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Incoming links: Jacobian matrix
The total derivative of sends a scalar increment toSince , the function composition is constantly , so . Directly, the Jacobian matrix product isThis verifies the chain rule at every .
At the gradient is . Therefore the total derivative is the linear maprepresented by the Jacobian matrix .
The total derivative of at is the linear map such thatThe Jacobian matrix is the matrix . When the total derivative exists, this is its matrix in standard bases: . Existence of partial derivatives alone need not imply existence of the total derivative.
The total derivative of at is the linear map such thatThe Jacobian matrix is the matrix . When the total derivative exists, this is its matrix in standard bases: . Existence of partial derivatives alone need not imply existence of the total derivative.
Third body and steady partition. The third molecule removes excess association energy and momentum, stabilizing newly formed ozone rather than being consumed in the net reaction. Its decreasing abundance with altitude suppresses three-body ozone formation strongly. Write , , , and , with held as reservoirs. The kinetic equations areThe fast association/photolysis subsystem conserves odd oxygen, , and has eigenvalues . Its partition-adjustment time is thereforeThis neglects the slower production and mutual-destruction terms while finding that fast mode. On its equilibrium partition, , and total odd-oxygen equilibrium gives . HenceWith fixed oxygen mole fraction and temperature, the atomic-oxygen mixing ratio scales as in this approximation, and thus increases upward as pressure decreases.
Lifetime versus relaxation. The distinction is formalized by Chapman family lifetime versus relaxation time. In the ozone-dominated regime , the family abundance is approximately , and its gross chemical lifetime isThis recovers the printed as a family turnover time under its additional partition assumption. It is not generally an e-folding relaxation time. With rapid partitioning, the family equation is . Linearizing about equilibrium gives relaxation rate but turnover rate , a factor of two difference. More generally the approximate family lifetime is , and its relaxation time is half that quantity. Neither can be replaced by when atomic oxygen dominates.
For a direct check using the complete two-species equations, the equilibrium Jacobian matrix isThe numerical high-altitude example below gives eigenmode times about and , whereas the printed is only . Thus interpreting both printed expressions as universal eigenmode time constants would be false, even though the fast partition and the ozone-dominated turnover approximations are useful.
Setting in the fast-partition approximation gives , henceRetaining mutual destruction gives and changes this estimate by less than one percent. The formal high-altitude limit of is : ozone photolysis, unlike three-body association, does not slow merely because collider density decreases. This is a partition timescale, not the lifetime of all odd oxygen. Extrapolating the fixed-reservoir approximation indefinitely upward would eventually cease to be self-consistent.
Day and night. At and , and . The exact equilibrium is conveniently obtained fromThe physical root givesTheir ratio is , consistent with the requested approximate factor ; the simpler fast-partition estimate gives approximately . The printed fast time is at this pressure.
For nighttime Chapman ozone, after sunset set both photolysis rates to zero. Ozone initially grows by converting atomic oxygen, but simply putting every initial oxygen atom into one ozone molecule neglects the simultaneous sink. The night equations areWhile sufficient atomic oxygen remains, the ozone balance tends towardThis is a useful long-night sunrise estimate, roughly thirty times the sunset ozone abundance, not an exact result for an unspecified night duration. The precise phase-plane relation isTogether with the night equations it supplies for any supplied duration. For illustration only, a twelve-hour dark interval gives ; no twelve-hour duration is stated by the paper. Even the infinite-dark-time endpoint is slightly below , because the finite initial atomic reservoir is consumed. This conditional estimate avoids inventing a unique sunset-to-sunrise duration.
Expand the vector field about the reference trajectory:Over a sufficiently short interval, the tangent-linear propagator isConsequentlyIf has and the perturbation is initially parallel to , its squared Euclidean norm obeysPositive means instantaneous norm-squared growth, negative decay, and zero no first-order change. The corresponding norm growth rate is , not . These are eigenvalues of the symmetric part governing the chosen error metric, not generally the eigenvalues of the nonsymmetric Jacobian matrix or the long-time Lyapunov exponents. The direction and the reference Jacobian matrix evolve during a finite forecast.
If one retains the Euler approximation , its exact norm matrix isThe additional term can mix the instantaneous symmetric-part eigenvectors, so they no longer generally diagonalize finite-time error amplification. However, when is material, a reliable physical propagator must itself be calculated beyond first-order Euler: a large Euler-step amplification can be a numerical artifact. For constant , the exact map is ; for a changing reference trajectory it solves , .
Use its singular value decomposition, . The initial orthonormal right singular vectors diagonalize and satisfyA singular value gives finite-time norm growth, decay, and the leading right singular vector is the initial perturbation with greatest amplification in the stated norm. The left singular vector describes the resulting final error pattern. Non-normal dynamics can give transient growth even if every Jacobian matrix eigenvalue has negative real part, so selecting observations from those eigenvalues alone is insufficient. Over a genuinely short interval, , recovering the symmetric-part result. Initial observations should target the relevant growing right singular projections, with prior covariance and observational precision included when they are not isotropic.
