
Egwald Mathematics: Nonlinear Dynamics:
One Dimensional Dynamics and Bifurcations
by
Elmer G. Wiens
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one dimensional dynamics  introduction  difference equation  fixed points  stability analysis  higher order maps bifurcations  saddlenode bifurcation  saddlenode example  flip bifurcation  second order map  saddlenode conditions flip bifurcation conditions  flip bifurcation example  transcritical bifurcation  transcritical example  transcritical conditions pitchfork bifurcation  pitchfork example  pitchfork conditions  references
One Dimensional Dynamics: Discrete Time Models.
Introduction
Continuous time, dynamic process are modelled with differential equations. Discrete time, dynamic processes are modelled with difference equations. Consider a dynamic process where the rate of change of a variable from one time period to the next is proportional to magnitude of the variable.
Writing x_{t} as the magnitude of the variable at time t, the dynamic process modelled as a difference equation is:
x_{t+1} = x_{t} + k * x_{t} = (1 + k) * x_{t}, x_{0} = x0, (1)
where the constant k is the rate of change, and x0 is the magnitude of the variable at the start, time t = 0.
The solution to this difference equation is the discrete function of time, x(t, x0), for t an integer:
x(t, x0) = {x_{0}, x_{1}, x_{2}, . . . x_{n}, ... },
providing the time profile of the magnitude of x with an initial value of x0.
The sequence of numbers, {x_{0}, x_{1}, x_{2}, . . . x_{n}, ... },
is called an orbit.
One can solve the difference equation, (1), by induction:
x_{1} = (1+ k) * x0,
x_{2} = (1 + k) * x_{1} = (1 + k) * ( (1 + k) * x0) =(1 + k)^{2} * x0,
x_{3} = (1 + k) * x_{2} = (1 + k) * ( (1 + k)^{2} * x0) = (1 + k)^{3} * x0,
. . . . . . . . .
x_{t} = (1 + k)^{t} * x0
. . . . . . . . .

Writing λ = (1 + k), the difference equation (2) and its solution (3) are:
x_{t+1} = λ * x_{t}, x_{0} = x0, (2)
x(t, x0) = λ^{t} * x0, for t an integer. (3)

The solution x(t, x0) = λ^{t} * x0 suggests that the variable x will grow or decay exponentially from its starting population, x0, at a rate depending on the value of the parameter λ. If x0 > 0, and if λ > 1 (k > 0), the variable x increases in value; but if x0 > 0, and if λ < 1 (k < 0), the variable x decreases in value. Conversely, if x0 < 0, and if λ > 1 (k > 0), the variable x decreases in value; but if x0 < 0, and if λ < 1 (k < 0), the variable x increases in value.
One can see the evolution of these solutions in the following staircase diagrams:
In the first set of diagrams, the green lines touch the xaxis at x0, x_{1}, x_{2}, x_{3}, etc. In both diagrams x0 > x_{1} > x_{2} > x_{3} > ... x_{t} > ... → 0, but in the second diagram the x_{t} alternate in sign.
Orbit Converges: 1 < λ < 1

Monotonic Convergence: 0 < λ < 1

Alternating Convergence: 1 < λ < 0

y = x versus
y = λ * x





The second set of diagrams above display the phase portraits of the difference equation, showing the movement of x_{t} along the xaxis for t = 0, 1, 2, ... etc. This sequence converges to x^{*} = 0 for any initial x0. The basin of attraction of x^{*} = 0 is said to be the interval (∞, ∞), since any initial x0 converges to 0.
In the first set of diagrams below, the green lines touch the xaxis at x0, x_{1}, x_{2}, x_{3}, etc. In both diagrams 0 < x0 < x_{1} < x_{2} < x_{3} < ... < x_{t} < ... → ∞ , but in the second diagram the x_{t} alternate in sign.
Orbit Diverges: 1 < λ

