A topological framework for identifying phenomenological bifurcations in stochastic dynamical systems

Abstract

Changes in the parameters of dynamical systems can cause the state of the system to shift between different qualitative regimes. These shifts, known as bifurcations, are critical to study as they can indicate when the system is about to undergo harmful changes in its behavior. In stochastic dynamical systems, there is particular interest in P-type (phenomenological) bifurcations, which can include transitions from a monostable state to multi-stable states, the appearance of stochastic limit cycles and other features in the probability density function (PDF) of the system’s state. Current practices are limited to systems with small state spaces, cannot detect all possible behaviors of the PDFs and mandate human intervention for visually identifying the change in the PDF. In contrast, this study presents a new approach based on Topological Data Analysis that uses superlevel persistence to mathematically quantify P-type bifurcations in stochastic systems through a “homological bifurcation plot”—which shows the changing ranks of 0th and 1st homology groups, through Betti vectors. Using these plots, we demonstrate the successful detection of P-bifurcations on the stochastic Duffing, Raleigh-Vander Pol and Quintic Oscillators given their analytical PDFs, and elaborate on how to generate an estimated homological bifurcation plot given a kernel density estimate (KDE) of these systems by employing a tool for finding topological consistency between PDFs and KDEs.

Appendix A

Consider the function

$$\begin{aligned} \begin{aligned} p(x_{1}, \cdots , x_{n}) = Ce^{-\sum _{i = 1}^{n} (x_{i}^2 - h)^2}, \end{aligned} \end{aligned}$$

where h is the bifurcation parameter between [0, 1], and C is the normalizing constant taken as 1 for the sake of simplicity since the topology would not change. The function is nonnegative, even and permutation-invariant. Given its Gaussian form, it will have peaks, and a bifurcation can be envisioned by estimating the number of maximas as the parameter h is varied.

The first derivative of the function is

$$\begin{aligned} \begin{aligned} p_{x_k} = -4e^{-\sum _{i = 1}^{n} (x_{i}^2 - h)^2}(x_{k}^2 - h)x_{k}. \end{aligned} \end{aligned}$$

The critical points of p exist where the n-dimensional gradient vector is all zero. Due to the permutation-invariance, only one such calculation needs to be made:

$$\begin{aligned} \begin{aligned} (x^2_k-h)x_k = 0 \implies x_k = 0, \pm \sqrt{h}. \end{aligned} \end{aligned}$$

Hence, the critical points are a combination of 0 and $\pm \sqrt{h}$ valued coordinates. The definiteness of the Hessian matrix

$$\begin{aligned} H = \begin{pmatrix} {\partial ^2 p}/{\partial x_1^2} &{} \cdots &{} {\partial ^2 p}/{\partial x_lx_n}\\ \vdots &{} \ddots &{} \vdots \\ {\partial ^2 p}/{\partial x_nx_l} &{} \cdots &{} {\partial ^2 p}/{\partial x_n^2} \end{pmatrix} \end{aligned}$$

at each critical point can reveal the point’s nature about whether it’s a maxima (negative-definite Hessian), a minima (positive-definite Hessian), a saddle (indefinite Hessian) or the test is inconclusive (semi-definite Hessian).

For the Hessian,

$$\begin{aligned} \begin{aligned} p_{x_kx_k} = 16e^{-\sum _{i = 1}^{n} (x_{i}^2 - h)^2}(x_{k}^2 - h)^2x^2_{k} \\ -4e^{-\sum _{i = 1}^{n} (x_{i}^2 - h)^2}(3x_{k}^2 - h), \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} p_{x_kx_l}&= 16e^{-\sum _{i = 1}^{n} (x_{i}^2 - h)^2}(x_{k}^2 - h)(x_{l}^2 - h)x_{k}x_{l} \\&= p_{x_lx_k}. \end{aligned} \end{aligned}$$

For values 0 and $\pm \sqrt{h}$ of $x_k$ and $x_l$, Table 1 shows that at each critical point, the Hessian is diagonal, and the eigenvalues are equal to the diagonal values. Hence, the definiteness only depends on the signs of $p_{x_kx_k}$. The following conclusions can be made

• 1. Hessian will have a combination of eigenvalues with a combination of 0 and $\pm \sqrt{h}$ for coordinates. Such critical points are neither maximas nor minimas.

• 2. Hessian will have all positive eigenvalues for all coordinates equal to 0. Therefore, in cases when $h \ne 0$, the origin is a minima.

• 3. Hessian will have all negative eigenvalues for all coordinates equal to $\pm \sqrt{h}$. This implies that there are $2^n$ maximas for an n-dimensional system when $h \ne 0$ and 1 maxima at origin when $h = 0$.

Table 1 Values for $p_{x_kx_k}$ and $p_{x_kx_l}$ for different values of critical points

Consider $n = 2$. At $h > 0$, $2^2 = 4$ peaks are expected. That is, the system bifurcates from a monostable state to a quadstable state. Figure 16 corroborates the proof above with the function having 1 peak at $h = 0$ and 4 peaks at $h = 1$.

Fig. 16

Monostable and quadstable PDFs for $h=0$ and $h=1$ for $n = 2$