Systems of Linear Equations and Phase Plane Analysis: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 8, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A collection of first-order equations for several unknown functions of the same variable, written compactly with a vector unknown and a coefficient matrix. Almost every method of the subject is stated most cleanly in this language.
Conversion of a higher-order equation to a system
Introducing the successive derivatives as new unknowns turns a single equation of order n into n first-order equations. This is why theory and numerical software need only handle first-order systems.
Vector solution
A vector-valued function whose components simultaneously satisfy every equation of the system. Solutions of a linear homogeneous system form a vector space of dimension equal to the number of unknowns.
Eigenvalue of the coefficient matrix
A number for which the matrix minus that number times the identity is singular. Substituting an exponential vector trial solution into a constant-coefficient system produces exactly the eigenvalue problem.
Eigenvector solution
An eigenvector multiplied by the exponential of its eigenvalue times the independent variable solves the constant-coefficient homogeneous system. Independent eigenvectors give independent solutions.
Distinct real eigenvalues in a system
When the matrix has a full set of real eigenvalues with independent eigenvectors, the general solution is a combination of the corresponding exponential vector solutions and behaviour is dominated by the largest eigenvalue.
Complex eigenvalue pair in a system
A complex eigenvalue and its conjugate yield spiralling solutions. Taking the real and imaginary parts of one complex vector solution produces two real independent solutions with no complex arithmetic left over.
Defective matrix
A matrix with fewer independent eigenvectors than its size, so the exponential vector trial solutions do not span the solution space. It arises when a repeated eigenvalue has too small an eigenspace.
Generalised eigenvector
A vector sent to an ordinary eigenvector by the matrix minus the eigenvalue times the identity. It supplies the missing solution, which carries an extra factor of the independent variable alongside the exponential.
Matrix exponential
The matrix-valued series in the coefficient matrix that solves the homogeneous system with identity initial data. Multiplying it by the initial vector gives the solution, which unifies all the eigenvalue cases.
Fundamental matrix
A matrix whose columns are independent solutions of the system. Any solution is that matrix times a constant vector, and its invertibility at one point is equivalent to independence of the columns.
Phase plane
The plane of the two dependent variables, in which a solution appears as a parametric curve rather than a graph against the independent variable. It displays the whole family of behaviours at once.
Trajectory
The oriented curve traced in the phase plane by a single solution as the independent variable increases. Under uniqueness two trajectories can never cross, though they may approach the same point.
Critical point of a planar system
A point where both right-hand sides vanish, giving a constant solution. Critical points are where all the interesting structure of the phase portrait is organised.
Node
A critical point whose eigenvalues are real and of the same sign, so all nearby trajectories enter or leave along eigenvector directions without spiralling. It is stable when both eigenvalues are negative.
Saddle point
A critical point with real eigenvalues of opposite sign, attracting along one eigenvector direction and repelling along the other. It is always unstable, and only two special trajectories actually reach it.
Spiral point
A critical point whose eigenvalues are complex with nonzero real part, so trajectories wind around it while approaching or receding. The sign of the real part decides which and the imaginary part sets the winding rate.
Centre
A critical point with purely imaginary eigenvalues, surrounded by closed trajectories representing periodic solutions. It is stable but not asymptotically stable, and nonlinear terms can destroy it.
Trace determinant classification
Reading the type of a planar critical point from the trace and determinant of the matrix, since these give the sum and product of the eigenvalues. Negative determinant means a saddle, and the discriminant separates nodes from spirals.
Stability from eigenvalue real parts
A linear system is asymptotically stable exactly when every eigenvalue has negative real part, unstable when some eigenvalue has positive real part, and borderline when the largest real part is zero.
Linearisation at a critical point
Replacing a nonlinear system near an isolated critical point by the linear system given by its matrix of partial derivatives. It predicts the local phase portrait whenever the resulting eigenvalues are not borderline.
Jacobian matrix
The array of first partial derivatives of the right-hand sides with respect to the dependent variables, evaluated at a critical point. Its eigenvalues drive the linearised classification of that point.
Borderline cases of linearisation
When the linearisation has purely imaginary eigenvalues or a zero eigenvalue, the nonlinear terms decide the true behaviour, so a predicted centre may actually be a slow spiral in either direction.
Nullcline
A curve on which one of the two rates vanishes, so trajectories cross it either vertically or horizontally. Intersections of nullclines of different types are exactly the critical points.
Predator prey and competition models
Nonlinear planar systems in which two populations interact through product terms. Their phase portraits show closed orbits or exclusion outcomes depending on whether the interaction is predation or competition for one resource.