What is Cobb’s maximum likelihood approach?
An approach to fit the cusp catastrophe to data consisting of cross-sectional measurements. The basic idea is to make catastrophe theory, a deterministic theory, stochastic by adding a stochastic term, called Wiener noise (with variance omega^2)
Explain Anomalous variance (catastrophe flag)
occurs because near a bifurcation point the second derivative diminishes, meaning that the minimum becomes less deep
What is the problem with early warning signals (in psychological research)?
the problem with early warning signals is that both type 1 and type 2 errors should be low for predicting transitions. This is challenging even in simulations, let alone in noisy psychological data.
Explain divergence of linear response and critical slowing down (catastrophe flags)
Divergence of linear response is the size of the effect of a small perturbation of the state, which will be greater near a bifurcation point. It will also take longer to return to equilibrium. This delay in return time is known as critical slowing down
How do you test for multimodality in frequency distributions and what is the advantage to this over testing for a sudden jump?
Finite mixture models. Advantage: can be done on cross-sectional data
What are the early warning signs (in the context of catastrophe flags)?
anomalous variance, divergence of linear response, and critical slowing down—are indicators that occur near the bifurcation lines.
What does Inaccessibility mean in the context of catastrophe flags and why is this relevant?
Certain values of the behavioral variable are unstable. Inaccessibility is relevant to reject the alternative hypothesis that the sudden jump and bimodality are due to an acceleration.
simple time series are not sufficient to distinguish accelerations from phase transitions. So how do we distinguish between the two processes?
Catastrophe flags: sudden jumps,multimodality (e.g bimodality), inaccessibility, divergence (e.g pitchfork?), hysteresis (+ the early warning signs)
What is the difference between mechanistic and phenomenological models?
In phenomenological models, we assume the presence of a cusp, and make hypotheses about the involved variables. In a mechanistic approach, the cusp is derived from basic assumptions or first principles
What are the normal and splitting variable in a rotated cusp model?
In such a rotation the normal variable is the difference between the 2 variables, and , while the splitting variable is the sum of the 2 variables.
What is the difference between a discontinuous jump and a continuous acceleration?
The main difference is that in the continuous case the intermediate values are stable. An acceleration can be understood as a quadratic minimum that changes its position quickly. If we stop the process by freezing the manipulated control variable in the process, the state will remain at an intermediate value. These intermediate values are all stable values. If we freeze the manipulated variable in a discontinuous process, it will continue to move to a stable state. In this case, the intermediate state is unstable.
What is hysteresis?
The delay in jumps dependent on the direction of the change in the normal variable (a)
What is the swallowtail catastrophe?
A higher-order catastrophe consisting of three surfaces of fold bifurcations meeting in two lines of cusp bifurcations, which in turn meet in a single swallowtail bifurcation point.
What is the cusp point?
The bifurcation lines meet at (0,0,0). At this point, the third derivative is also 0.
What can we see in the bifurcation diagram of the parameters a (assuming b is positive) and b, respectively?
a = hysteresis (jump + delay),
b = pitchfork bifurcation (bistability)
How is the cusp catastrophe expressed mathematically?
By setting the first derivative to 0: V’(X) = -a -bX + X^3 = 0. This type of equation is called a cubic equation
What 2 axes does the cusp catastrophe combine?
hysteresis along the normal axis (a) and the pitchfork (b) along the splitting axis
What is a degenerate critical point?
Both the first and second derivative is 0. Phase transitions can occur at these bifurcation points.
What is an inflection point?
A degenerate critical point (im guessing one of several types but idk, this book doesnt really explain the terms used)
What are critical points?
points where the first derivative of the potential function is 0. (minima and maxima, + degenerate critical points? idk im just putting together things they say in different paragraphs)
What is the difference between first and second-order phase transitions?
First order transitions are discontinuous
What kind of systems do catastrophe theory concern, and which points in these systems do they study?
Catastrophe theory is concerned with gradient systems. These are dynamic systems that can be described by a potential function. Catastrophe theory analyzes so-called degenerate critical points of the potential function. Phase transitions can occur at these bifurcation points.
What is the relationship between bifurcation and catastrophe
Catastrophe theory can be viewed as a branch of bifurcation theory, describing a subclass of bifurcations.
How can we know if a critical point is a minimum or a maximum?
The second-derivative test. If second derivative (V’’(X)) is positive, it is minimum.