Monotonic Divergence: 1 < λ

Alternating Divergence: λ < 1

y = x versus
y = λ * x





The second set of diagrams above are the phase portraits of the difference equation, displaying the movement of x_{t} along the xaxis for t = 0, 1, 2, ... etc. This sequence diverges for any initial x0 except x0 = 0. Thus, the basin of attraction of x^{*} = 0 is the single point 0.
One Dimensional Difference Equations
The general form of a one dimensional difference equation with a parameter r is:
x_{t+1} = f(x_{t}, r), x_{0} = x0. (4)
where f is a function of x and r.
Difference equations are also called maps, recursion relations, or iterated maps.
Fixed Points
For a specific value of the parameter r = r1 it is sometimes convenient to write:
f(x, r1) = f(x). (5)
A fixed point of the function f is a number x^{*} such that:
f(x^{*}) = x^{*}. (6)
Stability Analysis
One can analyze the local stability of the difference equation (4) about a fixed point by examining the first derivative of the function f with respect to x evaluated at x^{*} (the partial derivative of f with respect to x evaluated at x^{*}):
λ = f'(x^{*}) = f_{x}(x^{*}, r). (7)
where λ is called the multiplier or eigenvalue, determining the stability of (4) at x^{*}.
The discrete time, dynamic process determined by the function f is stable at x^{*} if:
λ = f'(x^{*}) < 1,
and superstable if:
λ = f'(x^{*}) = 0;
the process is unstable at x^{*} if:
λ = f'(x^{*}) > 1.
If f is stable at x^{*}, x^{*} is called an attractor. If f is unstable at x^{*}, x^{*} is called a repeller.
If λ ≠ 1, x^{*} is called a hyperbolic fixed point.
The stability of the process must be examined for each situation if:
λ = f'(x^{*}) = 1.
If λ = 1, x^{*} is called a nonhyperbolic fixed point.
Higher Order Maps
The second order map, f^{2}, for the function f in (4) is given by:
x_{t+2} = f(x_{t+1}, r) = f (f(x_{t}, r), r ) = f^{2}(x_{t}, r)
The third order map, f^{3}, is given by:
x_{t+3} = f( f (f(x_{t}, r), r), r) = f^{3}(x_{t}, r).
Repeat this sequence for the 4th, 5th, etc. order maps.
One Dimensional Bifurcations.
A dictionary definition of bifurcation states: Division into two branches; the point of division; the branches or one of them.
A nonlinear process experiences a qualitative change in its dynamics when the nature of its fixed points change. When a change in a parameter results in a qualitative change in the dynamics of a nonlinear process, the process is said to have gone through a bifurcation.
A one dimensional, discrete time, nonlinear dynamic process is modelled by the function f of two variables, x and r, where f is nonlinear in x, and r is a parameter. The difference equation governing the dynamics of the process is:
x_{t+1} = f(x_{t}, r), x_{0} = x0. (8)
Bifurcations are classified by the way in which the fixed points of the function f change their number, location, form, and stability. It is convenient to consider normal forms for the different types of bifurcations, and the conditions under which the dynamic process described by equation (8) will experience a specific type of bifurcation.
Discrete time dynamical systems may experience saddlenode, transcritical, and pitchfork bifurcations that are similar to their continuous time counterparts. The flip bifurcation is unique to discrete time dynamical processes.
A bifurcation can only occur at a nonhyperbolic fixed point.
SaddleNode Bifurcation
The normal form of a discrete time process with a saddlenode bifurcation with parameter r is:
f(x, r) = a * r + b * x^{2}, a ≠ 0, b ≠ 0. (9)
Fixed points of equation (9) satisfy:
f(x, r) = a * r + b * x^{2} = x. (10)
Fixed points are the roots of the quadratic equation:
b * x^{2}  x + a * r = 0. (11)
For a specific value of the parameter r, the roots of equation (11) are:
x_{1} = (1 + sqrt(1  4 * b * a * r)) / (2 * b) , (12)
x_{2} = (1  sqrt(1  4 * b * a * r)) / (2 * b). (13)
Linearized Stability:
The stability of f at a fixed point, x_{i}, is determined by the partial derivative of f with respect to x evaluated at the fixed point (ie the multiplier or eigenvalue of f):
λ_{i} = f_{x}(x_{i}, r) = 2 * b * x_{i}. (14)
Substituting (12) and (13) into (14) yields:
λ_{1} = 1 + sqrt(1  4 * b * a * r),
λ_{2} = 1  sqrt(1  4 * b * a * r)
If the parameter r takes on the critical value:
r_{c} = 1 / (4 * b * a),
the repeated fixed point is:
x_{1,2} = 1 / (2 * b),
with multiplier:
λ_{1,2} = 1 .
Consequently, ( x_{1,2}, r_{c}) = (1 / (2 * b), 1 / (4 * b * a) ) is a nonhyperbolic fixed point. In fact, it is a saddlenode bifurcation point.
Furthermore,
λ_{2} = 1  sqrt(1  4 * b * a * r) = 1, if
r_{c} = 3 / (4 * b * a).
The fixed points are:
x_{1} = (1 + sqrt(1 + 3)) / (2 * b) = 3 / (2 * b),
x_{2} = (1  sqrt(1 + 3)) / (2 * b) = 1 / (2 * b).
Consequently, ( x_{2}, r_{c}) = (1 / (2 * b), 3 / (4 * b * a) ) is also a nonhyperbolic fixed point. In fact, it is a flip bifurcation point.
SaddleNode Bifurcation: a = 1, b = 1; ( x_{1,2}, r_{c}) = ( 1 / 2,  1 / 4 )
The prototype function f is:
f(x, r) = r  x^{2}, (15)
whose fixed points satisfy:
r  x_{2} = x.
The fixed points and multipliers are:
x_{1} = (1  sqrt(1 + 4 * r)) / 2, λ_{1} = 1 + sqrt(1 + 4 * r)
x_{2} = (1 + sqrt(1 + 4 * r)) / 2, λ_{2} = 1  sqrt(1 + 4 * r)
The values of x_{1} and x_{2} are complex when r < 1 / 4, and the dynamic process has no fixed points here.
When r = 1 / 4, x_{1} = x_{2} = 1 / 2, and λ_{1} = λ_{2} = 1. The basin of attraction of the fixed point 1 / 2 is the closed interval [x_{1}, x_{1}] = [1/2, 1/2], since trajectories starting at x0 in this interval converge to 1 / 2.
When 1 / 4 < r < 3 / 4, λ_{1} > 1 and 1 < λ_{2} < 1. Consequently, x_{1}is a repeller (unstable), and x_{2} is an attractor (stable). The basin of attraction for the attractor x_{2} is [x_{1}, x_{1}].
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process.
The yellow lines delimit the basin of attraction of the stable fixed point given by the closed interval [x_{1}, x_{1}].
x_{t+1} = r  x_{t}^{2}, x_{0} = x0.
r =  .4 < 1 / 4 no real fixed points



