Show that the system \(x^{\prime}=x-y-x^{3}, y^{\prime}=x+y-y^{3}\), has at least one limit cycle by picking an appropriate \(\psi(x, y)\) in Dulac's Criteria.

A limit cycle is a closed trajectory in the phase space of a dynamical system that is approached by neighboring trajectories as time goes to infinity. Think of it as a path that the system's state follows, looping indefinitely without ever settling down to a single point or escaping to infinity.

For the system of differential equations given in the exercise, a limit cycle would represent a state where the system undergoes repetitive behavior. For example, it might represent an oscillation in a predator-prey model, voltage in a neuron model, or the cycle of a chemical reaction.

Identifying the existence of a limit cycle is crucial because it reveals the long-term behavior of the system under study. Systems with limit cycles are particularly interesting in the study of biological rhythms, electrical circuits, and many other scientific fields where periodic behavior is observed.

A system of differential equations is a set of equations that relate the rates of change of different variables to each other and possibly to the variables themselves. In the case of the exercise, the given system involves two differential equations, one for each variable, x and y.

The system demonstrates how x and y change over time based on their current values. Often, such systems require us to consider both equations simultaneously because the rate of change of one variable may affect the other. This interconnectedness is what produces the complex dynamics that can lead to phenomena like limit cycles when certain conditions are met.

When solving these systems, we look for solutions that can tell us how x and y behave, individually and in relation to each other, as time progresses. Such analysis allows us to predict long-term behavior and understand the underlying mechanism driving the system's evolution.

Divergence is a mathematical operator used in vector calculus that measures the magnitude of a vector field's source or sink at a given point. In simpler terms, it quantifies how much a vector field is spreading out or converging in space.

In the context of the exercise, the vector field represents the direction and magnitude of the change of the system state at each point. To apply Dulac's Criterion, we compute the divergence of the product of the Dulac function \(\psi(x, y)\) and the vector field \((x', y')\).

If the divergence is not identically zero over some region, Dulac's Criterion implies that the system does not have a fixed equilibrium in that region and may possess a limit cycle. In essence, the divergence helps us understand whether the system's state will converge to a point, diverge away, or continue in a periodic cycle, which is fundamentally important for studying the stability and long-term behavior of dynamical systems.