One-Dimensional Dynamical Systems

MATH 341 — Notes 3

University of Scranton

2026-06-16

Goals

What We Will Cover

Autonomous ODEs \(x'=f(x)\) as one-dimensional dynamical systems
Equilibrium solutions and stability via the phase line
Linearization — the algebraic stability criterion \(f'(x^*)\)
Bifurcations — qualitative changes in equilibrium structure
Bifurcation diagrams for the three classical cases

Note

Section 1.5 of (Logan 2015).

Autonomous Equations

The Setup

An autonomous ODE has no explicit dependence on \(t\): \[x' = f(x).\]

The rate of change of \(x\) depends only on the current state \(x\), not on when that state occurs.

Key structural property. The slope field is constant along every horizontal line \(x = c\).

\[\text{slope} = f(x) \quad \longrightarrow \quad \text{same value for all } t\]

This means the entire qualitative story lives on the \(x\)-axis alone — we never need to solve the ODE to understand the long-term behavior.

Examples: population size, temperature, voltage, chemical concentration.

Equilibrium Solutions

Definition

An equilibrium (steady state, fixed point) of \(x' = f(x)\) is a constant solution \(x(t) \equiv x^*\) satisfying \[f(x^*) = 0.\] Equilibria are the zeros of \(f\) — where the slope field is horizontal.

Geometric picture. In the \(tx\)-plane, an equilibrium is a horizontal line. On the phase line it is a single point.

If \(f(x) > 0\) near \(x^*\)	If \(f(x) < 0\) near \(x^*\)
Solutions increase	Solutions decrease
Flow moves right \(\rightarrow\)	Flow moves left \(\leftarrow\)

The Phase Line

Algorithm: Constructing the Phase Line

Solve \(f(x^*)=0\) — mark equilibria on the \(x\)-axis.
Determine the sign of \(f(x)\) on each interval:
- \(f>0\): arrow points right \(\rightarrow\) (increasing)
- \(f<0\): arrow points left \(\leftarrow\) (decreasing)
Draw flow arrows.

Reading stability from the phase line:

Arrows	Equilibrium type
Both point toward \(x^*\)	Stable (asymptotically stable)
Both point away from \(x^*\)	Unstable
One toward, one away	Semi-stable

Stability Definitions

Stability

Stable: solutions starting near \(x^*\) converge to \(x^*\) as \(t\to\infty\)
Unstable: solutions starting near \(x^*\) move away from \(x^*\)
Semi-stable: solutions approach from one side, diverge from the other

Phase Line Examples

Example 1 — Logistic Equation

\[x' = rx\!\left(1-\frac{x}{K}\right), \quad r=1,\; K=4\]

Equilibria: \(x^*=0\) and \(x^*=4\).

Sign analysis: \(f<0\) for \(x<0\); \(\;f>0\) for \(0<x<4\); \(\;f<0\) for \(x>4\).

Example 2 — Multiple Equilibria: \(x'=x^3-x\)

\[f(x)=x(x-1)(x+1), \quad \text{equilibria: } x^*=-1,\,0,\,1\]

Equilibrium	\(f'(x^)=3(x^)^2-1\)	Stability
\(x^*=-1\)	\(+2>0\)	Unstable
\(x^*=0\)	\(-1<0\)	Stable
\(x^*=1\)	\(+2>0\)	Unstable

Linearization

The Linearization Criterion

Write \(x(t) = x^* + u(t)\) where \(u\) is a small perturbation. Taylor-expand \(f\) about \(x^*\): \[u' = f(x^*+u) = \underbrace{f(x^*)}_{=0} + f'(x^*)\,u + O(u^2)\]

For small \(u\), drop the \(O(u^2)\) terms: \[\boxed{u' \approx f'(x^*)\,u \quad\Longrightarrow\quad u(t) = u(0)\,e^{f'(x^*)t}}\]

Stability Criterion (Hyperbolic Equilibria)

\(f'(x^*)\)	Behavior	Stability
\(< 0\)	\(e^{f'(x^*)t} \to 0\)	Asymptotically stable
\(> 0\)	\(e^{f'(x^*)t} \to \infty\)	Unstable
\(= 0\)	Test inconclusive	Need higher-order terms

The quantity \(\lambda = f'(x^*)\) is the eigenvalue of the equilibrium.

