Linear stability

Linear stability analysis classifies a hyperbolic equilibrium from the eigenvalues of the vector field's Jacobian.

Unit-modulus multipliers do not determine nonlinear stability ( $D F_{\pm} (0) = R_{θ}, F_{\pm} (z) = e^{i θ} (1 \pm ∣ z ∣^{2}) z$ )

For a discrete dynamical system, eigenvalues of the linearization strictly inside the unit circle imply local asymptotic stability; an eigenvalue outside implies instability. Eigenvalues on the unit circle require more information. A nontrivial Jordan block there can already make the linearized iterates unbounded.

Even a bounded linearization leaves nonlinear stability undecided. The two displayed maps of the complex plane have the same derivative, a rotation matrix. Their radial updates are

r \mapsto r (1 - r^{2})

and

r \mapsto r (1 + r^{2})

. For small positive radius, the first decreases to zero and the second moves out of every sufficiently small fixed neighborhood. Thus identical unit-modulus linear multipliers can accompany attraction or instability.

Recovery time of a stable equilibrium ( $τ = - 1/ f^{'} (N^{*})$ )

 0  0

For a scalar ordinary differential equation

N^{'} = f (N)

and an equilibrium point

N^{*}

with

f^{'} (N^{*}) < 0

, the linearization gives

n^{'} = f^{'} (N^{*}) n

. Thus small perturbations decay like

e^{- t / τ}

with

τ = - 1/ f^{'} (N^{*})

. This is a local exponential recovery time. In several dimensions, the slowest decaying stable eigenvalue typically controls asymptotic recovery, subject to additional polynomial factors for nontrivial Jordan blocks.

Stability threshold of a discrete exponential predator-prey model ( $r lo g r = r - b$ )

 0  0

For

n^{'} = r n e^{- p}

and

p^{'} = n (1 - b e^{- p})

, with

0 < b < 1 < r

, the positive equilibrium is

p_{c} = lo g r

n_{c} = r lo g r / (r - b)

. Its Jacobian has trace

1 + b n_{c} / r

and determinant

n_{c}

. The unit-disc root conditions reduce to

n_{c} < 1

. At

n_{c} = 1

the roots form a nonreal conjugate pair and cross the unit circle; their argument satisfies

2 cos θ = 1 + b / r

. The unique threshold lies between one and

e

because

n_{c}

increases strictly and

lo g r = 1 - b / r < 1

at threshold.

Stability matrix ( $J = D f (x_{*})$ )

 0  0

For

\overset{x}{˙} = f (x)

near an equilibrium point, the stability matrix is the Jacobian matrix of the vector field evaluated at that point. Perturbations obey

\overset{η}{˙} = J η

to first order. Its eigenvalues determine linear growth and decay: strictly negative real parts imply local asymptotic stability, while a positive real part gives instability. Zero real parts require further analysis. The stability matrix of a renormalization-group fixed point applies this construction to beta-function flows.

Saddle-centre equilibrium

 0  0

A conservative equilibrium with one real pair of linear eigenvalues

\pm λ

and one imaginary pair

\pm iω

has saddle and centre directions. Generic perturbations have an exponentially growing component, even though finely chosen initial conditions can remain in the centre-stable subspace. The local L2 Lagrange point dynamics is an example.

Sideband instability

 0  0

A sideband instability is growth of disturbances with wavenumbers slightly displaced from a carrier pattern's wavenumber. A real physical perturbation uses the displaced mode together with its complex conjugate; a complex amplitude equation can couple those two sidebands through its cubic term. The Eckhaus instability is a longitudinal phase-modulation example.

Bloch stability operator ( $L (q) = e^{- i q \cdot x} L e^{i q \cdot x}$ )

 0  0

For a spatially periodic nonlinear pattern, a linear stability disturbance is written

e^{λ t + i q \cdot x} v (x)

with

v

periodic on the pattern cell. The conjugated operator

L (q)

acts on that fixed cell and determines the sideband eigenvalues. Translation invariance supplies zero eigenfunctions at

q = 0

, namely spatial derivatives of the base pattern. Their small-

q

eigenvalues determine long-wavelength phase stability; other parts of the spectrum can give independent instabilities.

Phase-diffusion coefficient from a Bloch cell problem ( $D = ⟨ w, L_{2} v_{0} + L_{1} v_{1} ⟩$ )

 0  0

Suppose a Bloch stability operator has expansion

L (q) = L_{0} + i q L_{1} - q^{2} L_{2} + \dots

, with a simple neutral eigenfunction

v_{0}

and an adjoint neutral function

w

normalized by

⟨ w, v_{0} ⟩ = 1

. Expand

v (q) = v_{0} + i q v_{1} + \dots

and

λ (q) = i c q - D q^{2} + \dots

. The Fredholm alternative gives

c = ⟨ w, L_{1} v_{0} ⟩

and the cell problem

L_{0} v_{1} = c v_{0} - L_{1} v_{0}

⟨ w, v_{1} ⟩ = 0

. Applying the adjoint solvability condition at second order gives the displayed phase-diffusion coefficient. Multiple translation phases or a conserved mean require a coupled matrix problem rather than this scalar formula.

Spatial sideband

 0  0

A spatial sideband is a Fourier perturbation with wavenumber close to a pattern's carrier wavenumber, such as

k_{c} + p

k_{c} - p

. A real physical field also includes the conjugate Fourier component. Linearizing a nonlinear complex amplitude equation can couple the two displaced components, so testing only a single uniform amplitude misses their joint linear stability problem.