1.3 Classification of dynamical systems

Dynamical systems come in various flavors, each with unique characteristics. This section breaks them down into categories based on linearity, time dependence, predictability, and energy behavior. Understanding these classifications helps us analyze and model real-world systems more effectively.

By exploring linear vs. nonlinear, autonomous vs. non-autonomous, deterministic vs. stochastic, and conservative vs. dissipative systems, we gain insights into their behaviors. This knowledge forms the foundation for tackling more complex dynamical systems in later chapters.

System Linearity

Linear Systems

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• Satisfy the principle of superposition
  • The response to a linear combination of inputs is the same linear combination of individual responses
  • Enables the use of powerful mathematical tools for analysis (Fourier transforms, Laplace transforms)
• Exhibit proportional relationships between input and output
  • Doubling the input doubles the output (spring-mass system, resistor-capacitor circuit)
• Can be described by linear differential equations
  • Coefficients are constant and not dependent on the variables (first-order linear equation: $\frac{dx}{dt} + ax = b$)

Nonlinear Systems

• Do not satisfy the principle of superposition
  • The response to a combination of inputs is not the same as the combination of individual responses
  • Makes analysis more challenging and often requires numerical methods
• Exhibit complex behaviors and phenomena
  • Chaos, bifurcations, and emergent properties (logistic map, Lorenz system)
• Described by nonlinear differential equations
  • Coefficients or terms are functions of the variables (quadratic equation: $\frac{d^2x}{dt^2} + x^2 = 0$)

Time Dependence

Autonomous Systems

• The equations governing the system do not explicitly depend on time
  • The right-hand side of the differential equation does not contain the time variable $t$ (population growth model: $\frac{dP}{dt} = rP(1-\frac{P}{K})$)
• The system's behavior is determined solely by its current state
  • The future evolution depends only on the initial conditions (pendulum without external forcing)
• Equilibrium points and limit cycles are common features
  • Fixed points and closed trajectories in the phase space (Van der Pol oscillator)

Non-autonomous Systems

• The equations governing the system explicitly depend on time
  • The right-hand side of the differential equation contains the time variable $t$ (forced harmonic oscillator: $\frac{d^2x}{dt^2} + \omega^2x = F(t)$)
• External time-dependent inputs or forcing terms influence the system's behavior
  • The future evolution depends on both the initial conditions and the time-varying inputs (RLC circuit with time-varying voltage source)
• Can exhibit more complex behaviors, such as entrainment and synchronization
  • Coupled oscillators with time-dependent coupling strengths (Kuramoto model)

Predictability

Deterministic Systems

• The future state of the system can be precisely predicted given the initial conditions and governing equations
  • No randomness or uncertainty in the system's evolution (double pendulum with known initial angles and angular velocities)
• Sensitive dependence on initial conditions can lead to chaotic behavior
  • Small differences in initial conditions result in widely diverging trajectories over time (Lorenz system)
• Lyapunov exponents quantify the rate of separation of nearby trajectories
  • Positive Lyapunov exponents indicate chaos (logistic map with $r > 3.56$)

Stochastic Systems

• The future state of the system is described by probability distributions rather than precise values
  • Randomness or uncertainty is inherent in the system's dynamics (Brownian motion of particles)
• Governed by stochastic differential equations or random processes
  • Incorporate noise terms or random variables (Langevin equation: $\frac{dx}{dt} = f(x) + \xi(t)$, where $\xi(t)$ is a random noise term)
• Ensemble averages and probability densities characterize the system's behavior
  • Mean, variance, and higher-order moments of the state variables (diffusion process)

Energy and Dissipation

Conservative Systems

• The total energy of the system remains constant over time
  • No dissipation or external energy input (frictionless pendulum, ideal spring-mass system)
• Described by conservative forces and potential energy functions
  • The work done by conservative forces is path-independent (gravitational potential energy: $U(x) = mgx$)
• Exhibit reversible dynamics and preserve phase space volume
  • Trajectories can be traced backward in time (planetary motion)

Dissipative Systems

• The total energy of the system decreases over time due to dissipative forces
  • Friction, resistance, or energy dissipation mechanisms are present (damped harmonic oscillator, RLC circuit with resistance)
• Non-conservative forces perform work and convert energy into heat or other forms
  • The work done by dissipative forces is path-dependent (friction force: $F = -\mu N$)
• Attractors and limit cycles are common features
  • Trajectories converge to specific regions in phase space (Van der Pol oscillator with damping)

Hamiltonian Systems

• Described by generalized coordinates and momenta
  • The state of the system is represented by position and momentum variables $(q_i, p_i)$
• Governed by Hamilton's equations of motion
  • First-order differential equations relating coordinates and momenta ($\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}, \frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i}$)
• The Hamiltonian function $H(q, p, t)$ represents the total energy of the system
  • Sum of kinetic and potential energy (simple harmonic oscillator: $H = \frac{p^2}{2m} + \frac{1}{2}kq^2$)
• Exhibit symplectic structure and preserve phase space volume
  • Canonical transformations maintain the Hamiltonian form of the equations (action-angle variables)