Unpredictability in Complex Systems — Epoche B2
Beyond Smoothness: Unpredictability in Complex Systems The intuition classical mechanics leaves behind is that a smooth law of motion delivers a predictable world: write down a differential equation whose right-hand side is a nice function, supply the initial state, and the future follows. The intuition is not baseless — the existence-and-uniqueness theorem really does say something close to it — but it is far weaker than it feels. Hadamard's three conditions for a problem to be well posed are that a solution exists [1] , that it is unique, and that it depends continuously on the data, and each of the three can fail while the equation stays perfectly smooth. This essay takes them one at a time: a smooth ordinary differential equation whose solution ceases to exist after a finite time; a smooth conservation law whose solutions stop being unique the moment they stop being differentiable; and, in the case where existence and uniqueness both hold, a bound on continuous dependence that is exponential, so that "continuous" and "usable" are different claims. What smoothness actually buys Write the system as $$ \frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x},t),\qquad \mathbf{x}(t_0)=\mathbf{x}_0, $$ where $\mathbf{x}(t)$ is the state vector, listing every variable needed to specify the system, and $\mathbf{f}$ gives its rate of change. The Picard-Lindelöf theorem supplies the guarantee: if $\mathbf{f}$ is continuous in $t$ and Lipschitz in $\mathbf{x}$, meaning there is a constant $L$ with $|\mathbf{f}(\mathbf{x},t)-\mathbf{f}(\mathbf{y},t)|\le L\,|\mathbf{x}-\mathbf{y}|$ for all states in the region of interest, then a unique solution exists on some interval around $t_0$, and it is continuously differentiable in $t$. Note what is and is not claimed: the interval may be short, and "continuously differentiable" is not "smooth" in the sense of infinitely differentiable, which would require correspondingly more of $\mathbf{f}$. The theorem also controls how two solutions can separate, and this is the part usually misremembered. Let $\Delta\mathbf{x}(t)$ be the difference between two solutions of the same equation started a distance $|\Delta\mathbf{x}(0)|$ apart. Subtracting the two integral equations and applying the Lipschitz bound gives $|\Delta\mathbf{x}(t)|\le|\Delta\mathbf{x}(0)|+L\int_0^t|\Delta\mathbf{x}(\tau)|\,d\tau$, and Grönwall's inequality converts that into $$ \left|\Delta\mathbf{x}(t)\right|\;\le\;\left|\Delta\mathbf{x}(0)\right|\,e^{L\,t}. $$ Here $L$ is the Lipschitz constant, with units of inverse time. Continuous dependence on initial data is exactly this statement, and it is exponential . Smoothness does not forbid neighbouring trajectories from separating exponentially; it merely caps the rate at $L$. A system can be as smooth as one likes and still double the distance between two states every $\ln 2/L$ units of time. Whether the bound is attained or wildly pessimistic is a further question about the particular equation — and it is the question that separates the predictable systems from the rest. Where existence fails: finite-time blow-up Consider the smoothest possible nonlinearity: $$ \frac{dx}{dt}=x^{2},\qquad x(0)=x_0\gt0 . $$ The right-hand side is a polynomial, so it is Lipschitz on every bounded set and Picard-Lindelöf applies. Separating variables gives $-1/x=t+C$, and $x(0)=x_0$ fixes $C=-1/x_0$, so $$ x(t)=\frac{x_0}{1-x_0t},\qquad t_{*}=\frac{1}{x_0}. $$ The solution is perfectly well behaved until $t_{*}$ and does not exist at all afterwards: $x(t)\to\infty$ as $t\to t_{*}^{-}$. For $x_0=1$ the blow-up is at $t_{*}=1$ (Fig. 1); for $x_0=100$ it is at $t_{*}=0.01$. The generalisation is worth having: for $\dot x=x^{p}$ with $p\gt1$ the blow-up time is $t_{*}=x_0^{1-p}/(p-1)$, which is finite for every $p\gt1$ and infinite at $p=1$, where the equation is linear and the solution merely grows like $e^{t}$ forever. Nonlinearity of any degree above the first is enough. Notice what has and has not gone wrong. The Lipschitz condition was never violated locally ; what fails is that no single Lipschitz constant works on the whole trajectory, because the trajectory leaves every bounded set in finite time. Picard-Lindelöf promised a solution on some interval and delivered exactly that. The unpredictability here is not sensitivity — two nearby initial conditions give two nearby blow-up times, $t_{*}=1/x_0$ being a perfectly continuous function of $x_0$ — it is that the model has no state to report past $t_{*}$. Something outside the equation must take over: in a real system, the neglected physics that the model dropped. Time runs along the horizontal axis and the state x up the vertical axis. The amber curve starts at the value one when t is zero, rises gently at first, and then turns sharply upward, running off the top of the plot as it approaches the dashed vertical line at t equals one. Beyond that line the solution does not exist. t * = 1 no solution beyond 1 10 x(0) = 1 0 1 t x Fig. 1 — Finite-time blow-up. The solution $x(t)=1/(1-t)$ of $\dot x=x^{2}$, $x(0)=1$, is smooth and unremarkable up to $t_{*}=1$ and does not exist beyond it. Existence, the first of Hadamard's three conditions, has failed for an equation whose right-hand side is a polynomial. Where uniqueness fails: shocks and weak solutions Move from ordinary to partial differential equations and the second condition goes [2] . A one-dimensional conservation law reads $$ \partial_t u+\partial_x F(u)=0, $$ where $u(x,t)$ is the conserved density and $F(u)$ the flux carrying it. Take the standard case $F(u)=u^{2}/2$, so that $\partial_t u+u\,\partial_x u=0$: each parcel of fluid travels at its own speed $u$. Along a characteristic starting at $x_0$ the value of $u$ is constant, so the characteristic is the straight line $x=x_0+u_0(x_0)\,t$. Faster parcels behind slower ones must eventually catch them, and one can say exactly when: characteristics first cross at $$ t_b=\frac{-1}{\min_{x}u_0'(x)}, $