r =  1 / 4 saddlenode bifurcation



r = 0 unstable & stable fixed points



r = .5 unstable & stable fixed points



Flip Bifurcation: a = 1, b = 1; ( x_{2}, r_{c}) = ( 1 / 2, 3 / 4 )
As r increases past 3 /4, the fixed point x_{2} loses its stability, flipping from an attractor to a repeller. The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process. The green lines delimit the minimal trapping region. If x_{t0} lands in the trapping region for some value of t_{0}, x_{t} remains in the trapping region for t > t_{0}. This trapping region is the closed interval [f(0, r), f(f(0, r), r)], where f^{2}(x, r) = f(f(x, r), r) is the second order map discussed below.
x_{t+1} = r  x_{t}^{2}, x_{0} = x0.
r = 3 / 4 flip bifurcation unstable fixed points



r = .85 unstable fixed points



r = 1 unstable fixed points



r = 1.25 unstable fixed points



r = 1.75 unstable fixed points



The Second Order Map of f
The second order map of f as stated in (15) is the a fourthorder polynomial in x:
x_{t+2} = f(x_{t+1}, r) = f (f(x_{t}, r), r ) = r  (r  x_{t}^{2})^{2} = r  r^{2} + 2 * r * x_{t}^{2}  x_{t}^{4}. (16)
Denote this mapping by f^{2}:
f^{2}(x, r) = r  r^{2} + 2 * r * x^{2}  x^{4}, (17)
whose partial derivative with respect to x is:
f_{x}^{2}(x, r) = 4 * r * x  4 * x^{3}. (18)
Fixed points of (17) satisfy:
f^{2}(x, r) = r  r^{2} + 2 * r * x^{2}  x^{4} = x. (19)
The fixed points of f^{2} are:
x_{1} = (1  sqrt(1 + 4 * r)) / 2,
x_{2} = (1 + sqrt(1 + 4 * r)) / 2,
x_{3} = (1 + sqrt(3 + 4 * r)) / 2,
x_{4} = (1  sqrt(3 + 4 * r)) / 2
The first two fixed points of f^{2}are also fixed points of f. The stability of the fixed points of f^{2} provide information on the bifurcation points and solution trajectories of f.
The following graphs display the phase diagrams of f in blue and f^{2} in red.
r = .4 <  1 / 4

