Linearization of Nonlinear ODEs
Linearization of Autonomous Systems at Fixed Points
Conservative System in a Potential Field Dynamics
The study of linear ordinary differential equations is characterized the principle of superposition. This principle, a direct consequence of the linearity of the differential operator, allows for the construction of general solutions from a basis of fundamental solutions. However, the majority of physical systems exhibit nonlinear behavior, for which such general solution techniques are not available. The analysis of nonlinear systems often relies on qualitative methods, being the local analysis of dynamics in the vicinity of equilibrium states through linearization.
An operator \mathcal{L} is defined as linear if it satisfies the superposition property for any scalars \alpha, \beta and any elements \mathbf{x}, \mathbf{y} in its domain.
\mathcal{L}(\alpha \mathbf{x} + \beta \mathbf{y}) = \alpha \mathcal{L}(\mathbf{x}) + \beta \mathcal{L}(\mathbf{y})
The time derivative operator, \frac{\mathrm{d}}{\mathrm{d}t}, and the operation of matrix multiplication are examples of linear operators.
Consequently, a system of first-order ordinary differential equations of the form \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} is a linear system.
If \mathbf{s}_1(t) and \mathbf{s}_2(t) are two solutions to this system, then any linear combination \alpha \mathbf{s}_1(t) + \beta \mathbf{s}_2(t) is also a solution.
In contrast, functions such as trigonometric or polynomial functions of degree higher than one are nonlinear. An ODE containing such terms in its dependent variables, for example \ddot{x} + \sin(x) = 0, is a nonlinear equation, and the principle of superposition does not apply.
Linearization of autonomous systems at fixed points
Consider a general autonomous nonlinear system described by:
\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{f}(\mathbf{x})
where \mathbf{x} \in \mathbb{R}^n and \mathbf{f}: \mathbb{R}^n \to \mathbb{R}^n is a continuously differentiable vector field. A state \bar{\mathbf{x}} is termed a fixed point, or equilibrium point, of the system if it is a state of no change, meaning the time derivative is zero:
\mathbf{f}(\bar{\mathbf{x}}) = \mathbf{0}
To understand the dynamics in a small neighborhood of such a fixed point, we consider a small perturbation, \boldsymbol{\delta}(t) = \mathbf{x}(t) - \bar{\mathbf{x}}. The evolution of this perturbation is given by:
\frac{\mathrm{d}\boldsymbol{\delta}}{\mathrm{d}t} = \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} = \mathbf{f}(\bar{\mathbf{x}} + \boldsymbol{\delta})
We can analyze the right-hand side by performing a multivariate Taylor series expansion of \mathbf{f} around the point \bar{\mathbf{x}}.
\mathbf{f}(\bar{\mathbf{x}} + \boldsymbol{\delta}) = \mathbf{f}(\bar{\mathbf{x}}) + \frac{D\mathbf{f}}{D\mathbf{x}} \bigg\vert_{\mathbf{x} = \bar{\mathbf{x}}} \boldsymbol{\delta} + \mathcal{O}(\|\boldsymbol{\delta}\|^2)
The term \frac{D\mathbf{f}}{D\mathbf{x}} is the Jacobian matrix of the vector field \mathbf{f}, whose components are the partial derivatives of the components of \mathbf{f} with respect to the components of \mathbf{x}.
\mathbf{J}(\mathbf{x}) = \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \dfrac{\partial f_1}{\partial x_2} & \cdots & \dfrac{\partial f_1}{\partial x_n} \\ \\ \dfrac{\partial f_2}{\partial x_1} & \dfrac{\partial f_2}{\partial x_2} & \cdots & \dfrac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac{\partial f_n}{\partial x_1} & \dfrac{\partial f_n}{\partial x_2} & \cdots & \dfrac{\partial f_n}{\partial x_n} \end{bmatrix}
By definition of a fixed point, \mathbf{f}(\bar{\mathbf{x}}) = \mathbf{0}. For sufficiently small perturbations \boldsymbol{\delta}, the higher-order terms \mathcal{O}(\|\boldsymbol{\delta}\|^2) can be neglected. This yields a linear system that approximates the dynamics of the perturbation near the fixed point.
