Growth Dynamics Slides Discrete Dynamical Systems
Linear Growth &
Difference Equations
Lesson 1: Modeling Unrestricted Populations
The Exponential Trap
"If a bacteria colony doubles every hour, and it takes 24 hours to cover the entire Earth, when was the Earth only half-covered?"
The Result
Hour 23
Linear intuition fails us. We see 23 hours of "nothing" followed by total takeover.
The Model
\( x_{n+1} = 2x_n \)
Simple recursive rules drive massive global shifts.
Defining Discrete Systems
A First-Order Difference Equation relates the state at time \( n+1 \) to its state at time \( n \).
\( x_{n+1} = f(x_n) \)
\( x_n \): State at step \( n \)
\( f \): Evolution rule
\( x_0 \): Initial condition
Discrete vs. Continuous
Unlike differential equations, we observe the system at fixed intervals (breeding seasons, fiscal years).
Linear Form
For unrestricted growth: \( x_{n+1} = r x_n \). The ratio \( r \) is the key parameter.
Stability Analysis
The behavior of \( x_n = x_0 r^n \) is determined by the magnitude of \( r \):
|r| > 1
Unstable / Explosion
System diverges.
|r| < 1
Stable / Asymptotic
Decays to equilibrium.
r < 0
Oscillatory
Flips sign every step.
Case Study: Compound Interest
Financial Modeling
Consider a loan with principal \( P \), annual interest \( i \), and monthly payment \( M \).
\( B_{n+1} = (1 + \frac{i}{12}) B_n - M \)
This is an affine first-order equation. The fixed point occurs where \( B_{n+1} = B_n \), representing an interest-only payment.
Seminar Discussion
Why is this system fundamentally unstable?
What happens if \( M \) is less than the monthly interest?
How does this relate to eigenvalues?
Linear Growth Dynamics Worksheet Linear Dynamics Analysis
Problem Set 1.1: First-Order Recursive Systems
Name:
Date:
Core Concept
A first-order linear difference equation takes the form \( x_{n+1} = a x_n + b \). The general solution is \( x_n = a^n x_0 + b \frac{1-a^n}{1-a} \) for \( a \neq 1 \).
01
Asymptotic Behavior Analysis
Given the sequence defined by \( x_{n+1} = 0.85 x_n + 30 \) with an initial condition \( x_0 = 100 \).
a) Calculate the first three iterations \( (x_1, x_2, x_3) \):
b) Solve for the fixed point \( x^* \) where \( x^* = f(x^*) \). Interpret the physical meaning if this were a pollution model (units in ppm):
c) Prove stability using the eigenvalue of the system:
02
The Fibonacci Limit
Consider the second-order system \( F_{n+2} = F_{n+1} + F_n \). While not first-order, it can be written as a system of linear equations: \( \begin{bmatrix} F_{n+2} \\ F_{n+1} \end{bmatrix} = \mathbf{A} \begin{bmatrix} F_{n+1} \\ F_n \end{bmatrix} \).
a) Determine the transition matrix \( \mathbf{A} \):
b) Find the eigenvalues \( \lambda_1, \lambda_2 \) of matrix \( \mathbf{A} \). How does the dominant eigenvalue relate to the Golden Ratio?
03
Biological Modeling Application
A population of fish in a hatchery is modeled by \( P_{n+1} = 1.1 P_n - H \), where \( H \) is a constant annual harvest.
a) If the hatchery starts with 2000 fish, what is the maximum harvest \( H \) that prevents the population from crashing?
b) Sketch the long-term behavior of the system for \( H=250 \) vs \( H=150 \):
Work area for sketching phase plot
Logistic Map Slides Non-Linear Dynamics
The Logistic Map
Lesson 2: Limits to Growth and Complexity
Resource Exhaustion
Unrestricted growth is physically impossible. Real systems have Carrying Capacities .
The Logistic Rule:
\( x_{n+1} = r x_n (1 - x_n) \)
\( x_n \) represents the percentage of maximum capacity (\( 0 \leq x_n \leq 1 \)).
Growth Component
\( r x_n \)
Dominates when population is small.
Regulation Component
\( (1 - x_n) \)
Dominates as we approach carrying capacity.
Finding Equilibrium
To find fixed points \( x^* \), we solve:
\( x^* = r x^* (1 - x^*) \)
Two Solutions:
\( x^*_1 = 0 \) (Extinction)
\( x^*_2 = 1 - \frac{1}{r} \) (Sustainable State)
Impact of Parameter \( r \):
\( r < 1 \): Extinction is the only attractor.