r =  1 / 4

r = 0




r = .5

r = 3 / 4

r = .85




r = 1

r = 1.25

r = 1.75




At (x, r) = (1 / 2, 1 / 4), f and f^{2} have a slope of 1, and f saddlenode bifurcates into two fixed points, one unstable and the other stable. At (x, r) = (1 / 2, 3 / 4), f has a slope of 1, while f^{2} has a slope of 1. At this flip bifurcation, the stability of f flips from stable to unstable. Meanwhile, the fixed point of f^{2} splits into three fixed points, bracketing the unstable fixed point that it shares with f with two stable fixed points, x_{3} and x_{4}. At r = 1.25, the fixed points x_{3} and x_{4} of f^{2} flip from stable to unstable.
The following bifurcation diagram summarizes the behaviour of the fixed points. It shows the fixed points as a function of the parameter r. The blue curves represent the stable fixed points, while the red curves represent the unstable fixed points. The bifurcation points are marked with an asterisk, *.
This discrete time, saddlenode bifurcation diagram is similar to the continuous time, type two saddlenode bifurcation diagram.
The normal form of a discrete time process with a saddlenode bifurcation is:
x_{t+1} = f(x_{t}, r) = a * r + b * x_{t}^{2}, x_{0} = x0, a ≠ 0, b ≠ 0. (20)
SaddleNode Conditions
The discrete time, dynamic process described by equation (20) undergoes a saddlenode bifurcation at (x^{*}, r_{c}) if:
(SN1) f(x^{*}, r_{c})  x^{*}= 0 → (x^{*}, r_{c}) is a fixed point
(SN2) f_{x}(x^{*}, r_{c}) = 1 → (x^{*}, r_{c}) is nonhyperbolic
(SN3) a = f_{r}(x^{*}, r_{c}) ≠ 0
(SN4) b = (1/2) * f_{x,x}(x^{*}, r_{c}) ≠ 0
 (21) 
The bifurcation diagrams for the four types of saddlenode bifurcations are displayed below. The fixed points of f are x_{1} and x_{2}, while the fixed points of f^{2} are x_{1}, x_{2}, x_{3}, and x_{4}. These fixed points are graphed as functions of the parameter, r. The discrete time, dynamic process undergoes a flip bifurcation when the stability of the fixed point, x_{2}, flips. At this critical value of r, the stable fixed points x_{3} and x_{4} of f^{2} emerge and bracket x_{2}.
The blue curves represent the stable fixed points, while the red curves represent the unstable fixed points. The bifurcation points are marked with an asterisk, *.
a > 0 , b > 0

a > 0 , b < 0



a < 0 , b > 0

a < 0 , b < 0



Flip Bifurcation Conditions (Lorenz 111)
Consider the discrete time, dynamic process given by:
x_{t+1} = f(x_{t}, r), x_{0} = x0. (22)
If the fixed point x = x^{*} and parameter value r = r_{c} satisfy:
(F1) f(x^{*}, r_{c})  x^{*}= 0 → (x^{*}, r_{c}) is a fixed point,
(F2) f_{x}(x^{*}, r_{c}) = 1 → (x^{*}, r_{c}) is nonhyperbolic,
(F3) f_{r} * f_{x,x} + 2 * f_{x,r} ≠ 0,
at (x^{*}, r_{c}),
(F4) c = 2 * f_{x,x,x}  3 * (f_{x,x})^{2} ≠ 0, at (x^{*}, r_{c});
 (23) 
then, depending on the signs of the expressions in (F3) and F4),

the fixed point x^{*} is stable (unstable) for r < r_{c} (r > r_{c}), and

the fixed point x^{*} becomes unstable (stable) for r > r_{c} (r < r_{c}); moreover, a branch of stable (unstable) fixed points of order 2 (ie fixed points of f^{2}) emerges which bracket x^{*}.
Supercritical Flip Bifurcation (Period Doubling Bifurcation)
If c < 0 in expression F4, the emerging fixed points of f^{2} are stable and bracket an unstable fixed of f. The trajectory x_{t} switches between the two attracting fixed points of f^{2}.
Subcritical Flip Bifurcation
If c > 0 in expression F4, the emerging fixed points of f^{2} are unstable and bracket a stable fixed of f. See the supercritical pitchfork example for an example of a subcritical flip bifurcation.
Supercritical (Period Doubling) Flip Bifurcation Example
The discrete time, dynamic process:
x_{t+1} = f(x_{t}, r) = r  x_{t}^{2}, x_{0} = x0.
with:
f_{x}(x, r) = 2 * x
f_{x,x}(x, r) = 2
f_{x,x,x}(x,r) = 0

f_{r}(x, r) = 1
f_{x,r}(x,r) =0

undergoes a flip bifurcation at (x_{2}, rc) = (1 / 2, 3 / 4) as r increases, where the stability of the fixed point x_{2} of f flips from stable to unstable, and two stable fixed points x_{3} and x_{4} of f^{2} emerge bracketing x_{2}.
Checking the flip bifurcation conditions:
(F1) f(x_{2}, r_{c})  x_{2} = 3/4  (1 / 2)^{2}  1 / 2 = 0,
(F2) f_{x}(x_{2}, r_{c}) = 2 * 1 / 2 = 1,
(F3) f_{r} * f_{x,x} + 2 * f_{x,r} = 1 * (2) + 2 * 0 = 2 ≠ 0,
(F4) c = 2 * f_{x,x,x}  3 * (f_{x,x})^{2} = 2 * 0  3 * (2)^{2} =  12 ≠ 0.