\frac{\mathrm{d}\boldsymbol{\delta}}{\mathrm{d}t} \approx \mathbf{J}(\bar{\mathbf{x}}) \boldsymbol{\delta}
The stability and qualitative behavior of the nonlinear system in the vicinity of \bar{\mathbf{x}} can thus be investigated by analyzing the eigenvalues of the constant matrix \mathbf{A} = \mathbf{J}(\bar{\mathbf{x}}).
For example, let’s consider the following nonlinear ode:
\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) = \begin{bmatrix} f_1(x_1, x_2) \\ f_1(x_1, x_2) \end{bmatrix} = \begin{bmatrix} x_1 - x_2^2 \\ x_1 + x_2 \end{bmatrix}
it is possible to calculate the fix points:
\begin{aligned} & \mathbf{f}(\bar{\mathbf{x}}) = \bar{\mathbf{0}} \\ & \begin{bmatrix} x_1 - x_2^2 \\ x_1 + x_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \begin{bmatrix} x_1^1 \\ x_2^1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \begin{bmatrix} x_2^2 \\ x_2^2 \end{bmatrix} = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \end{aligned}
For each of these two fix points it is possible to linearize the system writing the Jacobian:
\begin{aligned} & \dot{\mathbf{x}} = \frac{D\mathbf{f}}{D\mathbf{x}} \bigg\vert_{\mathbf{x} = \bar{\mathbf{x}}} \Delta\mathbf{x} \\ & \\ & \begin{bmatrix} \dot{x_1} \\ \dot{x_2} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x_1} & \dfrac{\partial f_1}{\partial x_2} \\ \\ \dfrac{\partial f_2}{\partial x_1} & \dfrac{\partial f_2}{\partial x_2} \end{bmatrix} = \begin{bmatrix} 1 - 2x_1 & 0 \\ 1 & 1 \end{bmatrix} \end{aligned}
This matrix can be computed for each fix point. For the first point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}^1 = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix. As it is a diagonal, \lambda_{1,2}^{(1)}=1. The solution around this point is unstable. The two identical eigenvectors are:
\begin{aligned} & (\mathbf{A} - \lambda_{1,2}^{(1)} \mathbf{I}) \mathbf{x}_{1,2}^{(1)} = \mathbf{0} \\ & \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} x^{(1)}_{11,12} \\ x^{(1)}_{12,22} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \begin{bmatrix} x^{(1)}_{11,12} \\ x^{(1)}_{12,22} \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \\ \end{aligned}
For the second point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}^{(2)} = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} -1 & 0 \\ 1 & 1 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix. As it is also diagonal it is immediate to compute \lambda_1^{(2)} = -1 and \lambda_2^{(2)} =1. The solution around this point is a saddle, with one stable eigenvalue and an unstable one. The two eigenvectors are:
\begin{aligned} & (\mathbf{A} - \lambda_1^{(2)} \mathbf{I}) \mathbf{x}_1^{(2)} = \mathbf{0} \\ & \begin{bmatrix} -2 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_{11}^{(2)} \\ x_{12}^{(2)} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \begin{bmatrix} x_{11}^{(2)} \\ x_{12}^{(2)} \end{bmatrix} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \\ & (\mathbf{A} - \lambda_2^{(2)} \mathbf{I}) \mathbf{x}_2^{(2)}= \mathbf{0} \\ & \begin{bmatrix} 0 & 0 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} x_{21}^{(2)} \\ x_{22}^{(2)} \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \begin{bmatrix} x_{21}^{(2)} \\ x_{22}^{(2)} \end{bmatrix} = \begin{bmatrix} 2 \\ -1 \end{bmatrix} \end{aligned}
It is possible to graphically plot these local approximation in the phase plane.
The solution is a good approximation around the two fix points.
