| |
ISSN 1320-0682 | |||
| Volume 02 | April 1995 | |||
H. P. W. Gottlieb
School of Science, Griffith University
Nathan, Queensland 4111, Australia
Email: h.gottlieb@sct.gu.edu.au
The logistic map
with scaling parameter A has a number of interesting properties:
.All these familiar features are gainsaid by the generalised logistic map
with near-origin exponent a > 1 (and b > 0); that is,
with 
.
In particular, the attractor may depend on the initial
(or seed) value
. When the positive fixed point appears, it
occurs as a finite jump up from zero. Finally, the attractor
becomes zero at some cut-off value of the parameter less than
its maximum value. Some general features, and details for the
integer-exponent case a=2, b=1, have been presented in [1].
Some time ago, Stutzer [2], and, more recently, Tu [3],
analysed the bifurcations and chaotic dynamics in a discrete
time version of a simple continuous nonlinear macroeconomic
growth model of the form of (2) above with exponents a=1 and
b=1/2. This exhibited similar behaviour to the usual logistic
map (a=b=1 in (2)). The range of A is
. There is a
fixed point (period 1) for
with corresponding fixed
point value 
. The fixed point x=0 is unstable
for 
. Period doubling bifurcations appear, and chaos
sets in at about
(not 6.75 as suggested by a cursory
reading of [2] or [3]). A slightly differing behaviour is that
for the logistic map the slope of the fixed point curve dx/dA
at 
is unity, whereas for the Stutzer map this slope is
zero.
The simplest extension of this map to the case of both exponents fractional (with a > 1) is to have a=3/2 and b=1/2, and this will now be analysed in detail.
Consider the map
which has 
.
Thus, 
at
where the maximum value is
. The
maximum parameter value is
.
To find the value of the parameter
at which the positive
fixed point appears for maps such as this one, it clearly
suffices to solve the two simultaneous equations x=f(x);
, since the line and curve will just be tangent. This
gives x= 1/4 and
. The equation to a fixed point in the x,A
plane, f(x;A)=x, is satisfied by x=0 or by
so the
positive fixed points are given by
As derived above, this means the fixed points open up at
,
with 
. There is a positive jump from zero
up to the fixed point value .25 in the x,A attractor diagram,
unlike the logistic map case.
The fixed point x=0 has
, so it is stable for all
A, again unlike the logistic map case. (A good discussion of
the logistic map may be found in Froyland [4].)
To determine the stability ranges of the other fixed points, we have
Slope +1 gives
with
as above. Slope -1 for the
larger fixed point
yields
, with
corresponding
. At this point, a
bifurcation takes place.
For the lower branch fixed point
, it is evident from
the graphs of such curves that the slope is always greater
than unity for
. Explicitly for (5),
where the second term is positive
for 
. Thus, the slope is greater than unity and the
smaller fixed point is always unstable. From the curve for
such maps, for fixed A, any value
iterates to x=0.
Thus, between
and
there is bistablility, with a basin
boundary .
We may construct a Seed Plot as follows. For fixed A,
choose a seed value
and iterate, say several hundred times.
If the result is non-zero, record a dot at the corresponding
pixel in the 
plane; if zero, leave a blank. This is
repeated for the ranges
and
. By the above, such
a plot cuts on at
, and the lower bound there is
.
Consideration of iterative paths on the map of such f(x)
vis-a-vis the straight line f(x) = x shows that, to find the
upper bound of the seed plot at cut-on, we need the other pre-
image of 
there; that is,
satisfying
.
For our map (3), this gives
and thence, factorising out the known unwanted factor (X- 1/4 )
and setting Y=4X, we obtain
the cubic 
, whose real
solution gives
Any 
then iterates to x=0. Thus, to construct a
comprehensive orbit plot or attractor diagram, it is necessary
to choose a seed between
and
: that is, for the present
map, 
. For our attractor diagram we chose
.
The locus of the positive stable fixed point in the Orbit
Plot (or Attractor Diagram) is the curve
given by
Equation (4) above, over the range
. The slope
at cut-on is 
; that is, the tangent is
vertical. The locus of the positive unstable (smaller) fixed
point 
, Equation (4), is the lower boundary curve of
the Seed Plot over the range
, where
is a
cut-off value, to be determined later. (Recall that
iterates to zero.) Thus, this fixed point locus will not
normally appear in the attractor diagram. Since
is (for A > 4) positive or negative respectively, the locus of the positive stable fixed point in the Attractor Diagram is an increasing curve, and the lower boundary of the seed plot is a decreasing curve.