For a differentiable map , the Jacobian matrix represents its derivative and has entries . Differentiating the two components gives
The -th partial derivative is the vector limitwhen it exists. The total derivative is a linear map such thatThe Jacobian matrix is . When the total derivative exists, all partial derivatives exist and is its matrix in the standard bases; its -th column is . Existence of partial derivatives alone does not guarantee total differentiability.
The Fréchet derivative at a point is the unique linear map such thatThe error divided by tends to zero as . In standard coordinates this linear map is represented by the Jacobian matrix, but the derivative itself is the map acting on increments, not merely a list of directional rates.
A map is differentiable at when a linear map approximates its increment to first order:This Fréchet derivative is unique, and its matrix in the standard bases is the Jacobian matrix. Differentiability on a domain means differentiability at each point there.
A standard sufficient condition for a zero locus to be a submanifold is the regular level set theorem: if is on an open domain and is surjective at every , thenFor smooth , the submanifold is smooth. The full-rank condition is the relevant constraint qualification.
Under this regularity hypothesis, its tangent space isIndeed, if a differentiable curve lies in and passes through , the chain rule gives . Thus all tangent velocities lie in the kernel of a linear map. The rank-nullity theorem makes this kernel -dimensional, the same dimension as the tangent space, proving equality. In coordinates, solve the homogeneous linear equations given by the Jacobian matrix.
The regularity qualification matters: has zero set , a zero-dimensional submanifold, but . Its kernel is all of , whereas its tangent space is zero. This is the phenomenon that singular defining maps can overestimate tangent spaces.
A sufficient condition is that the derivative is surjective, equivalently that its Jacobian matrix has rank . Then the implicit function theorem makes the zero set near a embedded submanifold of dimensionThe rank condition persists nearby because the derivative is continuous. This is the local form of the regular level set theorem; only the rank at the specified point is needed for this local conclusion.
The map is differentiable at if there is a linear map such thatEquivalently, . This linear map is unique and is the Fréchet derivative . Its matrix in the standard bases is the Jacobian matrix, whose entry is at . A function is differentiable on its domain if this condition holds at every point.
At a fixed point the first equation gives . The second gives , so andThere is no disease-free fixed point: with , susceptibles still increase by each step.
Set . The Jacobian matrix of the update at the fixed point isIts characteristic equation and multipliers areFor , these are distinct complex conjugates with product one, so both have modulus one. The linearized iterates are bounded and oscillatory: neutral linear stability holds for , with no linear asymptotic attraction.
At , the repeated multiplier is , but . Writing gives , and , so generic linearized perturbations grow linearly. For , a real multiplier has modulus greater than one. Thus the equilibrium is linearly unstable for .
For the unit-modulus range, the nonlinear behavior is not determined by this calculation: unit-modulus multipliers do not determine nonlinear stability.
Take andIts Jacobian matrix and determinant areThus its total derivative is invertible everywhere, and the inverse function theorem makes it locally invertible. But , so it is not injective and has no global inverse on its image. The domain is open and connected as required.
Suppose were invertible at some . The inverse function theorem supplies a neighborhood of for which is a nonempty open subset of . But , and a one-dimensional embedded submanifold of has empty interior. To see this last assertion, a slice chart for an embedded submanifold takes locally into a coordinate line; a coordinate line cannot contain an open plane neighborhood, and a chart preserves openness. This is a contradiction.
Thus is singular everywhere. Part (c)(i) strengthens singularity to everywhere. By (b) and connectedness of ,The special form of the Jacobian matrix is essential: an arbitrary differentiable map can have nonconstant image in a curve with derivative of rank one.
Abbreviate , . The coordinate factors immediately give the three boundary equilibria . The Jacobian matrix isAt , the eigenvalues are , so it is a repelling equilibrium. At the other two points, triangularity givesTherefore is asymptotically stable for and a saddle point for ; is asymptotically stable for and a saddle point for . Equality gives a zero eigenvalue, so linearization alone is inconclusive there.
For later use, the coexistence point is physically positive exactly when if , or if . Its stability is as supplied in the question. If the displayed formula for does not apply: when also , the coincident non-axis nullclines give a whole segment of equilibria.
Conformality and analyticity. Use the orientation-preserving convention for a conformal map, with a continuously differentiable, nonzero differential. Its Jacobian matrix must be a positive scale times a rotation,Thus , , the Cauchy-Riemann equations; continuous partial derivatives then imply that is a holomorphic function. If angle preservation is allowed to reverse orientation, the statement needs this qualification: preserves unsigned angles but is an anticonformal map and is not a holomorphic function.
The mapping. Put . The displayed components combine to giveIts derivative does not vanish in the open half-strip , , so it is a conformal map there. Also . The lower edge maps to , the upper edge to , and the vertical edge to . The open half-strip maps bijectively onto the upper half-plane. Indeed, implies ; the half-strip allows only equality. Surjectivity follows by solving and choosing the root with and . The two corners are boundary critical points, not interior failures of conformality.
