Applications of Nonlinear Systems

MATH 341 — Notes 13

University of Scranton

2026-06-16

Goals

What We Will Cover

  1. Lotka–Volterra predator–prey model — periodic orbits, conserved quantity, the exceptional center case

  2. Competing species — coexistence vs. competitive exclusion depending on competition strength

  3. SIR epidemic model — disease-free equilibrium, the basic reproduction number \(R_0\)

  4. Chemical kinetics — law of mass action, consecutive reactions, asymptotically stable node

  5. Nonlinear pendulum — centers vs. saddles, energy conservation, the separatrix

  6. Van der Pol oscillator — unstable origin, attracting limit cycle, self-sustained oscillations

Note

Section 5.3 of (Logan 2015).

Tip

New vocabulary: predator–prey, conserved quantity, competitive exclusion, basic reproduction number \(R_0\), law of mass action, separatrix (pendulum), limit cycle.

Lotka–Volterra Predator–Prey

The Model

Let \(x(t)\) = prey, \(y(t)\) = predator. Parameters \(a,b,c,d > 0\).

\[x' = ax - bxy, \qquad y' = -cy + dxy. \tag{5.14--5.15}\]

Term Meaning
\(ax\) Prey grow exponentially without predators
\(-bxy\) Predation reduces prey
\(-cy\) Predators die without prey
\(dxy\) Predators grow by consuming prey

Critical points: \((0,0)\) and \(\left(\dfrac{c}{d},\dfrac{a}{b}\right)\).

Jacobian Analysis

\[J(x,y)=\begin{pmatrix}a-by & -bx \\ dy & -c+dx\end{pmatrix}.\]

At \((0,0)\): eigenvalues \(a>0\) and \(-c<0\) \(\implies\) unstable saddle.

At \(\left(\dfrac{c}{d},\dfrac{a}{b}\right)\): \[J=\begin{pmatrix}0 & -bc/d \\ da/b & 0\end{pmatrix}, \quad \lambda^2+ac=0, \quad \lambda=\pm i\sqrt{ac}.\]

Purely Imaginary Eigenvalues — Exceptional Case

Linearization gives a center. We cannot conclude whether the nonlinear system has a center or a spiral from linearization alone.

Resolution: find the conserved quantity \(H(x,y) = dx - c\ln x + by - a\ln y\). Its level curves are the orbits — they are closed, confirming the nonlinear system also has a center.

Conserved Quantity and Periodic Orbits

Divide \(y'\) by \(x'\) and separate variables:

\[\frac{-c+dx}{x}\,dx + \frac{a-by}{y}\,dy = 0.\]

Integrating:

\[\boxed{H(x,y) = dx - c\ln x + by - a\ln y = \text{const.}}\]

  • Level curves of \(H\) are closed curves encircling \((c/d,\, a/b)\).
  • Both populations oscillate periodically — the classic boom–bust cycle.
  • The orbit period depends on the energy level (orbit size).

Note

Volterra used this model to explain why fishing reductions during WWI increased the proportion of predatory fish (sharks) in the Adriatic — because removing both species equally shifts the oscillation average.

Phase Portrait and Time Series

Figure 1

Competing Species

The Model

Two species compete for the same resource. Logistic growth with interspecific competition:

\[x' = x(a_1 - b_1 x - c_1 y), \qquad y' = y(a_2 - b_2 y - c_2 x).\]

Four critical points:

Equilibrium Meaning
\((0,0)\) Both extinct (always unstable)
\((a_1/b_1,\; 0)\) \(x\) alone at carrying capacity
\((0,\; a_2/b_2)\) \(y\) alone at carrying capacity
\((x^*,y^*)\) (interior) Potential coexistence

Outcome depends on competition strengths \(c_1, c_2\) relative to \(b_1, b_2\):

  • \(c_1 c_2 < b_1 b_2\) (weak competition): interior equilibrium is stable — coexistence.
  • \(c_1 c_2 > b_1 b_2\) (strong competition): interior equilibrium is a saddle — competitive exclusion; one species wins depending on initial conditions.