The upper boundary curve of the Seed Plot is given by
which is the other pre-image of
; that is,
This gives
Factorising out the known unwanted factor
by
synthetic long division eventually yields
A real analytical solution for
could in principle be
obtained from this by the formula for solution of a cubic, but
this is very long and unilluminating. Values of
can be
found by numerical solution, or obtained graphically from the
seed plot's upper boundary curve.
For 
, the attractor diagram undergoes a sequence
of period-doubling bifurcations, until chaos is reached at
about
The straight line
is seen to be the upper bound of the whole Orbit Plot, in the following sense:
and
, it is the least upper bound of the iterated values.
For instance, for the period 1 curve (stable fixed point
locus) ,
one solves 
(equal
slopes). With A = (16/3)B, this gives
Clearly, B=1 is a solution; that is, A=16/3. To show that there are no other real solutions, factorise out (B-1) and complete a square to get
Since the left hand side is positive for B > 0, there are no
other real solutions for A > 0. For the solution A=16/3,
are also satisfied, so
is indeed tangent to
at
.
In the same sense as above (replacing "largest" and
"least upper" by "smallest" and "greatest lower"
respectively), the lower bound of the whole Orbit Plot, for
, is the iterate of the upper bound; namely, the decreasing,
concave down, curve
Figure 1: (a) Attractor diagram for map
(seed
value 0.8) 
Omit first 400 iterates;
plot next 400 points. (b) Attractor diagram,
,
zoomed for A near cut-off
. (c) Seed plot for map
.
401st iterate.
.
The chaotic region of the attractor diagram for maps such as
these does not extend all the way up to the maximum value
of
the map, and the extent of the seed plot is similarly
curtailed. This contrasts with the more familiar behaviour of
the logistic map.
Inspection of such maps and the straight line f(x) = x,
and consideration of the orbit paths, shows that the attractor
collapses to the point x=0 at a cut-off parameter value
which satisfies
By the definition (7) of
,
at . Thus,
This may be clearly interpreted as a boundary crisis as
defined by Grebogi et al. [5]: an unstable fixed point, here
, collides with the lower bound curve of the orbit plot,
here 
.
The cut-off parameter
must be found by numerical solution of
a suitable equation. Some possibilities depending on the
nature of the mathematical analysis are as follows:
is known
easily as an analytical function of A (as for example in an
integer exponent case considered in [1]), but this is not the
case here (compare with the discussion after (8) above). is not available
analytically but
is known analytically,
as in Equation (4) above. is a fixed point, this yields
This is useful if one does not have, or wish to use,
analytical expressions for
nor
.
For our map (3), after some cancellations (which also
get rid of the unwanted solution
), this is explicitly
Once has been found, the upper cut-off x-value in the
seed plot, and (by (9)) in the orbit plot, is given by
and the lower cut-off x-value in the seed plot, and (by (10) ) in the orbit plot, is given by
with solutions
,
,
.
For our map (3), methods (ii) (Equation (12)), (iii) (Equation (13) as well as the Equation (14)) and (iv) (Equations (17)) were all utilised. The values computed numerically for these parameters are found to be:
The generalised map (2) with a > 1 exhibits many features
which are quite different from the standard logistic map (1)
(a=b=1). The finite jump up of the Attractor Diagram at the
onset of the non-zero stable fixed point might be seen as a
definite switching mechanism. The collapse of the chaotic
attractor to zero at the computable cut-off value
might be
viewed as a "fuzzy off" switch from a chaotic range of
positive values back to zero. To achieve these phenomena
comfortably, a relatively large value of the exponent a may be
required (compare with [1] with a = 2).
On the other hand, in terms of a population model, the
zero slope of the map at origin which results in attractor
zero - that is, extinction, for
- could correspond to a
situation where a substantial initial population was required
(perhaps to utilise some co-operative behaviour) before the
species could persist. Favourable conditions for survival,
namely smaller
(and larger
), therefore would require
exponent a smaller than 2; that is, necessarily non-integer. The
fractional case a = 3/2 , considered above, is the simplest
such value for which substantial analytical progress may be
made.
I am grateful to the Department of Mathematics at the University of Queensland for the hospitality extended to me during my visit on an Outside Studies Program, and should like to thank members of the Mathematics and Physics Departments for some interesting discussions.
Properties of Some Generalised Logistic Maps with Fractional Exponents
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