The supercritical flip bifurcation conditions are satisfied with c < 0.
Transcritical Bifurcation
The normal form of a discrete time process with a transcritical bifurcation with parameter r is:
x_{t+1} = f(x_{t}, r) = a * r * x_{t} + b * x_{t}^{2}, x_{0} = x0, a ≠ 0, b ≠ 0. (24)
The fixed points of f satisfy:
f(x, r) = a * r * x + b * x^{2} = x. (25)
For a specific value of the parameter r, the roots of equation (25) are:
x_{1} = 0 , (26)
x_{2} = (1  a * r) / b. (27)
Linearized Stability:
The stability of f at a fixed point, x_{i}, is determined by the partial derivative of f with respect to x
f_{x}(x, r) = a * r + 2 * b * x, (28)
evaluated at the fixed point (ie the multiplier or eigenvalue of f):
λ_{i} = f_{x}(x_{i}, r) = a * r + 2 * b * x_{i}. (29)
Substituting (26) and (27) into (29) yields:
λ_{1} = a * r,
λ_{2} =  a * r + 2
If the parameter r takes on the critical value:
r_{c} = 1 / a,
the repeated fixed point is:
x_{1,2} = 0,
with multiplier:
λ_{1,2} = 1 .
Consequently, ( x_{1,2}, r_{c}) = (0, 1 / a) is a nonhyperbolic fixed point. In fact, it is a transcritical bifurcation.
Furthermore,
λ_{1} = a * r = 1, if
r_{c} = 1 / a.
The fixed points are:
x_{1} = 0,
x_{2} = 2 / b.
Consequently, ( x_{1}, r_{c}) = (0, 1 / a) is also a nonhyperbolic fixed point. In fact, it is a flip bifurcation.
Moreover,
λ_{2} =  a * r + 2 = 1, if
r_{c} = 3 / a.
The fixed points are:
x_{1} = 0,
x_{2} = 2 / b.
Consequently, ( x_{2}, r_{c}) = (2 / b, 3 / a) is also a nonhyperbolic fixed point. In fact, it is a flip bifurcation.
Transcritical Bifurcation: a = 1, b = 1; ( x_{1,2}, r_{c}) = (0, 1 )
The prototype function f is:
f(x, r) = r * x  x^{2}, (30)
whose fixed points satisfy:
r * x  x_{2} = x.
The fixed points and multipliers are:
x_{1} = 0, λ_{1} = r
x_{2} = (1  r), λ_{2} =  r + 2
The values of x_{1} and x_{2} are real for all r.
For 1 <= r <= 1, x_{1} is an attractor, while x_{2} is a repeller. Conversely, for 1 < r <= 3, x_{1} is a repeller, while x_{2} is an attractor.
When r < 1, x_{1} and x_{2} are repellers (unstable fixed points). Similarly, when r > 3, x_{1} and x_{2} are repellers.
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process:
x_{t+1} = r * x_{t}  x_{t}^{2}, x_{0} = x0.
The yellow linesdelimit the basin of attraction of the stable fixed point. The green lines delimit the minimal trapping region.
r =  2 unstable fixed points



r =  1 x_{1} flip bifurcation point



r = 0 unstable & stable fixed points



r = 1 transcritical bifurcation point



r = 2 unstable and stable fixed points



r = 3 x_{2} flip bifurcation point



r = 4 unstable fixed points



The Second Order Map of f
The second order map of f as stated in (30) is the a fourthorder polynomial in x:
x_{t+2} = f(x_{t+1}, r) = f (f(x_{t}, r), r ) = r * (r  x_{t}^{2})  (r  x_{t}^{2})^{2} = x_{t}^{4} + 2 * r * x_{t}^{3} + (rr^{2}) * x_{t}^{2} + r^{2} * x_{t}. (31)
Denote this mapping by f^{2}:
f^{2}(x, r) = r^{2} * x + (r  r^{2}) * x^{2} + 2 * r * x^{3}  x^{4}, (32)
whose partial derivative with respect to x is:
f_{x}^{2}(x, r) = r^{2} + 2 * (r  r^{2}) * x + 6 * r * x^{2}  4 * x^{3}. (33)
Fixed points of (32) satisfy:
f^{2}(x, r) = r^{2} * x + (r  r^{2}) * x^{2} + 2 * r * x^{3}  x^{4} = x. (34)
The fixed points of f^{2} are:
x_{1} = 0,
x_{2} = r  1,
x_{3} = (1 + r + sqrt(3  2 * r + r^{2})) / 2,
x_{4} = (1 + r  sqrt(3  2 * r + r^{2})) / 2

