🔄Dynamical Systems Unit 3 – Linear Systems and Stability

Key Concepts and Definitions

• Linear systems are mathematical models described by linear differential equations where the rate of change of variables depends linearly on the variables themselves
• Equilibrium points are steady-state solutions where the system remains unchanged over time and are found by setting the derivatives equal to zero
• Stability refers to the behavior of a system near an equilibrium point and whether small perturbations cause the system to return to (stable), move away from (unstable), or remain near (neutrally stable) the equilibrium
  • Asymptotic stability implies the system converges to the equilibrium point as time approaches infinity
  • Lyapunov stability means the system remains close to the equilibrium point for all time given a sufficiently small perturbation
• Eigenvalues $\lambda$ are scalar values that satisfy the equation $Av = \lambda v$ for a square matrix $A$ and nonzero vector $v$, determining the stability and behavior of the system near equilibrium points
• Eigenvectors $v$ are nonzero vectors that, when multiplied by a square matrix $A$, result in a scalar multiple of themselves: $Av = \lambda v$
• The phase plane is a graphical representation of a two-dimensional system's behavior, plotting the state variables against each other to visualize trajectories and equilibrium points
• Linearization approximates a nonlinear system's behavior near an equilibrium point by a linear system, enabling stability analysis using eigenvalues and eigenvectors

Linear Systems Overview

• Linear systems are described by linear differential equations of the form $\frac{dx}{dt} = Ax$, where $x$ is the state vector and $A$ is a constant matrix
  • The solution to a linear system is given by $x(t) = e^{At}x(0)$, where $x(0)$ is the initial condition
• Superposition principle holds for linear systems, meaning the response to multiple inputs is the sum of the responses to each input individually
• Homogeneity property states that scaling the initial condition by a constant factor scales the solution by the same factor
• Linear systems can be classified as homogeneous (no input or forcing term) or non-homogeneous (with input or forcing term)
• The behavior of a linear system is determined by the eigenvalues and eigenvectors of the matrix $A$
  • Real, distinct eigenvalues lead to exponential growth or decay along the eigenvector directions
  • Complex conjugate eigenvalues result in oscillatory behavior with exponential growth or decay
• The general solution to a non-homogeneous linear system is the sum of the homogeneous solution (response to initial conditions) and the particular solution (response to input or forcing term)

Equilibrium Points and Stability

• Equilibrium points $x^*$ are found by solving the equation $Ax^* = 0$ for homogeneous systems or $Ax^* + b = 0$ for non-homogeneous systems, where $b$ is the constant input or forcing term
• The stability of an equilibrium point is determined by the eigenvalues of the Jacobian matrix $J(x^*)$ evaluated at the equilibrium point
  • If all eigenvalues have negative real parts, the equilibrium is asymptotically stable
  • If any eigenvalue has a positive real part, the equilibrium is unstable
  • If all eigenvalues have non-positive real parts and at least one has a zero real part, further analysis is needed to determine stability (center manifold theory)
• Lyapunov's direct method provides a way to determine stability without explicitly solving the differential equations by constructing a Lyapunov function $V(x)$ that satisfies certain conditions
  • If $V(x)$ is positive definite and its time derivative $\dot{V}(x)$ is negative definite, the equilibrium is asymptotically stable
  • If $V(x)$ is positive definite and $\dot{V}(x)$ is negative semi-definite, the equilibrium is stable in the sense of Lyapunov
• Bifurcations occur when the stability of an equilibrium point changes as a parameter of the system is varied, leading to qualitative changes in the system's behavior (saddle-node, pitchfork, Hopf bifurcations)

Eigenvalues and Eigenvectors

• Eigenvalues and eigenvectors are computed by solving the characteristic equation $\det(A - \lambda I) = 0$, where $I$ is the identity matrix
  • The roots of the characteristic polynomial are the eigenvalues
  • For each eigenvalue, the corresponding eigenvector is found by solving $(A - \lambda_i I)v_i = 0$
• The eigenvalues determine the stability and behavior of the system near equilibrium points
  • Negative real eigenvalues imply stable nodes or spirals
  • Positive real eigenvalues imply unstable nodes or spirals
  • Zero eigenvalues indicate centers or non-isolated equilibria
  • Complex conjugate eigenvalues with negative real parts imply stable foci or centers
  • Complex conjugate eigenvalues with positive real parts imply unstable foci
• Eigenvectors form the basis for the solution space and determine the principal directions of growth or decay
• The general solution to a homogeneous linear system can be written as a linear combination of the eigenvectors: $x(t) = c_1e^{\lambda_1t}v_1 + c_2e^{\lambda_2t}v_2 + \ldots + c_ne^{\lambda_nt}v_n$, where $c_i$ are constants determined by initial conditions
• Degenerate eigenvalues (repeated roots of the characteristic polynomial) require generalized eigenvectors to form a complete basis for the solution space

