Critical Point Analysis
In the context of differential equations, a critical point, also known as an equilibrium point, is a solution where the system does not change over time, meaning that the first derivatives with respect to time are zero (i.e., \(x' = y' = 0\)). To find the critical points of a given system, one solves the equations obtained by setting the derivatives equal to zero. In the special case of the FitzHugh-Nagumo model, regardless of the value of the external stimulus parameter \(k\), the analysis shows there's always one critical point.
In practice, we solve the system of equations presented to find the precise location of the critical points. For example, with \(k = 0\) and \(k = 0.5\), we get different critical points, each holding distinct implications for the behavior of the system. The critical point changes with \(k\), indicating a shift in system dynamics, a phenomenon which becomes evident when observing the system's phase portraits at varying \(k\) values.
Stability of Differential Equations
The stability of a critical point in a system of differential equations can be described as the behavior of nearby points as time progresses. To determine stability, we assess the nature of the critical point by examining the eigenvalues of the Jacobian matrix evaluated at that point. If the real parts of all eigenvalues are negative, the critical point is asymptotically stable, meaning nearby solutions will tend toward this point as time approaches infinity. However, if there is an eigenvalue with a positive real part, the critical point is unstable, and trajectories will diverge away from it.
For the FitzHugh-Nagumo model's critical points at \(k = 0\) and \(k = 0.5\), the stability changes. The critical point is asymptotically stable for \(k = 0\), but becomes an unstable spiral point when \(k = 0.5\). This indicates that small perturbations will cause the trajectories to spiral away from the critical point.
Phase Portrait
A phase portrait provides a visual representation of the trajectories of a dynamical system in the phase plane. Each point on this plane represents a state of the system with a particular combination of variable values. For the FitzHugh-Nagumo equations, the phase portraits allow us to visualize how the system evolves over time for different values of \(k\).
These portraits illustrate the stable and unstable behaviors at the critical points. For instance, for \(k = 0\), the phase portrait will show trajectories spiraling toward the critical point, denoting stability. In contrast, for \(k = 0.5\), the trajectories spiral outwards, highlighting the critical point's instability. By continuously varying \(k\), we can see how the system dynamics evolve and how the stability of the critical point changes.
Hopf Bifurcation
A Hopf bifurcation occurs in a system of differential equations when a pair of complex conjugate eigenvalues crosses the imaginary axis as a parameter (in this case, \(k\)) is varied, leading to a change in stability of the critical point and typically giving rise to or annihilating a limit cycle. For the FitzHugh-Nagumo model, as \(k\) varies, the critical point changes from being asymptotically stable to unstable, indicating a Hopf bifurcation at the value \(k_{0}\).
It is this bifurcation that results in the emergence of an asymptotically stable limit cycle for \(k > k_{0}\), where the system's behavior changes from approaching a stable point to oscillating around it. The presence of this limit cycle can be confirmed through phase portraits showing closed loops at values of \(k\) just after \(k_{0}\).
Limit Cycle Behavior
Limit cycles represent periodic solutions to differential equations and manifest as closed trajectories in the phase plane. These cyclical paths indicate that the system's variables will repeat their values over time, typical of oscillatory systems such as those modeling biological rhythms or electrical circuits.
For the FitzHugh-Nagumo model, an asymptotically stable limit cycle emerges post-\(k_{0}\), where trajectories in the phase portrait are attracted to a closed loop, suggesting persistent oscillations. As \(k\) increases from \(k_{0}\), the size and shape of the limit cycle change until a second value \(k_{1}\) is reached, beyond which the limit cycle disappears, and the critical point regains stability. This outcome shows the richness of dynamical behaviors possible within such mathematical models.
