theorem 3.5.2 (Differentiability with respect to initial values). Let \(f\) and \(D_xf\) be continuous on \(D\), and let \(\phi (t; x_0)\) denote the solution with \(x(t_0) = x_0\), on a compact interval \(J \ni t_0\) on which it exists. Then \(\phi \) is continuously differentiable in \(x_0\), and \(Y(t) = \partial \phi (t; x_0)/\partial x_0\), an \(n \times n\) matrix, is the solution of the variational equation \[ Y' = A(t)Y, \qquad Y(t_0) = I, \qquad A(t) = D_xf\bigl (t, \phi (t; x_0)\bigr ) . \]
Proof. By continuous dependence (Theorem 2.4.2), for \(\|h\|\) small the solution \(\phi (t; x_0 + h)\) also exists on \(J\) and stays in a compact neighbourhood \(K\) of the graph of \(\phi (\cdot \,; x_0)\), on which \(f\) is Lipschitz with constant \(L\); and \(\psi _h(t) = \phi (t; x_0 + h) - \phi (t; x_0)\) satisfies \(\|\psi _h(t)\| \leq \|h\|e^{L|t - t_0|}\).
Write \(x(t) = \phi (t; x_0)\). By the integral form of the mean value theorem, \[ \psi _h' = f(t, x + \psi _h) - f(t, x) = A(t)\psi _h + r_h(t), \qquad r_h = \int _0^1\bigl [D_xf(t, x + \theta \psi _h) - D_xf(t, x)\bigr ]\psi _h\,d\theta . \] Since \(D_xf\) is uniformly continuous on \(K\) and \(\|\psi _h\|_\infty \leq \|h\|e^{L|J|}\), we have \(\|r_h(t)\| \leq \varepsilon (h)\|h\|\), with \(\varepsilon (h) \to 0\) as \(h \to 0\). The difference \(e_h = \psi _h - Yh\) satisfies \(e_h' = A(t)e_h + r_h\) and \(e_h(t_0) = 0\), so with \(\|A(t)\| \leq L\), \[ \|e_h(t)\| \leq \int _{t_0}^t\bigl (L\|e_h(s)\| + \varepsilon (h)\|h\|\bigr )ds , \] and Gronwall’s inequality gives \(\|e_h(t)\| \leq \varepsilon (h)\|h\|\,|J|e^{L|J|} = o(\|h\|)\). This is the statement that \(\phi \) is differentiable in \(x_0\) with derivative \(Y\). The derivative is continuous in \(x_0\) because \(Y\) solves a linear system whose coefficient matrix \(A(t)\) depends continuously on \(x_0\), and Theorem 2.4.2 applies to it. □
The variational equation is the linearisation of the system along a solution. Its solution \(Y(t)\) measures how an infinitesimal change in the initial state is stretched and rotated by time \(t\), and it underlies sensitivity analysis, Newton’s method for boundary value problems (the shooting method), the stability of periodic orbits through Floquet theory, and Lyapunov exponents in chaotic systems. Parameters are handled by enlarging the state: if \(f = f(t, x, \lambda )\), adjoin the equation \(\lambda ' = 0\); then \(Z = \partial x/\partial \lambda \) satisfies \[ Z' = D_xf\,Z + D_\lambda f, \qquad Z(t_0) = 0 . \] If \(f\) is \(C^k\), the solution is \(C^k\) in \((t, x_0, \lambda )\), by induction on \(k\).
Example 3.5.3. For \(x' = x^2\), \(\phi (t; x_0) = x_0/(1 - x_0t)\), so \(\partial \phi /\partial x_0 = (1 - x_0t)^{-2}\). The variational equation is \(Y' = 2\phi Y\), \(Y(0) = 1\), with solution \(Y = \exp \int _0^t\frac {2x_0}{1 - x_0s}\,ds = \exp \bigl (-2\log (1 - x_0t)\bigr ) = (1 - x_0t)^{-2}\), as it should be. Sensitivity to the initial value grows without bound as \(t\) approaches the blow-up time.
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