Discrete Dynamical Systems (DDS)

Introduction to DDS and Linear DDS

Discrete Dynamical Systems (DDS) are mathematical models that describe how a quantity evolves in discrete steps, often using recurrence relations. Linear DDS are a special class where the updating function is linear in the previous term.

• Explicit Sequence: A sequence where the nth term is given directly as a function of n, e.g., .

• Recursive Sequence: A sequence where each term is defined in terms of previous terms, e.g., .

• General Linear DDS:

• Equilibrium: A value such that . Solving gives (if ).

• Stability: The equilibrium is stable if , unstable if , and oscillates if .

Table: Equilibrium and Stability for Linear DDS

• Explicit Solution (for ):

• Equilibrium Form:

Malthusian Growth Model

• Basic Model: , where is the growth rate.

• With Immigration/Emigration:

• Both are linear DDS.

Non-linear DDS and the Logistic Model

Non-linear DDS have updating functions that are not linear, often leading to more complex behavior such as multiple equilibria.

• Logistic Model (Additive Form):

• Logistic Model (Quadratic Form):

• Parameters: (growth rate), (carrying capacity)

• Equilibria: (trivial, unstable), (non-trivial, stable)

Systems of Equations and Gauss-Jordan Method

Solving Systems of Linear Equations

A system of equations consists of multiple equations with several variables. Solutions can be found using substitution, elimination, or matrix methods.

• Solution Form:

• Augmented Matrix: Represents the system in matrix form, with the last column for constants.

• Gauss-Jordan Method: Uses elementary row operations to reduce the matrix to Reduced Row Echelon Form (RREF).

• Elementary Row Operations:

  • Swap two rows

  • Multiply a row by a non-zero constant

  • Add a multiple of one row to another

• Types of Solutions:

  • Unique Solution: Identity matrix on the left; solution is the last column.

  • No Solution: Row of zeros with a non-zero constant in the last column.

  • Infinitely Many Solutions: Row of all zeros; corresponding variable is free.

Matrices

Matrix Basics and Operations

Matrices are rectangular arrays of numbers, used to represent systems of equations and perform linear transformations.

• Notation: Capital letters (e.g., A, B) denote matrices.

• Dimensions: matrix has rows and columns.

• Square Matrix:

Addition and Subtraction

• Possible only if matrices have the same dimensions.

• Add corresponding entries.

• Example:

Multiplication

• Possible only if the number of columns in the first matrix equals the number of rows in the second.

• Resulting matrix has dimensions (rows of first) × (columns of second).

• Matrix multiplication is not commutative: in general.

• Example:

  • Multiply (3×2) by (2×3):

  • Result: (3×3)

Applications and Example Problems

Classifying Sequences

• Explicit: ,

• Recursive: ,

Classifying DDS Equations

Finding Equilibrium and Stability

• For :

  • Linear DDS: ,

  • Equilibrium:

  • Stability: (stable)

  • Explicit formula:

• For :

  • Linear DDS: ,

  • Equilibrium:

  • Stability: (unstable)

  • Explicit formula:

• For :

  • Non-linear DDS (quadratic term)

  • Equilibria: and (since )

  • Stability: is stable for , is unstable

Logistic Model Example

• Given :

• Quadratic form: ,

• Growth rate:

• Carrying capacity:

• Equilibria: (unstable), (stable)

Writing Systems from Augmented Matrices

• Given , the system is:

• Given , the system is:

General and Specific Solutions with Free Variables

• Given , the general solution is:

  • is free

  • General solution:

  • For :

Modeling Resource Constraints with Systems

Given the following table of resources needed per acre for apples (a), broccoli (b), and carrots (c):

• Constraints:

  • (nitrogen)

  • (phosphate)

  • (labor)

  • (total acres)

Matrix Dimensions and Operations

• Given:

  • (2×2)

  • (2×2)

  • (2×3)

  • (3×2)

• Matrix operations:

  • : Possible (both 2×2), result is 2×2

  • : Not possible (dimensions differ)

  • : Possible (2×2 × 2×2), result is 2×2

  • : Possible (2×2 × 2×2), result is 2×2

  • : Not possible (2×2 × 3×2)

  • : Possible (3×2 × 2×2), result is 3×2

Additional info: For matrix multiplication, the inner dimensions must match; the result has the outer dimensions.