TI-84 Plus CE: Logistic Map

TI-84 Plus CE: Logistic Map

It feels like forever since I with the TI-84 Plus CE.

Introduction

The program LOGISTIC plots the sequence

x = r * x * (1 – x)

where x is initially in the interval (0, 1) and r is a parameter. The sequence traces out the path of recursion.

Depending on the value of r, the sequence can eventually stabilize to a single point, fluctuate then stabilize, eventually bounce between two stable points, or going into the chaos. The higher the r, chance of the latter two possibilities increase. Typically, for any r < 3, the sequence will most likely stabilize.

TI-84 Plus CE Program: LOGISTIC.8xp

Type: TI-Basic Program

ClrHome

Disp “LOGISTIC MAP”

Disp “X1=R*X0*(1-X0)”

Seq

FnOff

Menu(“LAMBDA (R)”, “ENTER”, 1, “RANDOM”, 2)

Lbl 1

Input “0<R≤4: “, R

Goto A

Lbl 2

randInt(1,80) / 20 → R

Pause R

Goto A

Lbl A

Menu(“U(1)=”, ”ENTER”, 3, “RANDOM”, 4)

Lbl 3

Input “U(1)? “, U

Goto B

Lbl 4

rand → U

Pause U

Goto B

Lbl B

{U} → u(nMin)

SEQ(n+1)

“R*u(n)*(1-u(n))” → u

FnOn 1

1 → nMin

50 → nMax

0 → Ymin

1 → Ymax

0 → Xmin

50 → Xmax

DispGraph

Notes:

The Time setting is assumed: x-axis = n, y-axis = u__n

nMin: Sets the counter and the index of the first argument. Typically the counter starts with 0 or 1. Keystrokes: [ vars ], 1. Window, scroll to U/V/W, 4. nMin.

Seq: Sets Sequence mode.

I use R for λ since the TI-84 does not have Greek characters (except for π and σ).

randInt(1,80) / 20 → R: Selects random values from 0.05 and 4 with a tick mark of 0.05.

u(nMin): Store the initial conditions. The conditions are entered as a list, even there is only one initial condition. Keystrokes: [ vars ], 1. Window, scroll to U/V/W, 1. u(nMin)

SEQ(n+1): In the program editor, I had to select this from the catalog. It is listed under “SEQ(n+1) Type”. This allows for the recurring sequence to be in the format u(n+1) = f(u(n),n).

“R*u(n)*(1-u(n))” → u: The u is type from the keyboard: [ 2^nd ] [ 7 ].

FnOn 1: Turns the sequence u(n) on.

Examples

Left: r = 3.67, initial point = 0.2. This looks like total chaos as n grows.

Right: r = 3.1, initial point = 0.5. This sequence does not converge to a single value, but eventually provides a bifurcation between two points.

Caution: If r is too high, even if the u(1) is with in the interval [0, 1], the sequence can “escape” and an overflow can occur. I don’t recommend an r higher than 4.

Source

“Logistic map”. Wikipedia https://en.wikipedia.org/wiki/Logistic_map Retrieved October 20, 2025.

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

TI-84 Plus CE: Complex Recurring Sequences

TI-84 Plus CE:  Complex Recurring Sequences

The program CMPXSEQ generates a matrix table for the complex recurring series:

C = (P + B * Q) / (1 + A)

where:
P and Q are complex constants
A = z_0
B = z_1
C = z_2

Restated, this recursion series becomes:

z_2 = (P + z_1* Q) / (1 + z_0)

In its current operating system (5.4.0), the TI-84 Plus CE's recurring sequence graphing mode cannot accept complex numbers.   Therefore a program is required.  Furthermore, matrices on the TI-84 Plus CE do not handle complex numbers.   It is required to break the complex numbers into parts.   For completeness I added both the rectangular and the polar parts of complex numbers:  real part, imaginary part, radius, and angle (dependent on angle mode).

TI-84 Plus CE Program CMPXSEQ  (text)

"EWS 2020-04-28"
ClrHome
a+bi
Disp "C=(P+B*Q)/(1+A)","COMPLEX","C=Z(N), B=Z(N-1), A=Z(N-2)"
Prompt P,Q
Input "Z0?",A
Input "Z1?",B
Input "NO OF STEPS?",N
{5,N+1}→dim([E])
For(I,0,N)
(P+Q*B)/(1+A)→C
I→[E](1,I+1)
real(C)→[E](2,I+1)
imag(C)→[E](3,I+1)
abs(C)→[E](4,I+1)
angle(C)→[E](5,I+1)
B→A
C→B
End
[E]^T→[E]
Disp "RESULT = [E]","[REAL,IMAG,ABS,ANGLE]"


Results are stored in matrix E.  ^T is the transpose function. 

Examples

Tables generated using the LibreOffice Numbers desktop app and results verified with the TI-84 Plus CE. 

