An Introduction to Applied Mathematics - Modeling Strategy, Control, and Chaos

March 8, 2023 — #Mathematics #Applied Mathematics

While pure mathematics explores abstract structures and patterns for their own intrinsic beauty, applied mathematics focuses on developing mathematical tools to solve practical problems in science, engineering, business, and other fields. It provides a powerful bridge between theoretical concepts and the real world. This exploration provides a descriptive overview of four captivating and influential areas of applied mathematics: Game Theory, which models strategic interaction; Control Theory, the science of automation; and Chaos Theory, which finds patterns in seemingly random systems.

Game Theory - The Science of Strategy

Game theory is a branch of mathematics concerned with the analysis of strategies for dealing with competitive situations where the outcome of a participant's choice of action depends critically on the actions of other participants. It is the formal study of conflict and cooperation.

Key Components of a Game

Players: The rational decision-makers in the game.
Strategies: The set of possible actions that a player can take.
Payoffs: The outcome or consequence (e.g., utility, profit, reward) for each player for every possible combination of strategies chosen by all players. This is often represented in a payoff matrix.

Types of Games

Zero-Sum vs. Non-Zero-Sum: In a zero-sum game, the total benefit to all players adds up to zero; one player's gain is exactly another player's loss. In non-zero-sum games, players' interests are not always in direct conflict, and outcomes can be mutually beneficial or mutually detrimental.
Cooperative vs. Non-Cooperative: Cooperative games allow players to form binding commitments and alliances, whereas non-cooperative games do not.
Simultaneous vs. Sequential: In simultaneous games, players choose their actions at the same time, without knowledge of the others' choices. In sequential games, players have some knowledge about earlier actions.

Core Concepts and Solutions

Dominant Strategy: A strategy is dominant if it yields the best possible payoff for a player, regardless of the strategies chosen by the other players.
Nash Equilibrium: A foundational concept named after John Nash. A Nash Equilibrium is a set of strategies, one for each player, such that no player has an incentive to unilaterally change their strategy. Each player is playing their best possible response to the strategies of the other players.
The Prisoner's Dilemma: A classic example of a non-cooperative, non-zero-sum game that illustrates the conflict between individual and collective rationality.
- Scenario: Two suspects (prisoners) are arrested and held in separate cells, unable to communicate. The prosecutor lacks sufficient evidence for a major conviction unless one or both confess.
- Strategies: Each prisoner can either Cooperate (with their partner, by staying silent) or Defect (by confessing and implicating their partner).
- Payoffs (years in prison):
  - If both Cooperate (stay silent), they both get a minor sentence (e.g., 1 year).
  - If one Defects and the other Cooperates, the defector goes free (0 years) and the cooperator gets a severe sentence (e.g., 5 years).
  - If both Defect (confess), they both get a moderate sentence (e.g., 3 years).
Payoff Matrix: (Prisoner 1, Prisoner 2)

Prisoner 2 Cooperates Prisoner 2 Defects

Prisoner 1 Cooperates (-1, -1) (-5, 0)

Prisoner 1 Defects (0, -5) (-3, -3)

Analysis: From Prisoner 1's perspective:
- If Prisoner 2 Cooperates, my best move is to Defect (0 years is better than -1 year).
- If Prisoner 2 Defects, my best move is still to Defect (-3 years is better than -5 years).
Defecting is a dominant strategy for Prisoner 1. By symmetry, it is also the dominant strategy for Prisoner 2. The unique Nash Equilibrium is (Defect, Defect), leading to a (-3, -3) outcome. This is paradoxical because if both had cooperated, they would have achieved a better collective outcome of (-1, -1).

	Prisoner 2 Cooperates	Prisoner 2 Defects
Prisoner 1 Cooperates	(-1, -1)	(-5, 0)
Prisoner 1 Defects	(0, -5)	(-3, -3)

Control Theory - Engineering the Dynamics of Systems

Control theory is an interdisciplinary field of engineering and mathematics that deals with the behavior of dynamical systems with inputs, and how to design controllers that modify the system's behavior to achieve a desired output, ensuring stability and performance.

System Representation

Transfer Function ( $G(s)$ ): A key concept in classical (frequency-domain) control theory. For a linear time-invariant (LTI) system, the transfer function is the ratio of the Laplace transform of the output, $Y(s)$ , to the Laplace transform of the input, $U(s)$ , assuming zero initial conditions. $G(s) = \frac{Y(s)}{U(s)}$
State-Space Representation: A model in the time-domain used in modern control theory. It describes a system using a set of first-order differential equations.
- State Equations: $\dot{\mathbf{x}}(t) = A\mathbf{x}(t) + B\mathbf{u}(t)$
- Output Equation: $\mathbf{y}(t) = C\mathbf{x}(t) + D\mathbf{u}(t)$
Here, $\mathbf{x}(t)$ is the state vector, $\mathbf{u}(t)$ is the input vector, $\mathbf{y}(t)$ is the output vector, and $A, B, C, D$ are state matrices.

Feedback Control - The Core Idea

Feedback (or closed-loop) control is the process of using the measured output of a system to influence its input, thereby controlling its behavior. This is used to reduce the error between the desired output (reference) and the actual output, improve stability, and reject external disturbances.