Competing Species — Phase Portraits

Figure 2

SIR Epidemic Model

The Model

Compartments: \(S\) (susceptible), \(I\) (infected), \(R\) (removed). \(N = S+I+R\) constant.

\[S' = -\beta SI, \quad I' = \beta SI - \gamma I, \quad R' = \gamma I. \tag{5.18--5.20}\]

Parameter Meaning
\(\beta\) Transmission rate (contacts × infection probability)
\(\gamma\) Recovery rate (\(1/\gamma\) = mean infectious period)

The 2D system \((S,I)\) is autonomous. From \(I' = I(\beta S - \gamma)\):

  • \(I'>0\) (epidemic grows) \(\iff\) \(S > \gamma/\beta\).
  • \(I'<0\) (epidemic dies) \(\iff\) \(S < \gamma/\beta\).

The Basic Reproduction Number \(R_0\)

At the start of an outbreak, \(S \approx N\), so \(I' \approx I(\beta N - \gamma)\).

Basic Reproduction Number

\[R_0 = \frac{\beta N}{\gamma}\]

= average number of secondary infections from one infected individual in a fully susceptible population.

\(R_0 > 1\) | Epidemic occurs |
\(R_0 < 1\) | Infection dies out |
\(R_0 = 1\) | Threshold (endemic equilibrium) |

Stability of disease-free state \((S^*, 0)\):

\[J(S^*,0)=\begin{pmatrix}0 & -\beta S^* \\ 0 & \beta S^*-\gamma\end{pmatrix}, \quad \lambda_2 = \beta S^* - \gamma < 0 \iff R_0 < 1.\]

SIR Phase Portrait and Time Series

Figure 3

Chemical Kinetics

Law of Mass Action

The law of mass action states that the rate of a reaction is proportional to the product of the concentrations of the reacting species.

For consecutive reactions \(A \xrightarrow{k_1} B \xrightarrow{k_2} C\), letting \(x=[A]\) and \(y=[B]\):

\[x' = -k_1 x, \qquad y' = k_1 x - k_2 y. \tag{5.21--5.22}\]

Only equilibrium: \((0,0)\) — all material converts to \(C\).

\[J(0,0) = \begin{pmatrix}-k_1 & 0 \\ k_1 & -k_2\end{pmatrix}, \quad \lambda = -k_1,\,-k_2 < 0.\]

\(\implies\) Asymptotically stable node — all concentrations approach chemical equilibrium.

Analytical solution (first equation decouples):

\[[A](t) = x_0 e^{-k_1 t}, \quad [B](t) = \frac{k_1 x_0}{k_2-k_1}\left(e^{-k_1 t}-e^{-k_2 t}\right), \quad [C]=x_0-x-y.\]

\([B]\) has a characteristic intermediate peak before decaying to zero.

Chemical Kinetics — Plot

Figure 4

Tip

The intermediate peak in \([B](t)\) is a hallmark of consecutive reactions. Its height and time of occurrence depend on the ratio \(k_1/k_2\).

Nonlinear Pendulum

The Model

Newton’s law for a pendulum: \(\theta'' + \omega_0^2\sin\theta = 0\).

Let \(x=\theta\), \(y=\theta'\):

\[x' = y, \qquad y' = -\omega_0^2\sin x. \tag{5.23--5.24}\]

Equilibria: \(y=0\) and \(\sin x=0\), so \(x=n\pi\).

Jacobian: \(J(x,y) = \begin{pmatrix}0 & 1 \\ -\omega_0^2\cos x & 0\end{pmatrix}\).

