Show that linearization is a well-defined operation: the operator \(A\) does not depend on the coordinate system. The advantage of the linearized system is that it is linear and therefore easily solved: \(\mathbf{y}(t)=e^{A t} \mathbf{y}(0), \quad\) where \(e^{A t}=E+A t+\frac{A^{2} t^{2}}{2 !}+\cdots .\)

Understanding the Jacobian matrix is essential for analyzing dynamical systems. It represents how small changes in the state variables can affect the rate of change of the system itself. Formally, the Jacobian matrix is a multidimensional array of first-order partial derivatives. In the context of a dynamical system

\[\begin{equation}\frac{d \mathbf{x}}{dt} = \mathbf{F}(\mathbf{x}),\end{equation}\]
the Jacobian matrix at a fixed point \(\mathbf{x}^*\) is defined as
\[\begin{equation}A = \frac{\partial \mathbf{F}}{\partial \mathbf{x}}\bigg|_{\mathbf{x}=\mathbf{x}^*}.\end{equation}\]
In essence, the Jacobian quantifies the local behavior of the system near equilibrium points, and it can be used to determine the stability and nature of these points. A clear understanding of the Jacobian matrix is critical because it is extensively employed when linearizing and solving dynamical systems, giving insights into their behavior near specific points.

Coordinate transformation is a powerful tool used to simplify and solve complex problems in various scientific disciplines, including the study of dynamical systems. The main idea is to change the variables of the system from one set to another, which can make the problem easier to tackle. This is often achieved through a transformation matrix \(T\), establishing a relationship between the old coordinates \(\mathbf{x}\) and new coordinates \(\mathbf{v}\) via

\[\begin{equation} \mathbf{v} = T \mathbf{x}.\end{equation}\]
The transformation matrix must be invertible to ensure that the two coordinate systems are equivalent, and information is not lost in the process. After transforming, the dynamics of the system are expressed in the new variables, and any fixed points must also be translated accordingly. The ability to perform coordinate transformations allows the study of the system under different perspectives, potentially revealing conserved quantities or simplifying the analysis of stability and other properties.

The exponential of a matrix is a concept borrowed from the exponential function of real numbers. For a square matrix \(A\), the exponential \(e^{At}\) is defined via the power series

\[\begin{equation} e^{At} = E + At + \frac{A^{2} t^{2}}{2!} + \cdots\end{equation}\]
where \(E\) denotes the identity matrix. In the context of linearized dynamical systems, this expression provides the solution to the system of linear differential equations. The matrix exponential is particularly noteworthy in the study of linear systems because it relates directly to the system's flow and the evolution of states with time. If we know the initial state \(\mathbf{y}(0)\), then the future states can be computed using the matrix exponential, offering an explicit solution to the linearized system.

Stability in dynamical systems is a fundamental concept that helps predict the long-term behavior of those systems. The stability of a fixed point, or an equilibrium, can often be deduced from the linearized system around that point. Namely, if the real parts of all eigenvalues of the Jacobian matrix at the fixed point are negative, the equilibrium is stable—it will return to equilibrium after a small disturbance.

An understanding of stability aids in predicting whether a system will return to a steady state or diverge away from it over time, which is crucial for diverse scientific and engineering problems. The linearized system provides a simplified version of the dynamics close to equilibrium points, helping to capture essential features of the system's behavior without solving the full, often non-linear, equations.