3.2.1 Exponential growth
Discrete dynamical systems, as the name may suggest, unlike first order ODE models, consider discrete time steps, which are described by finite difference equations. They may be used, for example, to model populations in which reproduction occurs at a particular time of year.
Consider a population of animals whose size at generationt= 0,1,2, . . . is given byNt. Each animal lives for one year and has on averageroffspring. The
number of animals in the next generation is given by
Nt+1=rNt
By induction, whereN0=N(0)>0
Nt=rtN0
So as t → ∞, if r > 1, then the population tends to infinity, if r < 1, the population converges to 0, and ifr= 1, then the population is constant.
If r is not an integer, then Nt will typically not be an integer. Note the
number rbehaves differently to the growth rategseen in 3.1.1.
3.2.2 Logistic map
Similarly to logistic growth, for large population, reproduction may be lower due to factors such as competition for resources. For this the logistic map is used, where Kis a parameter:
Nt+1=rNt
1−NKt
Khas a related but not equivalent meaning as for the ODE based logistic growth model.
It is convinient to letxt= NKt for allt. Then
xt+1=rxt(1−xt)
It is always assumed that r ∈ (0,4]. Then x0 ∈ [0,1] =⇒ xt ∈ [0,1] for
t= 0,1,2, . . ..
3.2.3 Difference equations
A one-dimensional difference equation has the form
xt+1=f(xt)
for some functionf. The initial conditionx0also needs to be specified.
The cobweb method is a graphical method for constructing solutions of dif- ference equations. It works by plotting a graph of f on the same axes as the graph ofy=x. First construct a vertical line fromx0 to f. Then a horizontal
y=f(x) y=x y x x1 f(x2) x2 f(x3) x3 f(x4) x4 f(x5)
Often the long term behaviour of the system is of interest. The simplest behaviour is when allxi are equal.
Definition (Fixed point). A real number x∗ is an equilibrium or steady state or fixed point of f if
f(x∗) =x∗
These are solutions toxt+1=xt.
3.2.4 Stability of fixed points
The definition of stability is the same as in 3.1.4. To find stability, consider a small perturbation ytfrom a steady state and see whether it grows wheny0 is
small:
yt+1=f(x∗+yt)−x∗
By Taylor expansion off aroundx∗,
yt+1≈f(x∗) +ytf0(x∗)−x∗=ytf0(x∗)
ytf0(x∗) is a constant, so this equation is just exponential growth. So
yt= f0(x∗) t
y0
So if|f0(x∗)|<1, then the steady state is stable, and if |f0(x∗)|>1, then the steady state is unstable. Iff0(x∗) = 1 then the behaviour is unknown. The sign off0(x∗) indicates if the solution oscillates.
Definition (Bifurcation). Iff(x) depends on a parameterr, and the system’s long term behaviour changes significantly for small changes inr, then there is a bifurcation.
For discrete systems, if the stability of the fixed pointx∗ changes at some parameter value, such as |f0(x∗)| = 1 at r = r∗, then the system exhibits a bifurcation atr=r∗
Generally, a bifurcation means that a small change in a parameter leads to qualitatively different long term behaviour of the system.
y=x y x x0 x0 0< f0(x∗)<1 y=x y x x0 x0 1< f0(x∗) y=x y x x0 x0 −1< f0(x∗)<0 y=x y x x0 x0 f0(x∗)<−1
Example. Reconsider the logistic map.
f(x) =rx(1−x) It has fixed points atx∗0= 0 and ifr >1,x∗1= 1−1r.
f0(x) =r(1−2x) so f0(0) =randf0 1−1
r
= 2−r. Hence ifr <1,x∗0 is the only fixed point,
and it is stable. For 1< r <3,x∗
1 is a stable fixed point andx∗0 is an unstable
fixed point.
So there is a bifurcation atr= 1, when x∗
0 changes stability, and atr= 3,
whenx∗
r <1 y=x y x x0 r >1 y=x y x x0 3.2.5 Period doubling
Another possibility for the long-term dynamics of a finite difference equation are periodic orbits. This means that a sequence of values of xtrepeat themselves.
The simplest example is a system which oscillates between two values. This is called a period-2 cycle.
A period-2 cycle satisfies for allt,
xt+2=xt
Suppose there is a period-2 cycle{x∗
2, x∗3}. Thenx∗2 =f(x∗3) andx∗3 =f(x∗2).
So every point satisfies
x∗i =f f(x∗i)
So these points are steady states of the map x→f f(x)
. Reconsider the logistic map.
g(x) :=f f(x)
=rf(x) 1−f(x)
=r2x(1−x) 1−rx(1−x)
g has rootsx= 0,1−1
r and roots that satisfy
x2−x 1 +1 r +1 r+ 1 r2 = 0 x=r+ 1± √ r2−2r−3 2r
The first two roots are also roots of the logistic map, so the logistic map has a period-2 cycle whenx >3. This period-2 cycle must also be in [0,1].
The stability of a period-2 cycle of f is equivalently the stability of fixed points of the mapg. By the chain rule
g0(x) =f0 f(x)
f0(x) For the logistic map,
g0(x∗2) =f0 f(x∗2) f0(x∗2)f0(x∗3) =g0(x∗3) f0(x) =r(1−2x), andx∗ 2, x∗3 are roots ofx2−x 1 + 1r +1 r+ 1 r2 = 0, so their
sum and product must be 1 +1 r and
1 r +
1 r2. So
g0(x∗2) =r2(1−2x∗2)(1−2x∗3) =r2 1−2(x∗2+x∗3) + 4x∗2x∗3 =r2 1−2 1 +1 r + 4 1 r + 1 r2 = 5−(r−1)2
ghas bifurcations wheng0(x∗
2) = 1, which is whenr= 3,1+
√
6. So the period-2 cycle is stable if
3< r <1 +√6
Asrincreases pastr= 3, the steady state offloses stability and the period- 2 gains stability. This bifurcation is called a period doubling bifurcation. This in fact happens again atr= 1 +√6, where a period-4 cycle off gains stability. The logistic map undergoes infinitely many period-doubling bifurcations be- tweenr= 3 andr≈3.57. Asr increases,f goes from having a stable period-2 cycles, to stable period-4 cycles, to period-8,16, . . ..
Forr >3.57, there are no stable periodic orbits for many values of r. The dynamics are aperiodic and bounded, and have sensitive dependence on initial conditions (Butterfly effect). These three properties in a deterministic system define chaos. The bifurcations betweenr= 3 andr≈3.57 are therefore called the route to chaos.
There are some values ofr >3.57 that have dynamics that are not aperiodic. These can be seen as white bands in the diagrams below.
The diagrams below are called bifurcation diagrams, here drawn for the logistic map, where it is called the Feigenbaum Cascade. The x-axis recordsr, whilst they axis records the possible limiting values (long term behaviour).
0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 r Limit
3.0 3.2 3.4 3.6 3.8 4.0 0.0 0.2 0.4 0.6 0.8 1.0 r Limit 3.4 3.5 3.6 3.7 3.8 3.9 4.0 0.0 0.2 0.4 0.6 0.8 1.0 r Limit