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7: Continuous-Time Models II - Analysis
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- 7.1: Finding Equilibrium Points
- Finding equilibrium points of a continuous-time model dx/dt = G(x) can be done in the same way as for a discrete-time model, i.e., by replacing all x’s with xeq’s (again, note that these could be vectors). This actually makes the left hand side zero, because xeq is no longer a dynamical variable but just a static constant. Therefore, things come down to just solving the following equation
- 7.2: Phase Space Visualization
- A phase space of a continuous-time model, once time is discretized, can be visualized in the exact same way as it was in Chapter 5, using Codes 5.1 or 5.2. This is perfectly ﬁne. In the meantime, Python’s matplotlib has a specialized function called streamplot, which is precisely designed for drawing phase spaces of continuous-time models.
- 7.3: Variable Rescaling of Continuous-Time Models
- Variable rescaling of continuous-time models has one distinct difference from that of discrete-time models. That is, you get one more variable you can rescale: time. This may allow you to eliminate one more parameter from your model compared to discretetime cases.
- 7.4: Asymptotic Behavior of Continuous-Time Linear Dynamical Systems
- A general formula for continuous-time linear dynamical systems is given by dx /dt = Ax, where x is the state vector of the system and A is the coefﬁcient matrix. As discussed before, you could add a constant vector a to the right hand side, but it can always be converted into a constant-free form by increasing the dimensions of the system, as follows:
- 7.5: Linear Stability Analysis of Nonlinear Dynamical Systems
- Finally, we can apply linear stability analysis to continuous-time nonlinear dynamical systems.