Example 1:
P = 1
Q = i
z_0 = A = 2i
z_1 = B = -2

Example 2:
P = -i
Q = 1
z_0 = A = 2i
z_1 = B = -2

Example 3:
P = 1-i
Q = 1+i
z_0 = A = 2i
z_1 = B = -2

Have a great Sunday and week,

Eddie

All original content copyright, © 2011-2020.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

TI-34 II, TI-34 Multiview, TI-36X Pro: Stored Operations and Recurring Sequences

TI-34 II, TI-34 Multiview, TI-36X Pro: Stored Operations and Recurring Sequences 

Although not known as programming calculators, we can program small macros on the Texas Instruments TI-34 II, TI-34 Multiview, and TI-36X Pro calculators through the stored operation feature. You can store up to two operations on the TI-34 II and TI-34 Multiview, while only one operation can be stored on the TI-36X Pro.   The elementary calculator TI-15 also has this feature.  

Executing the stored operation types whatever is stored.  For example if op=+11, then pressing 12 [op]* shows 12+11 on the screen.  

There are limits to what can be programmed with this operation:

1.  For functions f(X), X is assumed to be entered, as once the operation is executed, what is stored is typed out. 

2. There are no loops or IF-THEN-ELSE structures, hence the extent of the the stored operation is meant for quick calculations. I believe there is a 40 character limit. 

3. For certain operations, you may need to enter certain functions or left hand parenthesis before executing the operation.  

4. With the last answer command (Ans), we can program recurring sequences with one initial condition.  We will discuss this later. 

General Steps for the TI-34 II and TI-34 Multiview:

1.  Press [2nd] [OP1] (set op1, >OP1) to edit your macro.  Press [Clear] if necessary.  

2. When done, press [ENTER]

3. Enter your variable, press [OP1].  The calculation is automatically made. 

The steps are similar for OP2. 

General steps for the TI-36X Pro (TI-30X Pro Multiview if you are outside the United States):

1.  Press [2nd] [ * ] (set op) to edit your macro.  Press [Clear] if necessary.  

2. When done, press [ENTER]

3. Enter your variable, press [2nd] [ ) ] (op). The calculation is automatically made.  

For the TI-15, the [OP1]/[OP2] has required repeated presses.  First for the edit mode, second to store the operation, subsequent presses execute the operation.  The TI-15 may have limited application, as in one step arithmetic macros.  I don't know if more complex macros can be programmed for the TI-15, since I do not own one.  For this post, I will focus on the TI-34 II, TI-34 Multiview, and TI-36X Pro.

Let's go over a few examples. 

Example 1: Area of a circle.   

The area of the circle is A = π * r^2.  Assume r will be in the display.  

Set up the operation.  Press [2nd], (set op 1/2)*, [x^2], [ * ], [ π ].  The screen will read: 

op=²*π, op1=²*π, or op2=²*π

Don't freak at this display, it is correct.  Press [ENTER].  Remember, this will be attached to the calculation whenever the stored operation is executed. 

*Refer to the instructions above for your particular calculator. 

Now enter r, and then press {op} to calculate the area.  

Test Data:

r = 0.76,  Result = 1.814583917

0.76 {op} (display should show 0.76² * π )

r = 9, Result = 81*π = 254.4690049

Example 2: Convert °C to °F. (Degrees Celsius to Fahrenheit) 

Formula: °F = °C * 9/5 + 32

Stored operation:  op = * 9 ÷ 5 + 32

Execute this stored operation in the same way as the above example.  

Test data:

32°C, result 89.6°F

24°C, result 75.2°F

Example 3: Hypotenuse of a Right Triangle

Calculate the hypotenuse of a right triangle where one side is 5 units and the other varies.  Formula:  √(x^2 + 5^2) = (x^2 + 5^2)^0.5

Here X is inside an operation, so we'll need to make some adjustments.  Store the operation as:

op = ² + 5² ) ^ 0.5

When executing the operation, we'll have to include the left parenthesis, then enter the number, then execute the operation ({op}).  Otherwise, a syntax error occurs. 

Test data:

x = 3.   Press [ ( ], 3, {op}.   The display should read (3² + 5²)^0.5 for the result of 5.830951895.  

x = 4.5. Result: 6.726812024. 

Example 4:  Percent Change

This will illustrate the use of variable registers.  The TI-34 II has five variable registers (A-E), the TI-34 Multiview has seven (x,y,z,t,a,b,c), and the TI-36X Pro has eight (x,y,z,t,a,b,c,d).  

For this example, I will use the variable x, but feel free to use which one you like. 

Percent change:  (n - x)/x * 100

where n = new value, x = old value.  

The stored operation will be:  op = - x) ÷ x * 100

To execute, store the old value in x, then press [ ( ], enter new value, then execute the operation.  

Test Data:

New = 50, Old = 36

36 [STO>] x, [ ( ], 50, {op}.   Result: 38.88888889

New = 26, Old = 36. Result: -27.77777778

Recurring Sequences:  u(n) = f( u(n-1) )

The stored operation format will be slightly different.  The variable Ans (last answer) represents u(n-1).  We have more freedom with the expression since Ans is used. 

Before execution the operation, enter the initial condition, press [ENTER] to store in Ans.  Press {op} and repeat as desired.  

Example 5:

Recurring sequence:  u(n) = 3 * u(n-1) - 1 with the initial condition u(1) = 1. 

Stored Operation:  op = 3 * Ans - 1

Find the next six terms.  

1, [ENTER], {op}

The next six terms are 2, 5, 14, 41, 122, 365. 

Hope you find the stored operation feature helpful.  Until next time,

Eddie

This blog is property of Edward Shore, 2016