Lecture 5–6 — Autonomous Dynamical Systems: Linearization, Classification, and Conservative Flows#
Source: NPTEL Classical Physics, Mod-01 Lec-05 (Autonomous dynamical systems, Part 1) and Lec-06 (Part 2), Prof. V. Balakrishnan.
Local solvability is (almost) automatic#
Write a general \(N\)-dimensional autonomous system as \(\dot{\vec x} = \vec f(\vec x)\), where \(\vec x = (x_1,\dots,x_N)\) is a point in phase space and \(\vec f\) is a vector field — at every point it hands you the phase-space velocity of the representative point sitting there. Chain enough of these infinitesimal velocity arrows together and you get a phase trajectory. Solving this exactly is, in general, hard: \(\vec f\) can mix all \(N\) variables in an arbitrarily complicated, nonlinear way.
Locally, though, it’s easy. Taylor-expand \(\vec f\) about any ordinary point \(\vec x_0\) and keep only the leading term:
Sufficiently close to \(\vec x_0\), \(\vec f(\vec x_0)\) is just a constant vector, and \(\dot{\vec x} = \text{const}\) integrates trivially: \(\vec x(t) \approx \vec x_0 + \vec f(\vec x_0)\,t\). Do this at the new point you’ve reached, then the next, and you can march the solution forward piecewise, patch by patch. The rigorous statement of this is the rectification theorem: in a small enough neighborhood of any ordinary point, there’s a change of variables \(\vec x \to \vec y\) in which the flow becomes dead simple — \(\dot y_1 = 1\), \(\dot y_2 = \dot y_3 = \cdots = 0\) — i.e. pure uniform translation along one axis.
This does not mean the system is solvable globally. The patch of variables that rectifies the flow near one point generally stops working a little further along, and you need a new change of variables for the next patch — and those patches can shrink to nothing, or simply never stitch together into one global formula. Local solvability is not global integrability; most systems are not globally integrable, which is precisely why the qualitative, geometric methods of this lecture — not explicit formulas — are the working tools of nonlinear dynamics.
There is exactly one place the rectification argument breaks down: where \(\vec f(\vec x_0) = \vec 0\). There the leading-order velocity vanishes, the local “translate at constant velocity” picture is trivial (nothing moves), and you’re forced to the next order in the Taylor expansion. A point where the vector field vanishes is a critical point (in mechanics, an equilibrium point) — and, like the position vector \(\vec r\) at the origin, a vector field is genuinely singular there: not because it blows up, but because its direction becomes undefined.
Linearizing at a critical point#
Take a critical point to be the origin without loss of generality (just shift coordinates), so \(\vec f(\vec 0) = \vec 0\). The Taylor expansion now starts at first order,
where \(L\) is the Jacobian matrix of \(\vec f\) evaluated at the critical point. As long as \(L\) is non-singular (a simple critical point — the generic case), the system near the origin is genuinely linear, and linear systems we already know how to solve:
Everything about the local flow — whether nearby trajectories are pulled in, pushed out, or orbit around — is therefore governed entirely by the eigenvalues of \(L\): positive real parts mean growth, negative real parts mean decay, and purely imaginary eigenvalues mean oscillation without either.
Classifying 2D linear flows: it’s all trace and determinant#
Restrict to two dimensions, \(\dot x = f(x,y)\), \(\dot y = g(x,y)\), with a critical point at the origin, \(L = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\). The characteristic (secular) equation is
The eigenvalues — and therefore the entire qualitative picture near the critical point — depend on the four numbers \(a,b,c,d\) only through these two combinations. That’s not an accident: \(T\) and \(D\) are invariant under any similarity transformation \(L \to SLS^{-1}\) (a linear change of coordinates), and a change of coordinates is exactly what separates two systems that “look different” on paper but have identical dynamics. This single fact — that only \(T\) and \(D\) matter — is what makes a complete classification of 2D linear flows possible with just six cases, laid out on the \((T,D)\) plane below the discriminant curve \(T^2 = 4D\) (real, distinct eigenvalues) and above it (complex-conjugate pair):
Region |
Eigenvalues |
Name |
|---|---|---|
\(D<0\) |
real, opposite sign |
Saddle (always unstable) |
\(D>0,\ T>0,\ T^2>4D\) |
real, both positive |
Unstable node |
\(D>0,\ T<0,\ T^2>4D\) |
real, both negative |
Stable node |
\(D>0,\ T>0,\ T^2<4D\) |
\(\lambda\pm i\mu,\ \lambda>0\) |
Unstable spiral (focus) |
\(D>0,\ T<0,\ T^2<4D\) |
\(\lambda\pm i\mu,\ \lambda<0\) |
Stable spiral (focus) |
\(D>0,\ T=0\) |
\(\pm i\mu\), pure imaginary |
Center (periodic) |
Step through the six cases: the saddle (\(D<0\)) is the only region that’s always unstable regardless of trace, because one eigenvalue is always positive. Crossing \(T^2=4D\) turns a node into a spiral without changing stability — real eigenvalues that collide and go complex. And the center is an infinitely thin razor’s edge: \(T\) must be exactly zero, so the slightest damping (\(T<0\)) or the slightest anti-damping (\(T>0\)) instantly turns closed periodic orbits into a spiral. This is the same fragility Lecture 2 found in the undamped harmonic oscillator, now seen as a single point balanced on a knife-edge in the \((T,D)\) plane.