Linearization: Two Examples

Logistic \(x'=rx(1-x/K)\), \(f'(x) = r(1-2x/K)\):

\[f'(0) = r > 0 \;\Rightarrow\; \text{unstable} \qquad f'(K) = -r < 0 \;\Rightarrow\; \text{stable} \checkmark\]

Cubic \(x'=x^3-x\), \(f'(x)=3x^2-1\):

\[f'(\pm 1) = 2 > 0 \;\Rightarrow\; \text{unstable} \qquad f'(0) = -1 < 0 \;\Rightarrow\; \text{stable} \checkmark\]

x_sym, r_sym, K_sym = sym.symbols('x r K', real=True, positive=True)
f_logistic = r_sym * x_sym * (1 - x_sym / K_sym)
fp = sym.diff(f_logistic, x_sym)
print("f'(x) =", fp)
print("f'(0) =", sym.simplify(fp.subs(x_sym, 0)))
print("f'(K) =", sym.simplify(fp.subs(x_sym, K_sym)))

f'(x) = r*(1 - x/K) - r*x/K
f'(0) = r
f'(K) = -r

Non-Hyperbolic Equilibrium: \(x'=x^2\)

At \(x^*=0\): \(f'(0)=0\) — test inconclusive. Must inspect \(f\) directly.

\(f(x)=x^2>0\) for all \(x\neq 0\) → arrows point toward zero from the left (solutions with \(x_0<0\) increase toward 0) and away from zero on the right (solutions with \(x_0>0\) increase away from 0) → semi-stable.

Bifurcations

What Is a Bifurcation?

In applications the ODE contains a parameter \(r\): \[x' = f(x;\,r)\]

As \(r\) varies, the equilibria may appear, disappear, or change stability.

Definition

A value \(r=r_c\) is a bifurcation point if the number or stability of equilibria changes as \(r\) passes through \(r_c\).

Bifurcation diagram: plot \(x^*\) vs. \(r\)

Solid curve = stable equilibria
Dashed curve = unstable equilibria
Dot = bifurcation point

Three classical bifurcations: saddle-node, transcritical, pitchfork.

Saddle-Node Bifurcation

Saddle-Node: \(x'=r-x^2\)

Equilibria: \(x^*=\pm\sqrt{r}\) (exist only for \(r\geq 0\)).

Linearization: \(f'(x)=-2x\)

\(r\)	Equilibria	Stability
\(r<0\)	None	—
\(r=0\)	\(x^*=0\)	Semi-stable (bifurcation point)
\(r>0\)	\(x^*=+\sqrt{r}\)	Stable
\(r>0\)	\(x^*=-\sqrt{r}\)	Unstable

Tip

A stable and unstable equilibrium collide and annihilate as \(r\) decreases through zero. This is the most generic way equilibria are created or destroyed.

Saddle-Node: Diagrams

Transcritical Bifurcation

Transcritical: \(x'=rx-x^2\)

Equilibria: \(x^*=0\) and \(x^*=r\) exist for all \(r\).

Linearization: \(f'(x)=r-2x\), so \(f'(0)=r\) and \(f'(r)=-r\).

\(r\)	\(x^*=0\)	\(x^*=r\)
\(r<0\)	Stable	Unstable
\(r=0\)	Non-hyperbolic	(same point)
\(r>0\)	Unstable	Stable

The two equilibria exchange stability at \(r=0\).

Tip

Arises in population models where \(x=0\) is always an equilibrium: the zero-population state loses stability when \(r\) exceeds a threshold, and a non-trivial population \(x^*=r\) becomes the attractor.

Transcritical: Bifurcation Diagram

Pitchfork Bifurcation

Pitchfork: \(x'=rx-x^3\) (Supercritical)

Arises in symmetric systems where \(f(-x;r)=-f(x;r)\).

Equilibria: \(x^*=0\) always; \(x^*=\pm\sqrt{r}\) when \(r>0\).