Conservative system in a potential field dynamics
We can use this method is the analysis of a particle of mass m moving in a one-dimensional conservative force field derived from a potential energy function V(x).
The equation of motion, from either Newtonian or Lagrangian mechanics, is:
m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - \frac{\mathrm{d}V}{\mathrm{d}x}
To analyze this system, we convert the second-order ODE into a first-order system by defining the state vector \mathbf{x} = [x, v]^T, where v = \dot{x} is the velocity.
\begin{aligned} \dot{x} &= v \\ \dot{v} &= -\frac{1}{m}\frac{\mathrm{d}V}{\mathrm{d}x} \end{aligned}
The fixed points (\bar{x}, \bar{v}) of this system must satisfy \bar{v}=0 and -\frac{1}{m}\frac{\mathrm{d}V}{\mathrm{d}x}\big\vert_{x=\bar{x}} = 0. This implies that fixed points in the phase space correspond to critical points of the potential energy function V(x).
The Jacobian matrix for this system is:
\mathbf{J}(x, v) = \begin{bmatrix} 0 & 1 \\ -\frac{1}{m}V^{\prime\prime}(x) & 0 \end{bmatrix}
At a fixed point (\bar{x}, 0), the linearized system is governed by \mathbf{J}(\bar{x}, 0). The characteristic equation is:
\det(\mathbf{J} - \lambda\mathbf{I}) = \lambda^2 + \frac{1}{m}V^{\prime\prime}(\bar{x}) = 0
The eigenvalues are therefore:
\lambda = \pm \sqrt{-V^{\prime\prime}(\bar{x})/m}
If V^{\prime\prime}(\bar{x}) > 0 (a local minimum of the potential), the eigenvalues are purely imaginary:
\lambda = \pm i\sqrt{V^{\prime\prime}(\bar{x})/m}
The linearized system predicts a center, corresponding to stable oscillations around the potential minimum.
If V^{\prime\prime}(\bar{x}) < 0 (a local maximum of the potential), the eigenvalues are real and of opposite sign:
\lambda = \pm \sqrt{|V^{\prime\prime}(\bar{x})|/m}
The linearized system predicts a saddle point, corresponding to an unstable equilibrium.
Let us consider a particle with unit mass (m=1) in the potential:
V(x, \dot{x}) = \dfrac {x^2}{2} - \dfrac{x^3}{3}
The equation of motion is:
\ddot{x} = - \dfrac {\partial V}{\partial x} = -x + x^2
Transforming this second order equation into a first order system:
\begin{aligned} & \dot{x} = v \\ & \ddot{x} = \dot{v} = -x + x^2 = x (x - 1) \end{aligned}
The fix points are [0, 0]^T and [1,0]^T.
For each of these two fix points it is possible to linearize the system writing the Jacobian:
\begin{aligned} & \dot{\mathbf{x}} = \frac{D\mathbf{f}}{D\mathbf{x}} \bigg\vert_{\mathbf{x} = \bar{\mathbf{x}}} \Delta\mathbf{x} \\ & \\ & \begin{bmatrix} \dot{x} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x} & \dfrac{\partial f_1}{\partial v} \\ \\ \dfrac{\partial f_2}{\partial x} & \dfrac{\partial f_2}{\partial v} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 + 2x & 0 \end{bmatrix} \end{aligned}
This matrix can be computed for each fix point. For the first point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_1 = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix. As it is a diagonal \lambda_{1}^{(1)}=i and \lambda_{2}^{(1)}=-i. The solution around this point is a center. The eigenvector are complex.
For the second point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\ & \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix: \lambda_{1}^{(2)}=1 and \lambda_{2}^{(2)}=-1. The solution around this point is a saddle, with one stable eigenvalue and an unstable one. The two eigenvectors are:
\begin{aligned} & (\mathbf{A} - \lambda_1^{(2)} \mathbf{I}) \mathbf{x}_1 = \mathbf{0} \\ & \begin{bmatrix} -1 & 1 \\ -1 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ v_1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \Rightarrow \begin{bmatrix} x_1 \\ v_1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \end{bmatrix} \\ & (\mathbf{A} - \lambda_2^{(2)} \mathbf{I}) \mathbf{x}_2 = \mathbf{0} \\ & \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} x_2 \\ v_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \Rightarrow \begin{bmatrix} x_2 \\ v_2 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \end{aligned}
It is possible to graphically plot these local approximation in the phase plane.