The first two fixed points of f^{2}are also fixed points of f. The stability of the fixed points of f^{2} provide information on the bifurcation points and solution trajectories of f.
The following graphs display the phase diagrams of f in blue and f^{2} in red.
At (x, r) = (0, 1), f and f^{2} have slopes of 1 and 1, respectively. Here f transcritical bifurcates into two fixed points x_{1} and x_{2}. For 1 < r < 1, x_{1} is stable and x_{2} is unstable. At (x, r) = (0, 1), f and f_{2} have slopes of 1 and 1, respectively. At this flip bifurcation, the stability of x_{1} flips from stable to unstable for r < 1. Meanwhile, the fixed point x_{1} of f^{2} splits into three fixed points, bracketing the unstable fixed point x_{1} it shares with f with two stable fixed points, x_{3} and x_{4}. For 1 < r < 3, x_{1} is unstable and x_{2} is stable. At (x, r) = (x_{2}, 3), f and f_{2} have slopes of 1 and 1, respectively. At this flip bifurcation, the stability of x_{2} flips from stable to unstable for r > 3. Meanwhile, the fixed point x_{2} of f^{2} splits into three fixed points, bracketing the unstable fixed point x_{2} it shares with f with two stable fixed points, x_{3} and x_{4}.
The normal form of a discrete time process with a transcritical bifurcation is:
x_{t+1} = f(x_{t}, r) = a * r * x_{t} + b * x_{t}^{2}, x_{0} = x0, a ≠ 0, b ≠ 0. (35)
Transcritical Conditions
(TC1) f(0, r_{c}) = 0 for all r
(TC2) f_{x}(0, r_{c}) = 1 → (0, r_{c}) is nonhyperbolic
(TC3) a = f_{x,r}(0, r_{c}) ≠ 0
(TC4) b = (1/2) * f_{x,x}(0, r_{c}) ≠ 0
 (36) 
The bifurcation diagrams for the four types of transcritical bifurcations are displayed below. The fixed points of f are x_{1} and x_{2}, while the fixed points of f^{2} are x_{1}, x_{2}, x_{3}, and x_{4}. These fixed points are graphed as functions of the parameter, r. The discrete time, dynamic process undergoes supercritical flip bifurcations when the stability of the fixed points, x_{1} and x_{2}, flip. At these critical values of r, the stable fixed points x_{3} and x_{4} of f^{2} emerge and bracket x_{1} or x_{2}.
The blue curves represent the stable fixed points, while the red curves represent the unstable fixed points. The bifurcation points are marked with an asterisk, *.
a > 0 , b > 0

a > 0 , b < 0



a < 0 , b > 0

a < 0 , b < 0



PitchFork Bifurcation
The normal form of a discrete time process with a pitchfork bifurcation with parameter r is:
x_{t+1} = f(x_{t}, r) = a * r * x_{t} + b * x_{t}^{3}, x_{0} = x0, a ≠ 0, b ≠ 0. (37)
Supercritical Pitchfork Bifurcation
In the normal form of a supercritical pitchfork bifurcation, the coefficient, b, in equation (37) is negative. Consequently, the cubic term stabilizes the dynamic process by pulling the trajectory of x(t) back toward x = 0.
Subcritical Pitchfork Bifurcation
In the normal form of a subcritical pitchfork bifurcation, the coefficient, b, in equation (37) is positive. Consequently, the cubic term destabilizes the dynamic process by driving the trajectory of x(t) toward infinity.
The fixed points of f satisfy:
f(x, r) = a * r * x + b * x^{3} = x. (38)
For a specific value of the parameter r, the roots of equation (38) are:
x_{1} = 0 , (39)
x_{2} = sqrt(b * (a * r  1)) / b. (40)
x_{3} = sqrt(b * (a * r  1)) / b. (41)
The fixed points x_{1} and x_{2} are real if:
b * (a * r  1) > 0. (42)
Linearized Stability:
The stability of f at a fixed point, x_{i}, is determined by the partial derivative of f with respect to x
f_{x}(x, r) = a * r + 3 * b * x^{2}, (43)
evaluated at the fixed point (ie the multiplier or eigenvalue of f):
λ_{i} = f_{x}(x_{i}, r) = a * r + 2 * b * x_{i}^{2}. (44)
Substituting (39), (40), and (41) into (44) yields:
λ_{1} = a * r,
λ_{2} = 2 * a * r + 3
λ_{3} = 2 * a * r + 3
If the parameter r takes on the critical value:
r_{c} = 1 / a,
the triple fixed point is:
x_{1,2,3} = 0,
with multiplier:
λ_{1,2,3} = 1 .
Consequently, ( x_{1,2,3}, r_{c}) = (0, 1 / a) is a nonhyperbolic fixed point. In fact, it is a pitchfork bifurcation.
Furthermore,
λ_{1} = a * r = 1, if
r_{c} = 1 / a,
with corresponding fixed point:
x_{1} = 0.
With r =  1 / a, the fixed points x_{2} and x_{3} are real if b > 0, with
x_{2} = sqrt(2 * b) / b,
x_{3} = sqrt(2 * b) / b.
Moreover,
λ_{2,3} =  2 * a * r + 3 = 1, if
r_{c} = 2 / a.
With r = 2 / a, the fixed points x_{2} and x_{3} are real if b < 0, with
x_{1} = 0,
x_{2} = sqrt( b) / b,
x_{3} = sqrt( b) / b.
Supercritical PitchFork Bifurcation: a = 1, b = 1; ( x_{1,2,3}, r_{c}) = (0, 1 )
The prototype function f is:
f(x, r) = r * x  x^{3}, (45)
whose fixed points satisfy:
r * x  x_{3} = x.
The fixed points and multipliers are:
x_{1} = 0, λ_{1} = r
x_{2} = sqrt(r  1), λ_{2} =  2 * r + 3
x_{3} = sqrt(r  1), λ_{2} =  2 * r + 3
The values of x_{2} and x_{3} are real for all r >= 1.
For r < 1, x_{1} is a repeller. At r = 1, x_{1} flips to attractor. For 1 < r < 1, x_{1} is an attractor. At r = 1, x_{1} flips to repeller. For 1 < r <= 2, x_{2} and x_{3} are attractors. At r = 2, x_{2} and x_{3} flip to repellers for r > 2.
The following diagrams display the phase diagrams and solution trajectories of x(t, x0) for the discrete time, nonlinear dynamic process:
x_{t+1} = r * x_{t}  x_{t}^{3}, x_{0} = x0.
r =  2 unstable fixed points



