Many real-world phenomena can be described by differential equations. When we try to capture more intricate behavior, nonlinear terms become necessary, but this also makes analytical solutions rare. In Vassar’s MATH 328 course with Professor Andy Borum, we focus on the qualitative techniques that give us insight into the dynamics without having to solve the equations explicitly.
Below are three examples of real-world phenomena that are described by nonlinear differential equations, followed by our qualitative analysis.
Modeling Bimanual Coordination with the Haken-Kelso-Bunz Model (HKB Model)
Bimanual coordination refers to the way two hands synchronize to achieve a common goal. A simple demonstration can be performed at home:
- Extend both arms in front of you, palms closed into fists, so that your forearms are roughly perpendicular to your torso.
- Rotate each fist about its forearm axis. When the two fists rotate in opposite directions (one clockwise, the other counter‑clockwise) the motion is called in‑phase. When both fists rotate in the same direction the motion is called anti-phase.
- Begin with slow rotations in the anti‑phase mode. You will find this easy to maintain.
- Gradually increase the rotation frequency. At some critical speed, you’ll notice that you subconsciously switched to the in-phase pattern.
This sudden loss of stability can be described using the HKB model, which describes the behavior of two oscillators coupled in the first- and second-harmonic.
$$ \begin{aligned} \dot{\theta}_1&=\omega_1+A_1\sin(\theta_2-\theta_1)+B_1\sin\left(2(\theta_2-\theta_1)\right)\\ \dot{\theta}_2&=\omega_2+A_2\sin(\theta_1-\theta_2)+B_2\sin\left(2(\theta_1-\theta_2)\right) \end{aligned} $$
where \(\theta_i\) represents the angle of each hand, \(A_i\) and \(B_i\) are the coupling strengths of the first- and second-harmonic terms, respectively, and \(\omega_i\) represents the natural frequency of oscillator \(i\).
By introducing the relative phase, \(\phi=\theta_2-\theta_1\) and non-dimensionalizing time via \(t=c_1\tau\), we can get the
more familiar version of the HKB model.
$$ \begin{aligned} \frac{d\phi}{d\tau}&=\mu-\nu\sin(\phi)-\sin(2\phi) \end{aligned} $$
where \(c_1=\frac{1}{B_1+B_2}, \mu=\frac{\omega_2-\omega_1}{B_1+B_2},\) and \(\nu=\frac{A_1+A_2}{B_1+B_2}\).
We can first analyze the scenario where the two oscillators share natural frequencies. In this case, \(\mu=0\). We can then plot the bifurcation diagram of this differential equation. Bifurcation diagrams allow us to view the stability of fixed points as a parameter changes, in this case, \(\nu\).
This does not look like our traditional subcritical pitchfork bifurcation, but we can visualize it a different way to see the pitchfork. If we think about the phase difference we are plotting on the vertical axis, this phase difference, \(\phi\in[-pi,\pi)\). Therefore, if we were to paste this bifurcation diagram to a cylinder, we can see the pitchfork, where the fixed point at \(\pi\) and \(-\pi\) are equivalent.
Understanding Neuron Activity through the FitzHugh-Nagumo Model
The FitzHugh-Nagumo model is a simplified version of the Hodgkin-Huxley model that describes how neurons send signals via a rapid change in membrane voltage. This model captures the exciting and recovering behavior of a single neuron.
$$ \begin{aligned} \dot{v}&=v-\frac{v^3}{3}-w+I(t)\\ \tau\dot{w}&= a+v-bw\end{aligned} $$
where \(a\) and \(b\) are constants, \(\tau\) is a scaling constant for recovery (in our case, \(\tau=1\)), \(v\) represents the membrane voltage in the neuron, \(w\) represents the recovery of the neuron after firing, and \(I(t)\) is an external stimulus that can vary with time.
Because the model is nonlinear and 2D, we are able to see more complex behaviors compared to the previous HKB model. 2D systems give rise to behaviors Hopf bifurcations, which is a point where limit cycles (repeated behavior) exist in the system. The following analysis is all for a single neuron that can be applied to organisms from C. elegans to humans. This analysis highlights how understanding a single neuron is a nontrivial problem, and the human brain, which has billions of coupled neurons is chaotic, and nearly impossible to analyze.
Chaos in Spacecraft Attitude Control
Rotating bodies have been a topic of interest in physics for centuries. A classic example is the intermediate axis theorem-aka tennis racket theorem.
This theorem describes that a rotation about the axis with the intermediate inertia is dynamically unstable. This phenomenon becomes especially
relevant for spacecrafts, whose attitude must be controlled.
To analyze this behavior, we first focus on the no-control scenario, where we find a visual description of the instability.
Later, we apply two scenarios of torque to the system. One of the scenarios is where the torque is dissipative, similar to air resistance.
The second is a state-feedback control that can paradoxically lead to chaotic behavior depending on the gains.
These dynamics are described by the following 3D differential equation. Since this system is now in three dimensions, it gives rise to even more complex behavior, such as chaos and strange attractors.
$$ \begin{aligned} I_1\dot{\omega}_1&=(I_2-I_3)\omega_2\omega_3+m_1(t)\\ I_2\dot{\omega}_2&=(I_3-I_1)\omega_3\omega_1+m_2(t)\\ I_3\dot{\omega}_3&=(I_1-I_2)\omega_1\omega_2+m_3(t) \end{aligned} $$
where \(I_1,I_2,\) and \(I_3\) are the principal moments of inertia (\((I_1>I_2>I_3\)), \(m_i(t)\) are applied torques, and \(\omega_i\) is the angular velocity about the ith principal axis.
This equation can also be conveniently written in vector form if we define the angular momenta as \(p_i(t) = I_i\omega_i(t)\). The simplified vector form of the differential equation is given below.
$$ \begin{aligned} \boldsymbol{\dot{P}}&=\boldsymbol{P}\times\boldsymbol{\Omega}+\boldsymbol{M}(t) \end{aligned} $$