Now, we introduce a linear damping force, -\alpha \dot{x} and consider the case \alpha=1. The system becomes:
\begin{aligned} & \dot{x} = v \\ & \dot{v} = -x + x^2 = x (x - 1) - v \end{aligned}
The fix points are not changed, [0, 0]^T and [1,0]^T.
The part which changes is the Jacobian:
\begin{aligned} & \dot{\mathbf{x}} = \frac{D\mathbf{f}}{D\mathbf{x}} \bigg\vert_{\mathbf{x} = \bar{\mathbf{x}}} \Delta\mathbf{x} \\ & \\ & \begin{bmatrix} \dot{x} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x} & \dfrac{\partial f_1}{\partial v} \\ \\ \dfrac{\partial f_2}{\partial x} & \dfrac{\partial f_2}{\partial v} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 + 2x & -1 \end{bmatrix} \end{aligned}
This matrix can be computed for each fix point. For the first point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_1^{(1)} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ -1 & -1 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix. In this case they will be complex conjugated of the form \lambda_{1}^{(1)}= -a + i and \lambda_{2}^{(1)}= -a -i. The solution around this point is a stable spiral center. The eigenvector are complex.
For the second point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ 1 & -1 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix. (-\lambda)(-1 -\lambda) -1=\lambda^2 - \lambda -1, \lambda^{(2)}_{1,2}=\frac{1\pm \sqrt{5}}{2} and it is a saddle in this case as well.
It is possible to graphically plot these local approximation in the phase plane.
Let’s now consider a different potential function:
\begin{gathered} V(x, \dot{x}) = \dfrac {x^4}{4} - \dfrac{x^2}{2} \\ \ddot{x} = - \dfrac {\partial V}{\partial x} = x - x^3 \end{gathered}
Transforming this second order equation into a first order system:
\begin{aligned} & \dot{x} = v \\ & \ddot{x} = \dot{v} = x - x^3 = x (1 - x^2) \end{aligned}
The fix points are [0, 0]^T, [1,0]^T and [-1,0]^T.
For each of these three fix points it is possible to linearize the system writing the Jacobian:
\begin{aligned} & \dot{\mathbf{x}} = \frac{D\mathbf{f}}{D\mathbf{x}} \bigg\vert_{\mathbf{x} = \bar{\mathbf{x}}} \Delta\mathbf{x} \\ & \\ & \begin{bmatrix} \dot{x} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x} & \dfrac{\partial f_1}{\partial v} \\ \\ \dfrac{\partial f_2}{\partial x} & \dfrac{\partial f_2}{\partial v} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 - 3 x^2 & 0 \end{bmatrix} \end{aligned}
This matrix can be computed for each fix point. For the first point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_1^{(1)} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix: \lambda_{1}^{(1)}=1 and \lambda_{2}^{(1)}=-1. The solution around this point is a saddle, with one stable eigenvalue and an unstable one.
For the second point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_2^{(2)} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ -2 & 0 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix: \lambda_{1}^{(2)}=2i, \lambda_{2}^{(2)}=-2i. The solution around this point is a stable center.
Since the Jacobian in x has only a single term x^2, the solution for [-1,0]^T are identical to [1,0]^T, with \lambda_{1}^{(3)}=2i and \lambda_{2}^{(3)}=-2i and the same behavior.
It is possible to graphically plot these local approximation in the phase plane.