Derivation of Closed-Loop Transfer Function: Consider a simple negative feedback system where $R(s)$ is the reference input, $Y(s)$ is the output, $C(s)$ is the controller transfer function, and $G(s)$ is the plant (the system being controlled). The error signal is $E(s) = R(s) - Y(s)$ . The controller acts on the error to produce the input to the plant, $U(s) = C(s)E(s)$ . The plant produces the output, $Y(s) = G(s)U(s)$ .
1. Substitute the expression for $U(s)$ : $Y(s) = G(s)C(s)E(s)$ .
2. Substitute the expression for $E(s)$ : $Y(s) = G(s)C(s)[R(s) - Y(s)]$ .
3. Distribute: $Y(s) = G(s)C(s)R(s) - G(s)C(s)Y(s)$ .
4. Solve for $Y(s)$ : $Y(s) + G(s)C(s)Y(s) = G(s)C(s)R(s) \implies Y(s)(1 + G(s)C(s)) = G(s)C(s)R(s)$ .
5. The closed-loop transfer function $T(s) = Y(s)/R(s)$ is: $T(s) = \frac{G(s)C(s)}{1 + G(s)C(s)}$

The PID Controller - The Industry Workhorse

The Proportional-Integral-Derivative (PID) controller is the most common feedback controller. It calculates an output $u(t)$ based on the error signal $e(t) = r(t) - y(t)$ .

Control Law: $u(t) = K_p e(t) + K_i \int_0^t e(\tau)d\tau + K_d \frac{de(t)}{dt}$
- Proportional ( $K_p$ ): Reacts to the present error. A larger $K_p$ gives a stronger reaction but can lead to instability.
- Integral ( $K_i$ ): Reacts to the past error by accumulating it. This term drives the steady-state error to zero.
- Derivative ( $K_d$ ): Reacts to the future error by considering its rate of change. This term dampens oscillations and improves stability.

Chaos Theory - Finding Order in Disorder

Chaos theory is the study of dynamical systems that are highly sensitive to their initial conditions, a property often referred to as the butterfly effect. These systems are deterministic, meaning their future behavior is fully determined by their initial conditions, with no random elements involved. However, their extreme sensitivity makes their long-term behavior impossible to predict in practice.

Key Characteristics of Chaotic Systems

Deterministic: The system follows precise rules with no randomness.
Sensitive Dependence on Initial Conditions (The Butterfly Effect): Tiny differences in the initial state of the system lead to vastly different outcomes over time.
Topological Mixing: The system evolves such that any given region of its phase space will eventually overlap with any other given region.
Dense Periodic Orbits: Within any chaotic system, there are an infinite number of periodic orbits, but they are all unstable.

The Logistic Map - A Simple Model of Chaos

A simple mathematical model that exhibits chaos is the logistic map, a recurrence relation that can model population dynamics. $x_{n+1} = r x_n (1 - x_n)$ Here, $x_n$ is a number between zero and one that represents the population at year $n$ as a fraction of the maximum possible population, and $r$ is a positive parameter controlling growth.

Behavior based on $r$ :
- For $0 < r \le 1$ , the population will eventually die out ( $x \to 0$ ).
- For $1 < r \le 3$ , the population will converge to a single stable value.
- For $3 < r \le 1+\sqrt{6} \approx 3.449$ , the population oscillates between two stable values (a period-doubling bifurcation).
- As $r$ increases further, the population oscillates between 4 values, then 8, 16, and so on. This cascade of period-doubling bifurcations occurs faster and faster.
- At approximately $r \approx 3.57$ , the system enters chaos. The population value varies unpredictably from year to year, never settling into a stable orbit.

A bifurcation diagram of the logistic map beautifully visualizes this "route to chaos" by plotting the long-term values of $x_n$ for each value of the parameter $r$ .

Strange Attractors

In the study of dynamical systems, an attractor is a set of states toward which a system tends to evolve. A strange attractor is an attractor that has a fractal structure. Trajectories on the attractor appear chaotic, but they are confined to this complex geometric object.

The Lorenz Attractor: A classic example, derived from a simplified model of atmospheric convection. Its equations are: $\begin{aligned} \frac{dx}{dt} &= \sigma(y-x) \\ \frac{dy}{dt} &= x(\rho-z) - y \\ \frac{dz}{dt} &= xy - \beta z \end{aligned}$ For certain parameter values (e.g., $\sigma=10, \rho=28, \beta=8/3$ ), the system exhibits chaotic behavior, with its trajectory tracing out a distinctive butterfly-shaped strange attractor.

Key Takeaways: The Mathematics of Dynamic Systems

The fields of game theory, control theory, and chaos theory provide powerful and distinct mathematical frameworks for analyzing complex, dynamic systems.

Game Theory: Models strategic interactions between rational agents. Core concepts like the payoff matrix, dominant strategies, and Nash Equilibrium allow for the analysis of conflict and cooperation in fields ranging from economics to biology.
Control Theory: Provides the engineering principles for making systems behave as desired. Using feedback and tools like transfer functions and state-space models, it allows for the design of stable, high-performance systems crucial to modern technology.
Chaos Theory: Reveals the intricate, ordered patterns hidden within systems that appear random. It shows that deterministic systems can exhibit profound unpredictability due to sensitive dependence on initial conditions, with their long-term behavior often characterized by fractal structures known as strange attractors.

Together, these disciplines showcase the power of applied mathematics to model and understand the strategic, engineered, and seemingly unpredictable dynamics that govern our world.