DQbee 0.1.1

DQode is a Python Library to compute the optimal inner-quadratic quadratization and dissipative quadratization of a given polynomial ODE system.

Dissipative-Quadratiation-Package

1. Introduction to the Problem

Definition 1 (Quadratization). Consider a system of ODEs $\mathbf{x}'=\mathbf{p}(\mathbf{x})$

$$ \begin{cases} x_1^{\prime}=f_1(\bar{x})\ \ldots \ x_n^{\prime}=f_n(\bar{x}) \end{cases} \tag{1} $$

where $\bar{x}=\left(x_1, \ldots, x_n\right)$ and $f_1, \ldots, f_n \in \mathbb{C}[\mathbf{x}]$. Then a list of new variables

$$ y_1=g_1(\bar{x}), \ldots, y_m=g_m(\bar{x}) $$

is said to be a quadratization of (1) if there exist polynomials $h_1, \ldots, h_{m+n} \in$ $\mathbb{C}[\bar{x}, \bar{y}]$ of degree at most two such that

• $x_i^{\prime}=h_i(\bar{x}, \bar{y})$ for every $1 \leqslant i \leqslant n$ which we define as $\mathbf{x}^{\prime}=\mathbf{q}_1(\mathbf{x}, \mathbf{y})$
• $y_j^{\prime}=h_{j+n}(\bar{x}, \bar{y})$ for every $1 \leqslant j \leqslant m$ which we define as $\mathbf{y}^{\prime}=\mathbf{q}_2(\mathbf{x}, \mathbf{y})$

Here we call the number $m$ as the order of quadratization. The optimal quadratization is the approach that produce the smallest possible order.

Definition 2 (Equilibrium). For a polynomial ODE system (1), a point $\mathbf{x}^* \in$ $\mathbb{R}^n$ is called an equilibrium if $\mathbf{p}\left(\mathbf{x}^*\right)=0$.

Definition 3 (Dissipativity). An ODE system (1) is called dissipative at an equilibrium point $\mathbf{x}^\ast$ if all the eigenvalues of the Jacobian $J(\mathbf{p})|_{\mathbf{x}=\mathbf{x}^\ast}$ of $\mathbf{p}$ and $\mathbf{x}^\ast$ have negative real part.

Definition 4 (Dissipative quadratization). Assume that a system (1) is dissipative at an equilibrium point $\mathbf{x}^* \in \mathbb{R}^n$. Then a quadratization given by $\mathbf{g}, \mathbf{q}_1$ and $\mathbf{q}_2$ is called dissipative at $\mathbf{x}^*$ if the system

$$ \mathbf{x}^{\prime}=\mathbf{q}_1(\mathbf{x}, \mathbf{y}), \quad \mathbf{y}^{\prime}=\mathbf{q}_2(\mathbf{x}, \mathbf{y}) $$

is dissipative at a point $\left(\mathbf{x}^\ast, \mathbf{g}\left(\mathbf{x}^\ast\right)\right)$.

2. Package Description

This python package is mainly used to compute the inner-quadratic quadratization and dissipative quadratization of multivariate high-dimensional polynomial ODE system. This algorithm also has tools to perform reachability analysis of the system. Here are some of the specific features of this package:

• Compute the inner-quadratic quadratization of polynomial ODE system.
• Compute the dissipative quadratization of polynomial ODE system at origin.
• Compute the dissipative quadratization of polynomial ODE system at an equilibrium point (input by the user), or at multiple equilibrium points (input by the user) with method numpy, sympy, or Routh-Huritwz criterion.
• Compute the weakly nonlinearity bound $\left| \mathcal{X_{0}} \right|$ of dissipative system.

3. Package Usage

We will demonstrate usage of Package on the example below. Other interactive examples you can find more in our example notebook.

3.1. Importing the Package