Now, we introduce a linear damping force, -\alpha \dot{x}. The system becomes:
\begin{aligned} \dot{x} &= v \\ \dot{v} &= -V^\prime(x) - \alpha v \end{aligned}
The fixed points remain unchanged, but the Jacobian is modified:
\mathbf{J}(x, v) = \begin{bmatrix} 0 & 1 \\ -V^{\prime\prime}(x) & -\alpha \end{bmatrix}
The characteristic equation becomes \lambda^2 + \alpha\lambda + V^{\prime\prime}(x) = 0. Considering the case \alpha=1:
\begin{aligned} & \dot{x} = v \\ & \dot{v} = x - x^3 = x (1 - x^2) - v \end{aligned}
The fix points are not changed, [0, 0]^T, [1,0]^T and [-1,0]^T.
The part which changes is the Jacobian:
\begin{aligned} & \dot{\mathbf{x}} = \frac{D\mathbf{f}}{D\mathbf{x}} \bigg\vert_{\mathbf{x} = \bar{\mathbf{x}}} \Delta\mathbf{x} \\ & \\ & \begin{bmatrix} \dot{x} \\ \dot{v} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_1}{\partial x} & \dfrac{\partial f_1}{\partial v} \\ \\ \dfrac{\partial f_2}{\partial x} & \dfrac{\partial f_2}{\partial v} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 1 - 3x^2 & -1 \end{bmatrix} \end{aligned}
This matrix can be computed for each fix point. For the first point the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_1 = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \\ & \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ 1 & -1 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix. (-\lambda)(-1 -\lambda) -1=\lambda^2 - \lambda -1, \lambda^{(1)}=\frac{1\pm \sqrt{5}}{2} and it is a saddle in this case as well.
For the second point and the third the matrix becomes:
\begin{aligned} & \bar{\mathbf{x}}_2 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \bar{\mathbf{x}}_3 = \begin{bmatrix} -1 \\ 0 \end{bmatrix} \\ & \\ & \frac{D\mathbf{f}}{D\mathbf{x}} = \begin{bmatrix} 0 & 1 \\ -2 & -1 \end{bmatrix} \end{aligned}
It is now possible to compute the eigenvalues of this matrix. In this case they will be complex conjugated of the form \lambda_{1}^{(2,3)}= -a + 2i and \lambda_{2}^{(2,3)}= -a -2i. The solution around this point is a stable spiral center. The eigenvector are complex.
It is possible to graphically plot these local approximation in the phase plane.
The validity of linearization
The practice of inferring the behavior of a nonlinear system from its linearization is given a mathematical foundation by the Hartman-Grobman theorem here.
This theorem addresses the topological equivalence between the phase portrait of the original nonlinear system and that of its linearization near a fixed point.
Let \bar{\mathbf{x}} be a fixed point of the system \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}). Let \mathbf{A} = \mathbf{J}(\bar{\mathbf{x}}) be the Jacobian matrix evaluated at this point. The fixed point \bar{\mathbf{x}} is classified as hyperbolic if none of the eigenvalues of \mathbf{A} have a zero real part.
\Re(\lambda_i) \neq 0, \quad \forall i
The Hartman-Grobman theorem states that if \bar{\mathbf{x}} is a hyperbolic fixed point, then there exists a neighborhood of \bar{\mathbf{x}} in which the flow of the nonlinear system \dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}) is topologically conjugate to the flow of the linear system \dot{\boldsymbol{\delta}} = \mathbf{A}\boldsymbol{\delta}.
This means there is a continuous mapping (a homeomorphism) that takes trajectories of the linear system to trajectories of the nonlinear system while preserving their orientation in time.
In essence, for hyperbolic fixed points (sinks, sources, saddles, spiral sinks, spiral sources), the linearization correctly predicts the qualitative local dynamics. However, the theorem does not apply to non-hyperbolic fixed points, such as centers, where \Re(\lambda) = 0.
In these marginal cases, the neglected higher-order terms of the Taylor expansion can qualitatively alter the dynamics, for example by turning a center into a stable or unstable spiral. Therefore, for non-hyperbolic equilibrium points, a more detailed analysis beyond linearization is required.