Linearization Analysis

Linear systems are much easier to analyze than nonlinear systems because they satisfy the superposition principle, which can be stated as follows: if R1 is the output of a linear system to input S1 and R2 is the output to input S2, then the output to the cumulative input a S1 + b S2 is a R1 + b R2. Applied iteratively, this property allows the inputs and outputs of a linear system to be decomposed in various ways to simplify analysis. Fully nonlinear systems do not share this useful property; thus the causal implications of each stimulation must be individually studied.

Linearization is an attempt to carry some of the attractive properties of linear systems into nonlinear domain. Linearization methods can be classified in two approaches: reductions of nonlinear systems to linearized approximations for small perturbations and extensions of linearized approximations to weakly nonlinear systems.

Linearization methods of the first approach are used for stability analyses of special solutions of a nonlinear system, such as critical points, periodic orbits, solitary waves, periodic waves, and so on. In the neighborhood of such special solutions of nonlinear systems, linear equations are derived for small perturbations, which are then studied using standard methods (Fourier or Laplace trandforms, Green functions, etc.).

Linearization methods of the second approach are based on specific parameters of a linearized system, such as dominant eigenvalues or resonant wavenumbers. Thus linearized systems are extended to weakly nonlinear systems such as normal forms and amplitude equations by means of asymptotic multi-scale expansion methods and the Fredholm's alternative theorem for linear non-homogeneous equations.

Linearization methods for systems of ordinary differential equations (ODEs) are illustrated with the example of an autonomous nonlinear dynamical system u'(t) = F(u), where u(t) and F(u) are vector functions. Critical points of the dynamical system occur for u = u0 such that F(u0) = 0. Linearizing the nonlinear systems at the critical point u0, one expands u(t) = u0 + v(t). Neglecting quadratic terms in v(t), we reduce the nonlinear problem to the linear system with constant coefficients:

v'(t) = J v,               J = F'(u0),

where J is the Jacobian matrix. Solutions of the linearized systems depend on eigenvalues and eigenvectors of the Jacobian matrix J w = λ w. If all eigenvalues are located in the left half-plane of λ for Re(λ) < 0, then the critical point u0 is asymptotically stable and the perturbation v(t) decays to zero exponentially in t. If there exists at least one eigenvalue in the right half-plane of λ, the critical point u0 is linearly unstable and the perturbation v(t) grows exponentially in t.

Some or all eigenvalues may be located at the imaginary axis of λ, where Re(λ) = 0. In such systems, when no other eigenvalues exist for Re(λ) > 0, the critical point u0 is neutrally (weakly) stable. Perturbations may however grow algebraically in t, if eigenvalues λ are multiple with algebraic multiplicity exceeding the geometric multiplicity. Local linearization is often insufficient for full description of such weak instability and the nonlinear stability analysis is desired.

When eigenvalues cross or coalesce on the axis Re(λ) = 0, bifurcations may occur in the spectrum of a linearized system. Normal forms for bifurcations can be derived by extending the linearized system into the nonlinear domain. For example, a Hopf bifurcation occurs when two simple eigenvalues λ = Γ(ε) + i Ω(ε) and λ* = Γ(ε) - i Ω(ε) cross the imaginary axis at the bifurcation parameter ε = 0, such that Γ(0) = 0 and Ω(0) = Ω0. The normal form for the Hopf bifurcation can be derived by the asymptotic multi-scale expansion method in the perturbation series:

u(t) = u0 + ε ( A(T) w exp(i Ω0 t) + A*(T) w* exp(- i Ω0 t) ) + O(ε2),

where T = ε2 t and w is the eigenvector of J w = λ w for λ = i Ω0 at ε = 0. The normal form for Hopf bifurcation follows from the Fredholm's alternative theorem at order O(ε3) in the form:

A'(T) = ( Γ'(0) + i Ω'(0) ) A - β(0) |A|2 A,

where β(0) is constant. The critical point bifurcates into a periodic orbit at the Hopf bifurcation when Γ'(0) Re(β(0)) >0. Thus, linearization methods allow for analysis of linear and nonlinear stability, resulting in simplifications of nonlinear dynamical systems.