\( 1 < r < 3 \): The non-zero point is stable.
\( r > 3 \): Bifurcation cascade begins.
Phase Transitions in Behavior
r = 2.5
Fixed Point
r = 3.2
Period-2
r = 3.5
Period-4
r = 4.0
Chaos
"Deterministic behavior does not guarantee long-term predictability."
Logistic Dynamics Activity Logistic Dynamics Lab
Simulation & Parameter Sensitivity Analysis
Investigator:
"The discrete logistic equation is perhaps the simplest system that can generate totally unpredictable behavior. In this lab, you will manually compute trajectories and observe the sensitivity to initial conditions."
Part 1: The Stability Threshold
Compute the first 8 iterations of \( x_{n+1} = r x_n (1 - x_n) \) for the following two cases. Use \( x_0 = 0.1 \).
Scenario A: \( r = 2.0 \)
Scenario B: \( r = 3.2 \)
Part 2: Qualitative Analysis
1. Describe the qualitative difference between Scenario A and Scenario B. What is the limit of \( x_n \) as \( n \to \infty \) for both?
2. Sensitivity to Initial Conditions: If \( r = 4.0 \), two trajectories starting at \( x_0 = 0.100 \) and \( x'_0 = 0.101 \) will diverge exponentially. Why is this catastrophic for weather forecasting or long-term economic modeling?
Part 3: The Linear Stability Criterion
A fixed point \( x^* \) is stable if \( |f'(x^*)| < 1 \). For the logistic map \( f(x) = rx(1-x) \), calculate the derivative \( f'(x) \) and determine the exact value of \( r \) where the fixed point \( x^* = 1 - \frac{1}{r} \) becomes unstable.
Stability Analysis Slides Phase Space Geometry
Stability Analysis &
Cobweb Plots
Lesson 3: The Visual Mechanics of Convergence
The Cobweb Algorithm
Cobwebbing allows us to track the evolution of a system \( x_{n+1} = f(x_n) \) visually on a plot of \( y = f(x) \).
The Steps:
Start at \( (x_0, 0) \).
Move vertically to the curve \( y = f(x) \).
Move horizontally to the line \( y = x \).
The new intersection is \( x_1 \). Repeat.
Visualizing the Intersection
Where \( f(x) \) crosses \( y=x \), we identify fixed points (equilibria).
The Jacobian Analysis
Stability is determined by the slope of \( f(x) \) as it crosses the diagonal \( y=x \).
|f'(x*)| < 1
Stable Attractor
|f'(x*)| > 1
Unstable Repeller
Linearization
Near the fixed point, the system behaves like a simple linear system:
\( x_{n+1} - x^* \approx f'(x^*) (x_n - x^*) \)
Visual Taxonomy of Stability
Monotonic Sink
\( 0 < f'(x^*) < 1 \)
Spiral Sink
\( -1 < f'(x^*) < 0 \)
Spiral Source
\( f'(x^*) < -1 \)
Cobweb Plotting Handout Cobweb Analysis Workshop
Visualizing Fixed Points & Stability
Analyst:
Instructions
For each graph provided, identify all fixed points (intersections with \( y=x \)), perform the cobwebbing algorithm starting from the indicated \( x_0 \), and determine the local stability (Stable/Unstable) based on the visual result.
1. Logistic Map: \( r = 2.8 \) (Stable Fixed Point)
Start at \( x_0 = 0.2 \)
Draw Cobweb Here
Identify Fixed Points \( x^* \):
Calculate \( f'(x^*) \):
Final Behavior:
Does it spiral or approach monotonically?
2. The Sine Map: \( x_{n+1} = \sin(\pi x_n) \)
Start at \( x_0 = 0.5 \)
Draw Cobweb Here
The sine map is often used in physical oscillators. Note how the slope at the intersection determines if the system returns to rest or enters a limit cycle.
Analytical Check:
Solve \( x = \sin(\pi x) \) approximately or graphically. Find the stability of the non-zero fixed point.
Bifurcation and Chaos Slides Deterministic Chaos
Bifurcation &
The Onset of Chaos
Lesson 4: How Patterns Shatter
The Sudden Qualitative Shift
A Bifurcation occurs when a tiny change in a parameter causes a major change in long-term behavior.
