1.1 Fundamentals of dynamical systems

Dynamical systems are the backbone of Ergodic Theory, describing how systems evolve over time. They're crucial for understanding everything from planetary motion to population dynamics, using math to predict future states based on current conditions.

In this section, we'll cover the basics: what makes a system "dynamical," different types of systems, and how they behave long-term. We'll also explore fixed points, orbits, and attractors – key concepts for grasping system behavior.

Dynamical Systems Defined

Mathematical Foundations

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• Dynamical system represents time evolution of a point in geometrical space governed by deterministic rules or equations
• State of system at any time represented by point in state space (discrete or continuous)
• Described by differential equations (continuous-time) or difference equations (discrete-time)
• Time parameter can be continuous (flows) or discrete (maps or cascades)

Key Components and Applications

• State space defines possible system configurations
• Time domain specifies how system evolves (continuous or discrete steps)
• Evolution rule determines how system changes over time (equations of motion)
• Real-world applications span physics (planetary motion), biology (population dynamics), economics (market trends), and engineering (control systems)

System Representations

• Continuous-time systems evolve smoothly, modeled by equations like $\frac{dx}{dt} = f(x,t)$
• Discrete-time systems change in distinct steps, represented as $x_{n+1} = f(x_n)$
• Phase space diagrams visualize system behavior (trajectories in state space)
• Vector fields illustrate flow of continuous systems in phase space

Types of Dynamical Systems

Linear vs Nonlinear Systems

• Linear systems obey superposition principle
  • Output proportional to input
  • Can be described by linear differential equations
  • Examples include simple harmonic oscillator, RC circuits
• Nonlinear systems exhibit more complex behaviors
  • May have multiple equilibria, limit cycles, or chaos
  • Described by nonlinear equations
  • Examples include pendulum with large swings, predator-prey models

Autonomous vs Non-Autonomous Systems

• Autonomous systems have evolution rules independent of time
  • Equations of motion do not explicitly contain time variable
  • Examples include simple pendulum, Lorenz system
• Non-autonomous systems have time-dependent evolution rules
  • Equations contain explicit time dependence
  • Examples include forced oscillators, seasonally-varying ecological models

Conservative vs Dissipative Systems

• Conservative systems preserve some quantity (energy) over time
  • Total energy remains constant
  • Examples include ideal pendulum, frictionless mechanical systems
• Dissipative systems lose energy or information over time
  • Energy decreases due to friction or other irreversible processes
  • Examples include damped oscillators, fluid systems with viscosity

Deterministic vs Stochastic Systems

• Deterministic systems have predictable future states given initial conditions
  • Same initial conditions always lead to same outcome
  • Examples include classical mechanical systems, simple population models
• Stochastic systems involve random elements
  • Probabilistic outcomes even with known initial conditions
  • Examples include Brownian motion, stock market fluctuations

Long-Term Behavior of Systems

Stability Analysis

• Examines how small perturbations affect system state over time
• Lyapunov stability theory provides framework for analyzing equilibrium stability
  • Lyapunov functions quantify "energy" of perturbations
  • Asymptotic stability implies perturbations decay over time
• Linear stability analysis uses eigenvalues of linearized system
  • Negative real parts indicate stability, positive parts instability

Bifurcation Theory

• Studies qualitative changes in system behavior as parameters vary
• Types of bifurcations include:
  • Saddle-node bifurcation (creation/annihilation of fixed points)
  • Hopf bifurcation (birth of limit cycle from fixed point)
  • Period-doubling bifurcation (route to chaos)
• Bifurcation diagrams visualize parameter-dependent behavior changes

Poincaré Maps and Limit Sets

• Poincaré maps analyze continuous systems through discrete snapshots
  • Reduces dimensionality of analysis
  • Useful for studying periodic orbits and chaos
• Limit sets characterize long-term trajectory behavior
  • Limit cycles represent periodic behavior (closed orbits)
  • Strange attractors characterize chaotic systems (fractal structure)
• Basin of attraction defines initial conditions leading to particular attractor

Fixed Points, Orbits, and Attractors

Fixed Points and Stability

• Fixed points (equilibria) satisfy condition that rate of change equals zero
• Stability classifications:
  • Stable node (all trajectories approach)
  • Unstable node (all trajectories move away)
  • Saddle point (some approach, some move away)
• Linear stability analysis uses Jacobian matrix eigenvalues
  • Negative real parts indicate stability
  • Positive real parts indicate instability

Periodic Orbits and Limit Cycles

• Periodic orbits represent cyclic behavior (closed trajectories in state space)
• Limit cycles are isolated periodic orbits that attract or repel nearby trajectories
  • Stable limit cycles attract neighboring trajectories
  • Unstable limit cycles repel neighboring trajectories
• Examples include self-sustained oscillations (heartbeats, predator-prey cycles)

Attractors and Repellers

• Attractors are sets of states system evolves towards
  • Can be fixed points, limit cycles, or strange attractors
  • Strange attractors exhibit fractal structure (Lorenz attractor)
• Repellers are sets system trajectories move away from
  • Unstable fixed points or limit cycles act as repellers
• Basins of attraction partition state space based on long-term behavior
  • Separatrices divide basins of different attractors