# 10: Dynamical Systems Analysis

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• 10.1: Finding fixed points in ODEs and Boolean models
• 10.2: Linearizing ODEs
If the ODE is nonlinear and not all of the operating parameters are available, it is frequently difficult or impossible to solve equations directly. Even when all the parameters are known, powerful computational and mathematical tools are needed to completely solve the ODEs in order to model the process. In order to simplify this modeling procedure and obtain approximate functions to describe the process, engineers often linearize the ODEs and employ matrix math to solve the linearized equations
• 10.3: Eigenvalues and Eigenvectors
Eigenvectors (mathbf{v}) and Eigenvalues ( λ ) are mathematical tools used in a wide-range of applications. They are used to solve differential equations, harmonics problems, population models, etc. In Chemical Engineering they are mostly used to solve differential equations and to analyze the stability of a system.
• 10.4: Using Eigenvalues and Eigenvectors to Find Stability and Solve ODEs
In this section, we will first show how to use eigenvalues to solve a system of linear ODEs. We will use the eigenvalues to show us the stability of the system. After that, another method of determining stability, the Routh stability test, will be introduced. For the Routh stability test, calculating the eigenvalues is unnecessary which is a benefit since sometimes that is difficult. The advantages and disadvantages of using eigenvalues to evaluate a system's stability will be discussed.
• 10.5: Phase Plane Analysis - Attractors, Spirals, and Limit cycles
We often use differential equations to model a dynamic system such as a valve opening or tank filling. Without a driving force, dynamic systems would stop moving. At the same time dissipative forces such as internal friction and thermodynamic losses are taking away from the driving force. Together the opposing forces cancel any interruptions or initial conditions and cause the system to settle into typical behavior.
• 10.6: Root Locus Plots - Effect of Tuning
Root locus plots show the roots of the systems characteristic equation, (i.e. the Laplacian), as a function of the control variables such as Kc . By examining these graphs it is possible to determine the stability of different values of the control variable.
• 10.7: Routh Stability - Ranges of Parameter Values that are Stable
The stability of a process control system is extremely important to the overall control process. System stability serves as a key safety issue in most engineering processes. If a control system becomes unstable, it can lead to unsafe conditions. For example, instability in reaction processes or reactors can lead to runaway reactions, resulting in negative economic and environmental consequences.

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