What if \(L\) doesn’t happen to be diagonal — off-axis coupling between \(x\) and \(y\)? Nothing new happens: any invertible \(2\times2\) linear transformation is some combination of a rotation, a dilation, and a shear (four parameters total — one angle, two scale factors, one shear angle — matching the four free entries of \(L\)). So a general saddle, node, or spiral is simply the canonical picture above viewed in skewed, rotated coordinates: topologically the same object, continuously deformable into the textbook picture. That’s exactly what the companion-matrix trick used to draw the figure exploits — off-diagonal entries are baked in by construction, and the trajectories still come out as the standard shapes, just tilted.
Beyond two dimensions#
For an \(N\times N\) Jacobian, the eigenvalues again depend only on combinations invariant under similarity transformations — but for \(N>2\) there are more of them than just “trace and determinant.” The right generalization is the set of \(N\) power sums \(\operatorname{tr}L,\ \operatorname{tr}L^2,\ \dots,\ \operatorname{tr}L^N\) (equivalent to the coefficients of the characteristic polynomial): each is invariant under \(L \to SLS^{-1}\) because in a basis where \(L\) is diagonal, \(\operatorname{tr}L^k = \sum_i \lambda_i^k\) manifestly doesn’t care how you got to that basis.
In three dimensions genuinely new geometry becomes possible. A pair of complex eigenvalues \(\lambda \pm i\mu\) with \(\lambda<0\), combined with one positive real eigenvalue \(\lambda_3>0\), produces a saddle-focus: trajectories spiral inward within a 2D invariant plane while simultaneously being ejected along the third direction.
Drag the surface around: for small \(z\) the picture is an ordinary stable spiral, but nothing keeps the trajectory pinned to that plane forever. Once nonlinear terms take over further from the critical point, a trajectory that has spiralled outward along \(z\) can get bent back and re-injected near the plane, spiral in again, escape again, and repeat — a mechanism known as homoclinic chaos, one of the standard routes by which higher-dimensional flows generate genuinely complex, aperiodic behavior. (It’s a preview, not a destination: the full story needs tools this course builds up to later.)
That same higher-dimensional richness is also where sensitive dependence on initial conditions becomes unavoidable. Two trajectories that start an arbitrarily small distance \(\epsilon\) apart can separate at a rate governed by the local eigenvalues; if that separation grows exponentially in time, \(\epsilon\, e^{\lambda t}\), then no finite amount of precision in the initial condition buys you more than a logarithmically short window of reliable prediction — this is the seed of chaos, and it recurs throughout the rest of the course.
Conservative versus dissipative flows#
Take a small volume element in phase space, \(\delta V = \delta x_1\,\delta x_2\cdots\delta x_N\), built from \(N\) neighboring trajectories each started an infinitesimal \(\delta x_i\) away from a reference point along axis \(i\). Each edge evolves according to the flow, \(\dot{(\delta x_i)} = \sum_j (\partial f_i/\partial x_j)\,\delta x_j\) to leading order (the same linearization as before, just applied edge by edge). Differentiate the product using the chain rule and only the “diagonal” terms survive after each \(\delta x_i\) factors back out:
So the divergence of the vector field directly measures the local fractional rate of change of phase-space volume. This motivates a clean definition: a dynamical system is conservative if \(\nabla\!\cdot\!\vec f \equiv 0\) everywhere — phase-space volume is preserved exactly, flowing the way an incompressible fluid does in ordinary space (the continuity equation \(\partial\rho/\partial t + \nabla\!\cdot\!(\rho \vec v)=0\) with \(\rho\) constant forces \(\nabla\!\cdot\!\vec v = 0\) — the same equation, just relabeled). A system with \(\nabla\!\cdot\!\vec f < 0\) everywhere is dissipative: phase-space volumes shrink, trajectories are drawn toward lower-dimensional attracting sets.