r =  1 x_{1} flip bifurcation



r = 0 x_{1} stable



r = 1 x_{1} pitchfork bifurcation



r = 1.5 stable and unstable and stable fixed points



r = 2 x_{2} and x_{3} flip from stable to unstable



r = 3 unstable fixed points



The Second Order Map of f
The second order map of f as stated in (45) is the a ninthorder polynomial in x:
x_{t+2} = f(x_{t+1}, r) = f (f(x_{t}, r), r ) = r * (r  x_{t}^{3})  (r  x_{t}^{3})^{3} =
x_{t}^{9}  3 * r * x_{t}^{7} + 3 * r^{2} * x_{t}^{5} + (r  r^{3}) * x_{t}^{3} + r^{2} * x_{t}. (46)
Denote this mapping by f^{2}:
f^{2}(x, r) = r^{2} * x + (r  r^{3}) * x^{3} + 3 * r^{2} * x^{5}  3 * r * x^{7} + x^{9}, (47)
whose partial derivative with respect to x is:
f_{x}^{2}(x, r) = r^{2} + 3 * (r  r^{3}) * x^{2} + 15 * r^{2} * x^{4}  21 * r * x^{6} + 9*x^{8}. (48)
Fixed points of (47) satisfy:
f^{2}(x, r) = r^{2} * x + (r  r^{3}) * x^{3} + 3 * r^{2} * x^{5}  3 * r * x^{7} + x^{9} = x. (49)
These fixed points of f^{2} are:
x_{1} = 0,

x_{2} = sqrt(r  1),
x_{3} = sqrt(r  1),
x_{4} = sqrt(r + 1,)
x_{5} = sqrt(r + 1),

x_{6} = sqrt(2 * r  2 * sqrt(r^{2}  4)) / 2,
x_{7} = sqrt(2 * r  2 * sqrt(r^{2}  4)) / 2,
x_{8} = sqrt(2 * r + 2 * sqrt(r^{2}  4)) / 2,
x_{9} = sqrt(2 * r + 2 * sqrt(r^{2}  4)) / 2.

