What does it mean for a linear approximation to be reliable?

In summary, linearized equations are considered reliable when they are linearly stable, meaning that the time dependent solution does not diverge from the nonlinear solution. This is determined by replacing the nonlinear solution with a time independent solution, and studying its eigenvalues. However, linear stability is a weak criteria for overall stability, as other types of perturbations can affect the system's long term behavior. Energetic stability, dynamical stability, and linear stability are all related, but the first two are more difficult to analyze and are typically only done for simple systems.
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Homework Statement


In regards to linearization of a nonlinear system in differential equations. What does it mean for a linear approximation to be reliable to describe the long term behavior of the non-linear system around the equilibrium point?

Homework Equations


jacobian matrix

The Attempt at a Solution


General question.
 
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Linearized equations are "reliable" when the equations are linearly stable, i.e. the time dependent solution of the linearized system do not diverge from the nonlinear solution. The linearized solution won't capture all features of the nonlinear solution but at least it gives you a rough idea about the time evolution. This is equivalent to saying that the equations are linearly stable.

To study the linear stability you replace, roughly speaking, the nonlinear solution as following [tex]\Phi(x)\rightarrow\Phi_0(x) + \delta(x)e^{\lambda t}[/tex] where ##\Phi_0(x)## is a time independent solution of the nonlinear equation. After plugging ##\Phi_0(x)+\delta(x)e^{\lambda t}## into the nonlinear equation one has to determine the eigenvalues ##\lambda##. If ##Re\{\lambda\}>0## then perturbation ##\delta## will grow with time and the solution ##\Phi_0(x)## it is said to be linearly unstable.
Very important: keep in mind that the linear stability depends on the (is associated with a) time independent solution of the nonlinear equation. It may happened that a solution may be linearly stable while others not.
See for instance the wikipedia page.

However, the linear stability is a weak criteria when deciding whether a system is stable or not. This means that even if solution is linearly stable don't imply that it will follow the long time behavior of the nonlinear equation. Aside from the linear perturbations there are other types of perturbations which may set in and affect the time development. The stability chain is as following [tex]Energetic\: stability \Rightarrow Dynamical\: stability\Rightarrow Linear\: stability[/tex] The linear stability is used to rule out the stability, is the system is not linearly stable then it won't be neither dynamical nor energetic stable. The energetic and dynamical stabilities are in general cumbersome to undertake, one should study the Hamiltonian structure and, something like, the Lyapunov stability (related directly to the time dependent evolution of the solution). They are performed only for simple nonlinear systems and solutions.

LE: You can follow, for instance, this notes as guide on linear stability analysis.
 
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1. What is a linear approximation?

A linear approximation, also known as a tangent line approximation, is a method used in calculus to estimate the value of a function at a specific point by using the slope of the function at that point.

2. How is a linear approximation calculated?

A linear approximation is calculated by finding the slope of the function at a specific point and using it to create a tangent line. The equation of the tangent line is then used to estimate the value of the function at that point.

3. Why is a linear approximation useful?

A linear approximation is useful because it allows us to estimate the value of a function at a specific point without having to use complicated calculus techniques. It is also used to approximate values in real-world applications where precise calculations are not necessary.

4. When is a linear approximation considered reliable?

A linear approximation is considered reliable when the function is differentiable at the point of approximation and the interval between the point and the approximation is small. Additionally, the approximation becomes more reliable as the interval decreases.

5. What are the limitations of a linear approximation?

Linear approximation is limited to functions that are differentiable at the point of approximation. It also becomes less accurate for functions with a high degree of curvature or when the interval between the point and the approximation is too large.

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