Qualitative analysis of systems of ordinary differential equations.

• Phase Space of the system, showing both the vector field and the trajectories corresponding to different initial conditions.

  Mathematically, consider an autonomous system of ordinary differential equations in \(\mathbb{R}^n\):

  \[ \frac{d\mathbf{x}}{dt} = \mathbf{f}(\mathbf{x}), \] where \(\mathbf{x} = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n\) represents the system's state, and \(\mathbf{f}:\mathbb{R}^n \to \mathbb{R}^n\) is a sufficiently regular vector function (e.g., continuous and continuously differentiable).

  The phase space is precisely \(\mathbb{R}^n\), and each point \(\mathbf{x}\) corresponds to a complete system state. The solutions \(\mathbf{x}(t)\) of the system define trajectories in phase space, illustrating how the state evolves over time.

• Equilibrium points (fixed points) using the Runge-Kutta method, facilitating the study of long-term system behavior.

  A fixed point (or equilibrium point) \(\mathbf{x}^*\) is where the dynamics stop, i.e.:

  \[ \mathbf{f}(\mathbf{x}^*) = \mathbf{0}. \]

  If the system reaches the state \(\mathbf{x}^*\), it will remain there indefinitely, as the rate of change is zero. Analyzing these points and their stability provides valuable information about the system's global behavior, helping determine if solutions tend to these points, move away, or exhibit other complex behaviors.

• Temporal evolution of the involved variables, enabling both quantitative and qualitative approaches to dynamic analysis.