Transformed boundary values. A conformal map preserves Laplace's equation up to a positive scale. Thus the transformed Dirichlet problem is to find a real harmonic function on withThe conflicting values at the original corners are interpreted as one-sided edge limits.
Logarithmic solution. Choose the complex logarithms of with arguments between zero and in the upper half-plane. A linear combination is a complex-valued harmonic function, since it is a holomorphic function; the real physical potential must be a real or imaginary part. In particular, setThenOn the three real boundary intervals, the argument difference is respectively , which gives the required values. This is the bounded harmonic solution, with at infinity; uniqueness holds in this bounded class by the maximum principle. Without a boundedness or far-field condition, the edge data alone do not give uniqueness: one may add .
Back substitution. Sincethe required continuous argument difference is in the open half-strip. ConsequentlyAs with , the arctangent tends to zero; as with , it tends to . These limits verify every open edge of the Dirichlet problem.
TODO: biological interpretations and organism-specific propagation are user-deferred. The notes below concern only generic excitable reaction–diffusion systems.
A front solution joins different asymptotic states in space. In a reaction–diffusion system, local nonlinear dynamics select possible states while diffusion couples neighboring points. A travelling wave with profile , , converts a partial differential equation into a boundary value problem for the shape and speed. A pulse instead returns to the same resting state on both sides and can contain leading and trailing fronts.
A useful nondimensional cubic fast-slow reaction-diffusion model, in a convention of the FitzHugh-Nagumo model, iswhere , , and . Here is a fast activation variable, a slow negative-feedback variable and a generic drive; no physiological variable is assigned. Without spatial coupling, nullclines are and . At , the resting state has Jacobian matrixIt has linear stability. With appropriate parameter choices a sufficiently large perturbation makes a fast excursion to the other outer branch of the cubic, followed by slow feedback and return. Thus a stable rest state can be excitable without being linearly unstable. Other parameters or applied drive can yield multiple equilibria or oscillations; these behaviours should not be asserted for every parameter set.
The fast-subsystem front can be obtained explicitly by freezing . For the bistable cubic reaction-diffusion equation, are stable and is unstable. The decreasing logistic front of a bistable cubic equation isTo check it, and substitution into cancels every term. For , the high state advances to the right; for , the direction reverses. At the front is stationary. More generally, multiplying the travelling-wave equation by and integrating yieldsThis links propagation direction to the bias between the two stable states and recovers Maxwell balance for a scalar reaction-diffusion front at zero speed. It is a bistable selected-speed problem, rather than a universal Fisher-type minimum-speed formula.
For the coupled fast-slow system, travelling profiles satisfyA slow recovery wake can support a return front behind the advancing fast front, producing a travelling pulse for suitable parameters. Its width reflects the slow recovery time and propagation speed, while each sharp edge has the faster diffusion–reaction scale. The frozen-recovery speed above is a controlled subsystem result, not an exact speed for every coupled pulse. A definite pulse speed requires parameters and connecting boundary condition; none are specified by this brief-note question.
Past exam of the systems biology course of the University of Cambridge 2024 Paper 3 3 c Solution 2026-10-20
At a fixed input the production terms are constant, so both linear ordinary differential equations can also be solved exactly:Each displacement from its equilibrium concentration decays with time constant one. The Jacobian matrix is , with two eigenvalues equal to , so the fixed point is globally asymptotically stable for the nonnegative state space.
Here is a numerical solver using the classical fourth-order Runge-Kutta method; it uses only the repository's existing NumPy and matplotlib dependencies. Continue the definitions from part (b):
def integrate(rhs, initial, times):
out = np.empty((len(times), len(initial)))
out[0] = initial
for j, dt in enumerate(np.diff(times)):
state = out[j]
k1 = rhs(state)
k2 = rhs(state + dt*k1/2)
k3 = rhs(state + dt*k2/2)
k4 = rhs(state + dt*k3)
out[j+1] = state + dt*(k1 + 2*k2 + 2*k3 + k4)/6
return out
def direct_rhs(state, X):
Y, Z = state
return np.array([2*hill(X, 1, 2) - Y,
3*hill(X, 2, 4) - Z])
times = np.linspace(0, 8, 801)
plt.figure(figsize=(7, 4))
for initial, style in [([0, 0], "-"), ([3, 4], "--")]:
trajectory = integrate(lambda s: direct_rhs(s, 2), initial, times)
exact = np.array([1.6, 1.5]) + (np.array(initial) - [1.6, 1.5]) \
* np.exp(-times[:, None])
assert np.max(np.abs(trajectory - exact)) < 1e-8
plt.plot(times, trajectory[:, 0], style, label=f"Y, initial {initial}")
plt.plot(times, trajectory[:, 1], style, label=f"Z, initial {initial}")
plt.xlabel("Time")
plt.ylabel("Concentration")
plt.legend()
plt.tight_layout()
plt.show()For , . The initial state approaches those values from below, while approaches from above. Over eight time units the initial displacement is reduced by , explaining why both trajectories nearly coincide with the equilibria from part (b). The two-panel figure above displays these simulations.