The first three fixed points of f^{2}are also fixed points of f. When r >= 1, x_{2} and x_{3} are real numbers. When r >= 1, x_{4} and x_{5} are real numbers. When r >= 2, x_{6}, x_{7}, x_{8}, and x_{9} emerge as real numbers.
The following graphs display the phase diagrams of f in blue and f^{2} in red.
At (x, r) = (0, 1), f and f^{2} each have a slope of 1; f pitchfork bifurcates with two new fixed points x_{2} and x_{3} emerging as r increases. At (x, r) = (0, 1), f and f_{2} have slopes of 1 and 1, respectively. At this flip bifurcation, the stability of x_{1} flips from unstable to stable as r increases. Meanwhile, the fixed point x_{1} of f^{2} splits into three fixed points, bracketing the stable fixed point x_{1} it shares with f with two unstable fixed points, x_{4} and x_{5}. For 1 < r < 1, x_{1} is stable and x_{4} and x_{5} are unstable. For 1 < r < 2, x_{1} is unstable, while x_{2} and x_{3} are stable. At r = 2, f and f_{2}evaluated at x_{2} and x_{3} have slopes of 1 and 1, respectively. At this flip bifurcation, the stabilities of x_{2} and x_{3} flip from stable to unstable for r > 3. Meanwhile, the fixed points x_{2} and x_{3} of f^{2} each split into three fixed points, bracketing the unstable fixed point shared with f with stable fixed points, x_{6} and x_{8} around x_{2}, x_{7} and x_{9} around x_{3}.
The normal form of a discrete time process with a pitchfork bifurcation is:
x_{t+1} = f(x_{t}, r) = a * r * x_{t} + b * x_{t}^{3}, x_{0} = x0, a ≠ 0, b ≠ 0. (35)
Note that f is an odd function since:
f(x, r) = f(x, r),
and therefore function f does not have a transcritical bifurcation since:
f_{x,x}(0, r) = 0.
Pitchfork Conditions
(PF1) f(x, r) = f(x, r) → f is an odd function → f(0, r) = 0
(PF2) f_{x}(0, r_{c}) = 1 → (0, r_{c}) is nonhyperbolic
(PF3) a = f_{x,r}(0, r_{c}) ≠ 0
(PF4) b = (1/6)* f_{x,x,x}(0, r_{c}) ≠ 0
 (36) 
Bifurcation Diagrams
The bifurcation diagrams for the two types of supercritical and two types of subcritical pitchfork bifurcations are displayed below. The fixed points of f are x_{1}, x_{2}, and x_{3}, whereas the fixed points of f^{2} are x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, and x_{9}. These fixed points are graphed as functions of the parameter, r.
Supercritical Pitchfork Bifurcation Diagrams (b < 0)
The discrete time, dynamic process undergoes a pitchfork bifurcation at r = 1 / a where x_{1} bifurcates into x_{1}, x_{2}, and x_{3}. A subcritical flip bifurcation occurs at r = 1 / a where the stability of the fixed point x_{1} flips, and the unstable fixed points x_{4} and x_{5} of f^{2} emerge bracketing the stable fixed point x_{1} of f. Supercritical flip bifurcations occur at r = 2 /a where the stabilities of x_{2} and x_{3} flip from stable to unstable. Concurrently, the stable fixed points x_{6} and x_{8} of f^{2} emerge bracketing x_{2}, and the stable fixed points x_{7} and x_{9} of f^{2} emerge bracketing x_{3}.
The blue curves represent the stable fixed points, whereas the red curves represent the unstable fixed points. A bifurcation point is marked with an asterisk, *.
a > 0 , b < 0

a < 0 , b < 0



Subcritical Pitchfork Bifurcation Diagrams (b > 0)
The discrete time, dynamic process undergoes a pitchfork bifurcation at r = 1 / a where x_{1} bifurcates into x_{1}, x_{2}, and x_{3}. A supercritical flip bifurcation occurs at the critical value of r = 1 / a where the stability of the fixed point x_{1} flips. Concurrently, fixed points of f^{2} emerge bracketing x_{1} with x_{4} and x_{5}. At r = 2 / a, the fixed point x_{4} of f^{2} bifurcates into x_{4}, x_{6} and x_{8}; meanwhile the fixed point x_{5} of f_{2} bifurcates into x_{5}, x_{7} and x_{9}.
The blue curves represent the stable fixed points, whereas the red curves represent the unstable fixed points. A bifurcation point is marked with an asterisk, *.
a > 0 , b > 0

a < 0 , b > 0



References.
 Burden, Richard L. and J. Douglas Faires. Numerical Analysis. 6th ed. Pacific Grove: Brooks/Cole, 1997.
 Demmel, James W. Applied Numerical Linear Algebra. Philadelphia: Siam, 1997.
 Devaney, Robert L. An Introduction to Chaotic Dynamical Systems. Menlo Park, CA: Benjamin/Cummings, 1986.
 Intriligator, M. D. Mathematical Optimization and Economic Theory. Englewood Cliffs: PrenticeHall, 1971.
 Kaplan, Wilfred. Ordinary Differential Equations. Reading: AddisonWesley, 1958.
 Lipschutz, Seymour. Linear Algebra. New York: Schaum McGrawHill, 1968.
 Lorenz, HansWalter. Nonlinear Dynamical Economics and Chaotic Motion. Berlin: SpringerVerlag, 1993.
 Mathews, John H. and Kurtis D. Fink. Numerical Methods Using MATLAB. 3rd ed. Upper Saddle River: Prentice Hall, 1999.
 Nagata, K. Wayne. Nonlinear Dynamics and Chaos: Mathematics 345 Lecture Notes. Vancouver: University of B.C., 2006
 Press, William H., Brian P. Flannery, Saul A. Teukolsky, and William T. Vetterling. Numerical Recipes: The Art of Scientific Computing. Cambridge: Cambridge UP, 1989.
 Strang, Gilbert. Linear Algebra and Its Applications. 3d ed. San Diego: Harcourt, 1976.
 Strogatz, Steven H. Nonlinear Dynamics and Chaos. Cambridge MA: Perseus, 1994.
 Wan, Henry Y. Jr. Economic Growth. New York: Harcourt, 1971.
 Watkins, David S. Fundamentals of Matrix Computations. New York: John Wiley, 1991.