Phase Plane Analysis

• The phase plane is a graphical tool for analyzing two-dimensional systems by plotting the state variables $x_1$ and $x_2$ against each other
• Nullclines are curves in the phase plane where $\frac{dx_1}{dt} = 0$ or $\frac{dx_2}{dt} = 0$, and their intersections represent equilibrium points
• Vector field plots show the direction and magnitude of the system's rate of change at each point in the phase plane, indicated by arrows
• Trajectories are curves in the phase plane that represent the system's evolution from different initial conditions
  • Trajectories cannot cross each other in autonomous systems (no explicit time dependence)
• Stable and unstable manifolds are special trajectories that approach or depart from saddle points, separating regions of different behavior in the phase plane
• Limit cycles are isolated closed trajectories in the phase plane that represent periodic solutions, and their stability is determined by the behavior of nearby trajectories (stable if attracting, unstable if repelling)
• Bifurcations can be visualized in the phase plane as the creation, destruction, or change in stability of equilibrium points or limit cycles as a parameter is varied

Stability Criteria and Classification

• The Routh-Hurwitz criterion provides a necessary and sufficient condition for the stability of a linear system based on the coefficients of the characteristic polynomial
  • Construct the Routh array using the coefficients and analyze the signs of the entries in the first column
  • If all entries in the first column have the same sign, the system is stable; otherwise, it is unstable
• The trace-determinant plane is a graphical method for classifying the stability of a 2D linear system based on the trace $\tau$ and determinant $\Delta$ of the matrix $A$
  • The parabola $\Delta = \frac{\tau^2}{4}$ and the lines $\tau = 0$ and $\Delta = 0$ divide the plane into regions corresponding to different types of equilibria (nodes, saddles, foci, centers)
• Lyapunov's indirect method (linearization) determines the stability of a nonlinear system's equilibrium point by analyzing the stability of the linearized system
  • If the linearized system is asymptotically stable (unstable), the equilibrium point of the nonlinear system is locally asymptotically stable (unstable)
  • If the linearized system is stable but not asymptotically stable, no conclusion can be drawn about the nonlinear system's stability
• The Hartman-Grobman theorem states that the behavior of a nonlinear system near a hyperbolic equilibrium point (no eigenvalues with zero real part) is qualitatively the same as that of its linearization
• Poincaré-Bendixson theorem provides criteria for the existence of limit cycles in 2D nonlinear systems based on the properties of the region containing the trajectory (bounded, closed, and containing no equilibrium points)

Applications and Examples

• Population dynamics models, such as the Lotka-Volterra predator-prey model, use linear systems to describe the interaction between species
  • The model consists of two coupled differential equations representing the populations of prey and predators
  • Equilibrium points and their stability provide insights into the long-term behavior of the populations (coexistence, extinction, or oscillations)
• Mechanical systems, such as coupled mass-spring-damper systems, can be modeled using linear differential equations
  • The state variables represent the positions and velocities of the masses
  • Eigenvalues and eigenvectors determine the natural frequencies and modes of vibration of the system
• Electrical circuits, such as RLC (resistor-inductor-capacitor) circuits, are described by linear differential equations
  • The state variables are the currents through the inductors and the voltages across the capacitors
  • The stability and transient behavior of the circuit can be analyzed using the techniques for linear systems
• Consensus protocols in multi-agent systems, such as flocking birds or robot swarms, use linear systems to model the interaction and coordination among agents
  • The state variables represent the positions, velocities, or other attributes of the agents
  • The stability of the consensus equilibrium determines whether the agents achieve a common state or exhibit other collective behaviors
• Epidemic models, such as the SIR (Susceptible-Infected-Recovered) model, use linear systems to describe the spread of infectious diseases in a population
  • The state variables represent the fractions of the population in each compartment (susceptible, infected, or recovered)
  • The stability of the disease-free and endemic equilibria provides insights into the long-term behavior of the epidemic (eradication or persistence)

Advanced Topics and Extensions

• Nonlinear systems exhibit rich and complex behaviors that cannot be captured by linear systems, such as multiple equilibria, limit cycles, and chaos
  • Lyapunov stability theory provides tools for analyzing the stability of nonlinear systems without explicitly solving the differential equations
  • Bifurcation theory studies the qualitative changes in the behavior of a nonlinear system as parameters are varied, such as the creation or destruction of equilibria or limit cycles
• Time-delayed systems involve differential equations with delayed arguments, where the rate of change of the state variables depends on their values at previous times
  • The stability analysis of time-delayed systems requires the use of functional differential equations and infinite-dimensional techniques
  • Time delays can lead to instabilities and oscillations that are not present in the corresponding delay-free systems
• Stochastic systems incorporate random noise or uncertainty into the differential equations, leading to stochastic differential equations (SDEs)
  • The stability and behavior of stochastic systems are described using concepts such as almost sure stability, stability in probability, and moment stability
  • Stochastic Lyapunov functions and Itô's calculus are used to analyze the stability of SDEs
• Adaptive control systems involve the design of controllers that can adapt their parameters or structure in response to changes in the system or environment
  • Lyapunov-based adaptive control methods use Lyapunov functions to ensure the stability of the closed-loop system while estimating unknown parameters or adapting the controller gains
  • Model reference adaptive control (MRAC) aims to make the closed-loop system behave like a reference model by adjusting the controller parameters based on the error between the system and the model
• Synchronization of coupled oscillators or chaotic systems is a phenomenon where two or more systems adjust their rhythms or behaviors to match each other due to weak interactions
  • The master stability function approach analyzes the stability of the synchronized state by studying the variational equations around the synchronization manifold
  • Synchronization has applications in various fields, such as neuroscience (brain rhythms), biology (circadian clocks), and engineering (power grids, communication networks)