Period-Doubling Cascade
A stable fixed point splits into two values (a 2-cycle). Then each splits again (4-cycle), and so on until infinity.
x* → {x_a, x_b}
Symmetry Breaking
Visualizing the Cascade
Mapping the parameter \( r \) (x-axis) against the asymptotic states \( x_\infty \) (y-axis).
\( r < 3 \): Single branch (stability).
\( r = 3 \): First fork (2-cycle).
\( r \approx 3.57 \): Point of accumulation.
Beyond: Deterministic Chaos.
CHAOS
The Three Marks of Chaos
1. SDIC
Sensitive Dependence on Initial Conditions. The "Butterfly Effect".
2. Density
Periodic orbits are dense within the chaotic set.
3. Transitivity
The system eventually visits every region of the phase space.
"Chaos is not randomness. It is infinite order compressed into a finite space."
Bifurcation Lab Activity Chaos & Bifurcation Lab
Investigating the Point of Accumulation
Investigator:
"The Feigenbaum Constant \( \delta \approx 4.669 \) describes the universal rate at which bifurcations occur in a wide class of maps. In this lab, we identify the specific parameter values where the logistic map shatters its periodicity."
Task 1
The Period-Doubling Sequence
Recall that the first bifurcation (\( 1 \to 2 \)) occurs at \( r_1 = 3.0 \). Using a calculator or computational tool, verify the following bifurcation points \( r_n \) where the system moves from a \( 2^{n-1} \)-cycle to a \( 2^n \)-cycle.
Calculate \( r_2 \) (2 to 4 cycle):
Hint: Look between 3.4 and 3.5.
Calculate \( r_3 \) (4 to 8 cycle):
Hint: Look near 3.54.
Calculate the ratio of the interval lengths:
\( \frac{r_2 - r_1}{r_3 - r_2} = \)
Does this value approach the Feigenbaum Constant?
Task 2
Window of Order
Within the chaotic region (\( r > 3.57 \)), there are "islands of stability" or windows where periodic behavior briefly returns.
Identify the behavior at \( r = 3.83 \). Is it periodic? If so, what is the period?
Discussion: Why does order spontaneously emerge from chaos in these windows? How does this relate to the concept of a "strange attractor"?
Modeling Capstone Brief Capstone Project
Discrete Systems Modeling & Asymptotic Prediction
Weight
40%
Objective
Select a real-world dynamic system and develop a First-Order Discrete Difference Equation to model its evolution. You must analyze the system's stability, identify fixed points, and predict long-term asymptotic behavior under varying parameters.
Suggested Modeling Tracks
Macro-Economics
Model national debt evolution relative to GDP growth and interest rates. Determine the "Debt Trap" bifurcation point.
Pharmacokinetics
Model drug concentration in the bloodstream with repeated dosing and metabolic decay. Solve for the steady-state concentration.
Information Viral
Model the spread of a "meme" or news article through a network with a saturation term. Identify when a post goes "chaotic".
Submission Deliverables
1. Technical Report
Definition of variables, the recursive equation, derivation of fixed points, and stability analysis.
2. Phase Portrait / Cobweb Analysis
Visual proof of the system's behavior for at least three different parameter regimes.
3. Bifurcation Study
Identify critical thresholds where the system transitions from stability to oscillation or chaos.
Draft Topic & Proposal:
Describe your system and proposed recursive rule here...
Project Evaluation Rubric Project Rubric
System Dynamics Evaluation Framework
Level: Graduate
Criteria Exemplary (4) Proficient (3) Developing (1-2) Mathematical Rigor Model is perfectly defined with rigorous notation. Fixed points and stability are analytically proven via derivatives. Model is well-defined. Fixed points are calculated, but stability analysis may lack formal derivative proof. Model contains logical errors or notation is inconsistent. Stability is only qualitatively discussed. Visual Representation Cobweb plots and phase diagrams are precise, clear, and effectively illustrate convergence or chaos. Visuals are included and mostly accurate, though some scaling or labeling issues exist. Visuals are missing or do not correctly represent the mathematical model. Sensitivity Analysis Thorough exploration of parameter sensitivity and bifurcation points. Clear distinction between regimes. Bifurcation is mentioned and one threshold is identified. Parameter effects are ignored or misunderstood. Synthesis & Context Exceptional translation of math to real-world implications. Prediction of "black swan" events or catastrophic shifts. Connection to real-world context is clear but safe. Basic asymptotic predictions provided. The model remains abstract with little connection to the chosen scenario.
Instructor Feedback
Total Score: ____ / 16
A: 15-16 B: 12-14 C: 9-11 F: < 9
