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In 2nd year differential equations, we define a centre to be the graphical phenomena we witness around a critical point with complex eigenvalues in form $\pm bi$.

But, we also throw around this term "non-linear centre," saying that for a conservative system, a linear centre is often a non-linear centre.

I'm having a tough time differentiating between a "non-linear centre" and a "linear centre," being only familiar with what a "centre" is based on my earlier definition.

What's the difference? Could you give an example?

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A linear center is a type of critical point where the eigenvalues of the Jacobian have purely imaginary values, $\pm bi$, as you mentioned. Around a linear center, the system exhibits a periodic behavior, typically resulting in closed orbits or ellipses. The solutions of the system oscillate around the critical point but do not usually grow or decay over time.

On the other hand, a non-linear center refers to a critical point where the behavior of the system around the point is similar to that of a linear center, but the underlying equations are nonlinear. In other words, a non-linear center retains the periodic behavior observed in a linear center but may have additional nonlinear effects. The trajectories of the system still form closed orbits or ellipses, but the shape or size of the orbits may change due to the nonlinearity of the equations.

For an example, consider the following:

Linear Center: Consider the system of differential equations: $$\begin{cases} \dfrac{{dx}}{{dt}} = -y\,,\\ \\ \dfrac{{dy}}{{dt}} = x\,. \end{cases}$$

The critical point at the origin $(0, 0)$ is a linear center. The linearization of this system yields a purely imaginary eigenvalue of $\pm i$. The trajectories of this system will form closed orbits around the origin, akin to circular motion.

non-Linear center: Now, let's modify the previous linear system slightly by introducing a nonlinear term as follows: $$\begin{cases} \dfrac{{dx}}{{dt}} = -y + x^2\,,\\ \\ \dfrac{{dy}}{{dt}} = x + y^2\,. \end{cases}$$

The critical point at the origin remains a center, but it is now a non-linear center. The additional nonlinear terms ($x^2$ and $y^2$) introduce deformations to the circular orbits observed in the linear case. The trajectories will still be closed orbits, but they might be (slightly) distorted due to the nonlinear effects. The size, shape, or orientation of the orbits may change as the system evolves.

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