# 5: Discrete-Time Models II - Analysis

- 5.1 Finding Equilibrium Points
- When you analyze an autonomous, ﬁrst-order discrete-time dynamical system (a.k.a. iterative map) one of the ﬁrst things you should do is to ﬁnd its equilibrium points (also called ﬁxed points or steady states), i.e., states where the system can stay unchanged over time.

- 5.2 Phase Space Visualization of Continuous-State Discrete-Time Models
- Once you ﬁnd where the equilibrium points of the system are, the next natural step of analysis would be to draw the entire picture of its phase space (if the system is two or three dimensional).

- 5.3 Cobweb Plots for One-Dimensional Iterative Maps
- One possible way to solve the overcrowded phase space of a discrete-time system is to create two phase spaces, one for time t−1 and another for t, and then draw trajectories of the system’s state in a meta-phase space that is obtained by placing those two phase spaces orthogonally to each other. In this way, you would potentially disentangle the tangled trajectories to make them visually understandable.

- 5.4 Graph-Based Phase Space Visualization of Discrete-State Discrete-Time Model
- The cobweb plot approach discussed above works only for one-dimensional systems, because we can’t embed such plots for any higher dimensional systems in a 3-D physical space. However, this dimensional restriction vanishes if the system’s states are discrete and ﬁnite. For such a system, you can always enumerate all possible state transitions and create the entire phase space of the system as a state-transition graph, which can be visualized reasonably well even within a 2-D visualization space.

- 5.5: Variable Rescaling of Discrete-Time Models
- Variable rescaling is a technique to eliminate parameters from your model without losing generality. The basic idea is this: Variables that appear in your model represent quantities that are measured in some kind of units, but those units can be arbitrarily chosen without changing the dynamics of the system being modeled. This must be true for all scientiﬁc quantities that have physical dimensions—switching from inches to centimeters should not cause any change in how physics works!

- 5.6: Asymptotic Behavior of Discrete-Time Linear Dynamical Systems
- One of the main objectives of rule-based modeling is to make predictions of the future. So, it is a natural question to ask where the system will eventually go in the (inﬁnite) long run. This is called the asymptotic behavior of the system when time is taken to inﬁnity, which turns out to be fully predictable if the system is linear.

- 5.7 Linear Stability Analysis of Discrete-Time Nonlinear Dynamical Systems
- All of the discussions above about eigenvalues and eigenvectors are for linear dynamical systems. Can we apply the same methodology to study the asymptotic behavior of nonlinear systems? Unfortunately, the answer is a depressing no. Asymptotic behaviors of nonlinear systems can be very complex, and there is no general methodology to systematically analyze and predict them. We will revisit this issue later.