Equilibrium \(\cos x\) \(\lambda\) Type
\((0,0),(2\pi,0),\ldots\) \(+1\) \(\pm i\omega_0\) Center (pendulum down)
\((\pi,0),(3\pi,0),\ldots\) \(-1\) \(\pm\omega_0\) Saddle (pendulum up)

Energy and the Separatrix

The system is conservative with energy

\[H(x,y) = \tfrac{1}{2}y^2 - \omega_0^2\cos x = \text{const.}\]

Orbits lie on level curves of \(H\):

Energy level Orbit type Motion
\(H < \omega_0^2\) Closed Oscillation (swings back and forth)
\(H = \omega_0^2\) Separatrix Approaches inverted position asymptotically
\(H > \omega_0^2\) Open Full rotation (swings over the top)

The separatrix is the boundary between oscillatory and rotational motion. It passes through the saddle points \((\pm\pi, 0)\).

Pendulum Phase Portrait

Figure 5

Van der Pol Oscillator

The Model

Nonlinear electronic circuit oscillator (van der Pol, 1926):

\[\mu'' - \varepsilon(1-\mu^2)\mu' + \mu = 0, \quad \varepsilon > 0.\]

Let \(x=\mu\), \(y=\mu'\):

\[x' = y, \qquad y' = \varepsilon(1-x^2)y - x. \tag{5.25--5.26}\]

Nonlinear damping \(\varepsilon(1-x^2)y\):

  • \(|x|<1\): negative damping — energy added (system amplifies small oscillations)
  • \(|x|>1\): positive damping — energy removed (system suppresses large oscillations)

This self-regulating mechanism produces a stable limit cycle — all orbits converge to the same periodic orbit regardless of initial conditions.

Critical Point Analysis

Only critical point: \((0,0)\).

\[J(0,0)=\begin{pmatrix}0&1\\-1&\varepsilon\end{pmatrix}, \quad \text{tr}\,J=\varepsilon>0, \quad \det J=1>0.\]

\(\implies\) Unstable spiral (for small \(\varepsilon\)). Orbits spiral outward from the origin.

Limit Cycle

A limit cycle is an isolated closed orbit. The van der Pol system has a stable (attracting) limit cycle:

  • Orbits inside the cycle spiral outward toward it.
  • Orbits outside the cycle spiral inward toward it.
  • This represents self-sustained oscillations — independent of initial conditions.

Limit cycles are a purely nonlinear phenomenon: they cannot exist in linear systems.

Van der Pol Phase Portrait

Figure 6

Summary

Key Takeaways

  • Lotka–Volterra: the non-trivial equilibrium \((c/d,\, a/b)\) is a center for the linearization and a true center for the nonlinear system, confirmed by the conserved quantity \(H = dx - c\ln x + by - a\ln y\). Both populations oscillate periodically.

  • Competing species: if \(c_1 c_2 < b_1 b_2\) (weak competition) the interior equilibrium is stable and both species coexist; if \(c_1 c_2 > b_1 b_2\) (strong competition) the interior equilibrium is a saddle and competitive exclusion occurs — which species wins depends on initial conditions.

  • SIR epidemic: the key quantity is the basic reproduction number \(R_0 = \beta N/\gamma\). An epidemic occurs if and only if \(R_0 > 1\). The disease-free equilibrium is stable when \(R_0 < 1\), unstable when \(R_0 > 1\).

  • Chemical kinetics: consecutive reactions \(A\to B\to C\) give an asymptotically stable node at the origin; the intermediate species \(B\) exhibits a characteristic peak before decaying to zero.

  • Nonlinear pendulum: centers at the downward equilibria and saddles at the inverted equilibria. The conserved energy \(H = \tfrac{1}{2}y^2 - \omega_0^2\cos x\) gives the orbits; the separatrix (\(H = \omega_0^2\)) bounds oscillatory from rotational motion.

  • Van der Pol: the origin is an unstable spiral; all orbits are attracted to a stable limit cycle — a purely nonlinear phenomenon representing self-sustained oscillations.

Tip

Looking Ahead: The existence of limit cycles and their stability is addressed by the Poincaré–Bendixson theorem, which provides criteria for when planar autonomous systems must have periodic orbits — a fundamental result in nonlinear dynamics.

Note

Next: Numerical methods — Logan §6.2.

References

Logan, J David. 2015. A First Course in Differential Equations, Third Edition.