14.2 Delay differential equations

Delay differential equations are a fascinating twist on ordinary differential equations. They introduce time delays, making the system's future depend on its past. This adds complexity, leading to oscillations and instabilities that ordinary equations don't capture.

Analyzing these equations is tricky. The state space becomes infinite-dimensional, and stability analysis gets more complex. But new tools like Lyapunov-Krasovskii functionals help us understand these systems and their unique behaviors.

Fundamental Concepts

Time Delay and Retarded Functional Differential Equations

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• Time delay refers to the phenomenon where the rate of change of a system depends on the state of the system at a previous time
• Delay differential equations (DDEs) incorporate time delays into the mathematical model
• Retarded functional differential equations are a type of DDE where the highest order derivative depends on the state of the system at a previous time
  • The term "retarded" indicates that the derivative depends on past states
• The presence of time delays can significantly impact the behavior and stability of a system compared to ordinary differential equations (ODEs)
  • Time delays can introduce oscillations, instabilities, and complex dynamics

State Space Representation and Stability Analysis

• The state space of a DDE is infinite-dimensional due to the dependence on past states
  • This is in contrast to ODEs, which have a finite-dimensional state space
• The state of a DDE at time $t$ is represented by a function $x_t(\theta)$ defined on the interval $[-\tau, 0]$, where $\tau$ is the maximum delay
  • $x_t(\theta) = x(t + \theta)$ for $\theta \in [-\tau, 0]$
• Stability analysis of DDEs involves studying the behavior of solutions near an equilibrium point
  • Asymptotic stability: Solutions starting close to the equilibrium converge to it as $t \to \infty$
  • Exponential stability: Solutions converge to the equilibrium exponentially fast

Advanced Analysis Techniques

Characteristic Equation and Stability Criteria

• The characteristic equation of a linear DDE with a single delay $\tau$ is given by $\det(\Delta(s)) = 0$, where $\Delta(s) = sI - A_0 - A_1 e^{-s\tau}$
  • $A_0$ and $A_1$ are constant matrices, and $I$ is the identity matrix
• The roots of the characteristic equation, called characteristic roots or eigenvalues, determine the stability of the system
  • If all characteristic roots have negative real parts, the system is asymptotically stable
  • If any characteristic root has a positive real part, the system is unstable
• The characteristic equation of a DDE is transcendental and has infinitely many roots, making stability analysis more challenging than ODEs

Lyapunov-Krasovskii Functionals and Stability Analysis

• Lyapunov-Krasovskii functionals are an extension of Lyapunov functions used for stability analysis of DDEs
• A Lyapunov-Krasovskii functional $V(x_t)$ is a positive definite functional that decreases along solutions of the DDE
  • If $\dot{V}(x_t) \leq -\alpha V(x_t)$ for some $\alpha > 0$, then the system is exponentially stable
• Constructing suitable Lyapunov-Krasovskii functionals can be challenging and often requires insight into the specific system
• Examples of Lyapunov-Krasovskii functionals include the energy functional and the Lyapunov-Razumikhin function

Bifurcations in Delay Differential Equations

• Bifurcations in DDEs occur when the stability of an equilibrium changes as a parameter varies
• Hopf bifurcation: A pair of complex conjugate characteristic roots crosses the imaginary axis, leading to the emergence of periodic solutions
  • The period of the periodic solutions is related to the delay and the imaginary part of the characteristic roots
• Fold bifurcation: A real characteristic root crosses zero, resulting in the creation or destruction of equilibria
• Bifurcation analysis helps understand the qualitative changes in the dynamics of DDEs and the parameter values at which these changes occur
• Examples of systems exhibiting bifurcations include the delayed logistic equation and the Mackey-Glass equation