Two quick checks confirm the definition does what you’d expect. The undamped harmonic oscillator, \(\dot x = v,\ \dot v = -\omega^2x\), has \(\partial \dot x/\partial x + \partial \dot v/\partial v = 0 + 0 = 0\) — conservative, trivially, matching the centers of Lecture 2’s ellipse-laminated phase portrait. Add linear damping, \(\dot v = -\omega^2 x - \gamma v\), and \(\nabla\!\cdot\!\vec f = -\gamma < 0\) identically: every phase-space volume shrinks as \(e^{-\gamma t}\), consistent with every trajectory spiraling down into the single point at the origin.
The Lotka–Volterra predator–prey model#
Here’s an example far from mechanics that shows off the same machinery. Let \(x\) be a rabbit population and \(y\) a fox population, large enough to treat as continuous:
rabbits breed exponentially in isolation (\(\alpha x\)) but are eaten at a rate proportional to encounters with foxes (\(-\beta xy\)); foxes starve without rabbits (\(-\gamma y\)) but grow by eating them (\(\delta xy\)). Set all four rate constants to 1 for illustration: \(\dot x = x(1-y),\ \dot y = y(x-1)\).
There are exactly two critical points. At the origin, linearizing keeps only the leading terms \(\dot x \approx x,\ \dot y \approx -y\) — eigenvalues \(+1\) and \(-1\), a saddle: extinction is an unstable equilibrium, reachable only along a knife-edge approach. At \((1,1)\), substituting \(x=1+u,\ y=1+v\) and linearizing gives \(\dot u = -v,\ \dot v = u\) — eigenvalues \(\pm i\), a center: rabbits and foxes oscillate around a coexistence point, out of phase, forever (in this simplified model with no other limiting effects).
Every orbit in the positive quadrant is a closed loop encircling \((1,1)\) — near the center they’re the ellipses linearization predicts, and further out they distort into the lopsided loops the phase portrait shows, but they never cross (the non-crossing theorem again) and never spiral in or out. The time series makes the “out of phase” claim concrete: the fox population peaks visibly after the rabbit population does, each overshoot setting up the next collapse. (The model is a famous piece of applied math history: Vito Volterra derived it in the 1920s to explain oscillating fish catch data reported by the biologist Umberto D’Ancona, independently of very similar work by Alfred Lotka on chemical kinetics.)
Is Lotka–Volterra conservative? Only on average#
Compute the divergence: \(\nabla\!\cdot\!\vec f = \partial_x[x(1-y)] + \partial_y[y(x-1)] = (1-y) + (x-1) = x - y\). That is not identically zero, so by the strict definition above Lotka–Volterra is not conservative — phase-space area genuinely contracts while \(y>x\) and expands while \(x>y\). But every orbit is periodic, crossing the line \(x=y\) exactly twice per cycle, and the contracting and expanding episodes are equal and opposite: averaged over one full period, the net change in area is exactly zero.
This can be checked quantitatively rather than just argued qualitatively. The fractional rate of change of an infinitesimal patch of area is \(\det\Phi(t)\), where \(\Phi(t)\) is the variational (tangent) matrix satisfying \(\dot\Phi = J(\vec x(t))\,\Phi,\ \Phi(0)=I\), with \(J\) the (now time-dependent, since the system is nonlinear) Jacobian evaluated along the actual trajectory — the direct nonlinear generalization of \(e^{Lt}\) from the linear case above. Integrating this alongside a representative periodic orbit and comparing against the exact result for the damped oscillator, \(\det\Phi(t) = e^{-\gamma t}\) (constant divergence \(-\gamma\) integrates trivially), makes the contrast explicit:
The damped oscillator’s curve decays monotonically and never comes back — genuinely dissipative. The Lotka–Volterra curve dips as low as \(\sim\!0.32\) mid-cycle (area actively contracting while foxes outnumber rabbits) and climbs back above \(1\) later, landing within \(0.003\%\) of exactly \(1\) at \(t=T\), the numerically located period — confirming “conservative on average” isn’t just a verbal argument but an identity a direct calculation reproduces. (It’s not a coincidence: a later change of variables turns Lotka–Volterra into an honest Hamiltonian system, for which \(\nabla\!\cdot\!\vec f \equiv 0\) exactly, without needing to average over anything — a preview of where this course is headed.)