91Ó°ÊÓ

Equilibrium points in a dynamical system are locations where the system does not change, meaning the derivative terms are zero. For a given system of differential equations, these points indicate where the variables remain constant over time. To find the equilibrium points, set each derivative equal to zero and solve the resulting equations.
In the exercise, we have the differential equations:

• \(x^{\prime}=y\)
• \(y^{\prime}=-\sin x-y\)

To find an equilibrium point, both derivatives must be zero at the same time. This occurs when:

• \(y = 0\)
• \(-\sin x - y = 0\)

Substituting \(y = 0\) into the second equation gives \(\sin x = 0\). The solutions for \(x\) are any integer multiples of \(\pi\), so the equilibrium points are at \((n\pi, 0)\), where \(n\) is an integer. These points represent the system's steady states.

Nullclines are curves in the phase plane where one of the derivatives of a system of differential equations is zero. They help visualize how the phase plane is divided and predict the behavior of the trajectories.
For the given system, we determined the nullclines by setting each derivative to zero:

• The y-nullcline: Set \(x' = y = 0\). This yields the x-axis; every point along this line has a zero y-derivative.
• The x-nullcline: Set \(y' = -\sin x - y = 0\) to solve for \(y = -\sin x\). This is a sinusoidal curve reflecting and shifting the sine wave pattern along the y-axis.

By sketching these nullclines, we understand that trajectories cross the y-nullcline horizontally and interface with the x-nullcline vertically. Where these lines intersect are the equilibrium points, giving a clear picture of how trajectories behave around these areas.

The Jacobian matrix in the context of dynamical systems is a matrix that describes how the system dynamics change in the vicinity of an equilibrium point. It is composed of the first-order partial derivatives of the system's functions.
For our system:

• \(x^{\prime}=y\)
• \(y^{\prime}=-\sin x-y\)

The Jacobian matrix \(J\) of the system is:\[\begin{bmatrix} 0 & 1 \ -\cos x & -1 \end{bmatrix}\]
This matrix is evaluated at each equilibrium point \((n\pi, 0)\). The cosine function alternates its value based on the integer \(n\): \(\cos(n\pi) = (-1)^n\). Thus, the stability of each equilibrium depends on whether \(n\) is even or odd, leading to either stable or unstable points. Checking the eigenvalues will indicate the type of stability: nodes, saddles, spirals, etc. Therefore, analyzing \(J\) provides deep insight into how the system network reacts at equilibrium, effectively classifying each equilibrium point in terms of stability and type.