First-Order Difference Equations

A difference equation advances a sequence one index at a time. Given a rule

and a starting value $y_0$, the whole sequence is generated by repeated substitution: $y_1=f(y_0)$, $y_2=f(y_1)=f(f(y_0))$, and in general $y_n=f^n(y_0)$, the $n$-th iterate. Where a first-order differential equation runs a solution continuously in $t$, a first-order difference equation steps a solution discretely in $n$, and the central question is the same: how does $y_n$ behave as $n\to \infty$?¹

The discrete analogue of a constant solution is a sequence that never changes.

Equilibria are the fixed points of $f$, found by setting $y_{n+1}=y_n$.

Linear difference equations

Take a population whose size in year $n+1$ is a fixed multiple of its size in year $n$, plus a constant net migration $b$:

The constant $\rho$ is the reproduction ratio, and $b$ the yearly immigration ($b>0$) or emigration ($b<0$). Iterating from $y_0$,

and the pattern is a geometric sum. For $\rho \ne 1$,

The first term is the descendants of the original population; the second collects the migration of every preceding year. Rewriting it around the equilibrium makes the long-time behavior visible:

Setting $y_{n+1}=y_n$ in the equation gives the same equilibrium directly, $y^{\ast }=b/(1-\rho )$. Everything hinges on the factor ${\rho }^n$:

• $|\rho |<1$. The factor ${\rho }^n\to 0$, so $y_n\to b/(1-\rho )$ from any start. The equilibrium is asymptotically stable.
• $|\rho |>1$. The factor ${\rho }^n$ grows without bound and $y_n$ diverges, unless $y_0=b/(1-\rho )$ exactly. The equilibrium is unstable.
• $\rho =1$. The formula degenerates; iterating directly gives $y_n=y_0+nb$, growing linearly without bound (unless $b=0$).

For the pure model $y_{n+1}=\rho y_n$ (no migration) the equilibrium is $y^{\ast }=0$, stable for $|\rho |<1$ and unstable for $|\rho |>1$: the same dichotomy the continuous equation $y^{\prime }=ky$ shows through the sign of $k$, now read off the magnitude of $\rho$.

The same linear model covers many financial calculations. There $y_n$ is an account balance in period $n$, $\rho =1+r$ with $r$ the interest rate per period, and $b$ is the deposit ($b>0$) or payment ($b<0$) made each period.

The logistic difference equation

Nonlinear difference equations behave far more richly than linear ones. The sharpest example is the discrete counterpart of the logistic model. Replacing the derivative in $\frac{\mathbf{d}y}{\mathbf{d}t}=ry(1-y/K)$ by a difference quotient and rescaling the variable turns it into the logistic map

with $\rho >0$ a single parameter. The equilibria come from $u=\rho u(1-u)$, that is $u=\rho u-\rho u^2$, so

The second equilibrium is positive only for $\rho >1$. Stability is tested by linearizing near each fixed point. Near $u=0$ the quadratic term is negligible and the map reduces to $u_{n+1}\approx \rho u_n$, stable for $0<\rho <1$. Near $u^{\ast }=(\rho -1)/\rho$, writing $u_n=u^{\ast }+v_n$ and dropping the quadratic term gives

stable for $|2-\rho |<1$, that is $1<\rho <3$. The two ranges meet at $\rho =1$, where the equilibria coincide and stability passes from one branch to the other.

A cobweb diagram displays the iteration geometrically. Plot the map $y=\rho x(1-x)$ and the line $y=x$ on the same axes; their intersections are the equilibria. Starting at $u_0$ on the horizontal axis, a vertical segment to the parabola computes $u_1=\rho u_0(1-u_0)$, a horizontal segment to the line $y=x$ transfers that value back to the axis, and repeating traces a staircase or spiral toward the stable equilibrium.

Collecting the stability results: $u=0$ is stable for $0\le \rho <1$ and $u=(\rho -1)/\rho$ is stable for $1<\rho <3$. Plotting both equilibria against $\rho$, the stable portions form a continuous path that jumps from one branch to the other at $\rho =1$, an exchange of stability.

Period doubling and chaos

For $\rho >3$ neither equilibrium is stable, and the iterates never settle to a single value. Just past $\rho =3$ the sequence approaches a two-cycle: after a transient it alternates between two values. At $\rho =3.2$ the alternation is between $0.5130$ and $0.7995$.

The cobweb of a two-cycle is a rectangle traced repeatedly: the map now folds the interval so that two points map to each other, and the fixed point at $(\rho -1)/\rho =0.6875$ sits inside the loop, unstable.

Raising $\rho$ further, each state of the two-cycle splits in two. At about $\rho =3.449$ the orbit becomes a four-cycle; near $\rho =3.544$ it becomes an eight-cycle, and periods $16,32,\dots$ follow in quick succession. This cascade of period doublings accumulates at a finite limit ${\rho }_{\infty }\approx 3.57$. The parameter gaps between successive doublings shrink by a nearly constant factor, the Feigenbaum constant$\delta \approx 4.6692$, which recurs across a wide class of maps.

Beyond $\rho \approx 3.57$ the solutions are chaotic: they stay in a bounded range but follow no periodic pattern. For $\rho =3.65$ the iterates wander between roughly $0.3$ and $0.9$ with no discernible order.

A hallmark of chaos is sensitivity to initial conditions. Two orbits of the same map starting at $u_0=0.3$ and $u_0=0.305$ stay close for about fifteen iterations, then diverge completely; after that, neither can be used to predict the other. The logistic map was one of the first simple systems in which this was recognized, in Robert May's 1974 study of insect populations: a growth parameter too large makes long-range prediction impossible in principle, not merely in practice.

Summary

• A first-order difference equation $y_{n+1}=f(y_n)$ generates a sequence by iteration; its equilibria are the fixed points $y^{\ast }=f(y^{\ast })$.
• The linear equation $y_{n+1}=\rho y_n+b$ solves in closed form, with equilibrium $b/(1-\rho )$ that is stable for $|\rho |<1$ and unstable for $|\rho |>1$; the $\rho =1$ case grows linearly. Compound interest, savings, and loan amortization are direct applications.
• The logistic map $u_{n+1}=\rho u_n(1-u_n)$ has equilibria $0$ and $(\rho -1)/\rho$, with an exchange of stability at $\rho =1$ and loss of stability of the nonzero branch at $\rho =3$.
• Past $\rho =3$ come period doublings at a geometric rate set by the Feigenbaum constant $\delta \approx 4.6692$, accumulating at $\rho \approx 3.57$, beyond which the dynamics are chaotic and sensitive to the initial value.