Linearization: \(f'(x)=r-3x^2\)

\(r\leq 0\)	\(x^*=0\) stable	No other equilibria
\(r>0\)	\(x^=0\) unstable*	\(x^=\pm\sqrt{r}\) stable*

As \(r\) increases through zero, the origin loses stability and two new symmetric stable equilibria are born — the bifurcation diagram has the shape of a pitchfork.

Supercritical vs. Subcritical

Supercritical \(x'=rx-x^3\): smooth “soft” transition — new stable states appear gradually.
Subcritical \(x'=rx+x^3\): “hard” dangerous transition — all equilibria disappear at once.

Pitchfork: Subcritical \(x'=rx+x^3\)

The subcritical normal form reverses the cubic sign.

Equilibria: \(x^*=0\) always; \(x^*=\pm\sqrt{-r}\) when \(r<0\).

Linearization: \(f'(x)=r+3x^2\)

\(r\)	\(x^*=0\)	\(x^*=\pm\sqrt{-r}\)
\(r<0\)	Stable	Unstable (exist for \(r<0\))
\(r=0\)	Non-hyperbolic	Disappear
\(r>0\)	Unstable	None

Why “Hard” or “Dangerous”?

For \(r\) slightly below 0 there are three equilibria: one stable (\(x^*=0\)) flanked by two unstable ones (\(\pm\sqrt{-r}\)). As \(r\) increases through 0, the two unstable equilibria collide with the origin and destroy it — the only remaining behavior is divergence.

There is no gradual escape: the system jumps catastrophically. This is why the subcritical pitchfork appears in engineering contexts such as buckling, where a structure that seems stable can suddenly fail without warning.

Pitchfork: Diagrams (Supercritical)

All Three Bifurcations

Side-by-Side Comparison

Bifurcation	What happens at \(r_c\)
Saddle-node	Stable + unstable pair created/destroyed
Transcritical	Two equilibria exchange stability
Pitchfork	Origin loses stability; symmetric pair born (supercritical) or unstable pair annihilates origin (subcritical — dangerous)

Application: Spruce Budworm

The Model

The spruce budworm model (Ludwig–Jones–Holling 1978): \[x' = rx\!\left(1-\frac{x}{K}\right) - \frac{x^2}{1+x^2}\]

\(x\): budworm population density
\(rx(1-x/K)\): logistic growth
\(x^2/(1+x^2)\): predation (Type III functional response)

Bistability and hysteresis: for certain \(r\), there are two stable equilibria — a low refuge state and a high outbreak state. The system can jump suddenly from refuge to outbreak when \(r\) crosses a threshold, and will not return unless \(r\) is reduced well below the forward bifurcation point.

This is an example of hysteresis — the forward and backward transitions occur at different parameter values.

Budworm: Solutions

Note

The two stable states are clearly visible: initial populations below the unstable separator converge to the low refuge state; those above converge to the outbreak state.

Summary

Key Takeaways

An autonomous ODE \(x'=f(x)\) defines a 1D dynamical system. The slope field is constant along horizontal lines.
Equilibria are zeros of \(f\). Stability is read from the phase line: arrows pointing inward → stable; outward → unstable.
Linearization: the perturbation \(u=x-x^*\) satisfies \(u'\approx f'(x^*)u\), so \(\lambda=f'(x^*)\) determines stability when \(\lambda\neq 0\).
When \(f'(x^*)=0\) (non-hyperbolic), higher-order terms decide — the linearization is inconclusive.
A bifurcation is a qualitative change in equilibrium structure. The three normal forms are saddle-node, transcritical, and pitchfork. The pitchfork comes in two flavors: supercritical (soft, gradual transition) and subcritical (hard, catastrophic transition — important in engineering applications such as buckling).
Real applications like the spruce budworm exhibit bistability and hysteresis — phenomena impossible to see from a single solution curve, but immediately apparent from the bifurcation diagram.

Tip

Next: These ideas extend to systems of ODEs in 2D. The phase line becomes the phase plane, \(f'(x^*)\) becomes the Jacobian matrix, and the pitchfork becomes the Hopf bifurcation — Chapter 3 of Logan (2015).

Note

Next: Second-order linear equations — Logan §2.2.

References

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