CSYS5040 Criticality in Dynamical Systems Assignment 2
Due Date: This is assignment is due in TurnItIn by Sunday at the end of week 7. This assignment is worth 25% of your final mark.
You must do all of your working in a Mathematica notebook that I can run (no pdfs of Mathematica notebooks).
The * for some questions indicate the relative difficulty of the question. This is an individual assessment; your answers must reflect your own work. Marks will be based on the correctness of each answer, the effort put into exploring each question, and the originality of the examples you choose to look at. You are strongly encouraged to read beyond the class material to get a higher grade.
Question 1 (7.5%): The dynamics of a stochastic differential equation
a. Choose the constant valued parameters (i.e. μ, x! and E ) of a linear stochastic
differential equation of the form. dx = μ dt + E dw where E is the strength of the stochastic (noise) term, x! is the initial starting point and then plot the time series of the solution, making sure μ is not equal to zero (it can be positive or negative though). Choose some numerical value b (for ‘boundary’) that has the same sign as μ, run some simulations of your solution, does your solution ever cross the boundary b and if so at what value of t does your solution cross it? Without simulating your solution, how can you know when to expect the solution to cross the boundary?
b. * Repeat part a. except that the stochastic differential equation is now non-linear, i.e. dx = (αx + μ) dt + E dw or uses even higher order terms in x, e.g. x 2 or x 3 such as dx = (βx2 + αx + μ) dt + E dw you will score higher the more sophisticated your model is in this part and consequently for part 1 c. below. Do not implement the same equations here that are needed for Question 2.a.
c. Write a single paragraph on one application of the methods you used in parts a. and
b. For example you can look up: “drift diffusion” and “neural network”, “two alternative forced choice task” (e.g. here:https://tinyurl.com/yygku3oa), “Ornstein- Uhlenbeck process”, or see here: https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_process
Question 2 (7.5%): Plotting non-linear functions for a non-linear map
a. From the article we looked at in Week 4 I want you to implement one of the non-
linear stochastic neural models. Do not implement the same equations you used for 1.b. For this first step all you have to do is replicate the work I showed you in class but using either 1 or 2 (depending on which model you chose to implement) stochastic non-linear equations that you will you find in the articles listed below where a decision is reached once a “decision variable” crosses a boundary threshold. This question is not intended to be difficult to understand but it can still be tricky to implement, just modify the code I’ve already given you in class to reflect the non-linear model you’re implementing. For the different equations for each system see pages 705 to 707 of the article here
https://sites.engineering.ucsb.edu/~moehlis/moehlis_papers/psych.pdf
Or Iook at the different equations Iisted on the Wikipedia page under “Other ModeIs” here:https://en.wikipedia.org/wiki/Two–aIternative_forced_choice
b. * Find an articIe (using GoogIe SchoIar etc.) that has used the modeI you
impIemented in part 2a. and iIIustrate some aspect of the resuIts from that articIe using the modeI you’ve just impIemented.
c. Discuss the impact of the modeI you’ve used in the context of the articIe you’ve
found, e.g. you might discuss why this stochastic modeI was used rather than some other modeI, or what the parameters mean in a practicaI setting, or what new interpretation the modeI has provided in the area of study etc.
Question 3 (5% + 5%): Parameters in non–Iinear dynamicaI systems
a. **Based on your answer to Question 1 for the non-linear system, write a
Mathematica function that shows what happens when a parameter vaIue, e.g. 阝 or α changes and the system switches from one equiIibrium state to another. See the Mathematica notebook from Week 1, at the end of the notebook there is a stochastic diffusion processes with more than one stationary state, we didn’t discuss this modeI but I want you to base your answer on the ideas outIined there. Note that the stationary state the system was attracted to (settIed in) depended on the initiaI starting point x。. For this question, using a system with more than one stationary state, I want you to Iet the system find its stationary state (i.e. it is in equiIibrium) and then change the parameter vaIues so that the system switches from one equiIibrium point to another by going through a tipping point. The minimum you need to do to pass this question is to pIot the time series of the system passing through this tipping point.
b. ** Process Design and Methodology
You are given an unknown univariate time series that is suspected to contain both deterministic nonIinear dynamics and stochastic components. Design a comprehensive anaIyticaI framework that wouId aIIow you to:
1. Identify the presence and nature of any nonIinear dynamics
2. Distinguish between deterministic part and stochastic noise
3. Detect potentiaI bifurcation points or regime changes
4. Characterize the underIying dynamicaI structure
Your answer shouId outIine the sequentiaI steps you wouId take, justify the choice of each anaIyticaI method, and expIain what each step wouId reveaI about the system. Consider methods such as (but not Iimited to): phase space reconstruction, correIation dimension anaIysis, Lyapunov exponent estimation, recurrence anaIysis, and tests for nonIinearity vs. stochastic processes.