Plot phase portraits for the Lienard system $$ \begin{aligned} x^{\prime} &=y-\mu\left(x^{3}-x\right) \\ y^{\prime} &=-x \end{aligned} $$ for a small and a not so small value of \(\mu\). Describe what happens as one varies \(\mu\).

A dynamical system is a concept used to model the behavior of complex systems that change over time. In mathematics, this typically refers to a system of equations that define how the state of the system evolves. The state is represented by a set of variables, and the evolution is governed by rules or equations that describe how these variables interact.

For example, the Lienard system provided in the exercise is a twodimensional dynamical system with state variables \(x\) and \(y\). It consists of two differential equations that describe the time evolution of these variables. In the context of the exercise, the system models the behavior of the variables based on their rate of change. For a small value of \(\mu\), the system behaves differently than for a large value of \(\mu\), and studying these behaviors is essential in understanding how dynamical systems respond to changes in parameters.

An essential aspect of studying dynamical systems is simulating their behavior under various initial conditions to observe patterns, transitions, and the emergence of different behaviors as parameters are varied. This study often results in a better understanding of the system's dynamics and the prediction and control of its future state.

Phase portrait analysis is a powerful tool used to visualize and understand the behavior of dynamical systems. A phase portrait is a graphical representation where the solutions to the differential equations are plotted in the state space of the system. For the Lienard system, this means we plot trajectories in the \(x-y\) plane, where each point on this plane represents a state of the system with specific \(x\) and \(y\) values.

Constructing phase portraits involves plotting trajectories corresponding to different initial conditions. These trajectories help to identify critical points, also known as equilibrium points, where the system does not change over time. In the case of varying \(\mu\), analyzing the phase portrait allows one to see how these critical points move and how their stability might change.

The structure of a phase portrait can convey a wealth of information about the system, such as whether the system settles into a steady state, oscillates periodically, or behaves chaotically. Notably, phase portraits do not typically represent time explicitly; instead, they show how the system's state changes relative to different starting conditions, revealing the long-term behavior of the system.

The stability of critical points is an essential characteristic to analyze in any dynamical system. These are points in the system's phase space where the system does not exhibit any change over time; in other words, they are the points where the system's state variables cease evolving. Understanding the stability of these points helps predict long-term behavior and the system's response to small perturbations around these points.

In the Lienard system, critical points are found by setting the time derivatives of \(x\) and \(y\) to zero and solving the resulting equations. Stability is then determined by analyzing the signs and magnitudes of the eigenvalues of the Jacobian matrix evaluated at the critical points. If all eigenvalues have negative real parts, the point is considered stable (an attractor). Conversely, a critical point is unstable (a repeller) if any eigenvalue has a positive real part.

As one varies \(\mu\) in the Lienard system, the number and type of critical points can change, and their stability can shift. For instance, a critical point might transition from stable to unstable or vice versa, fundamentally altering the dynamics. This understanding is crucial, especially in systems where stability signifies a well-behaved, predictable outcome, while instability might indicate chaotic or unpredictable behavior.