Bounds on Positive Invariant Sets

We consider autonomous systems of the form \[ \dot{x}=f(x) \] with a vector field $f:\mathbb{R}^n\to\mathbb{R}^n$. Furthermore, consider a continuously differentiable Lyapunov-type function $V:\mathbb{R}^n\to[0,\infty)$ with a constant $\gamma>0$ as \[ \limsup_{t\to\infty} V(x(t))\leq\gamma. \] For polynomial systems, the bound $\gamma$ could be computed by \[ \forall x\in\mathbb{R}^n:\quad V(x)>\gamma\;\implies\;\dot{V}(x)<0 \] or \[ \exists \alpha>0\; \forall x\in\mathbb{R}^n:\quad \dot{V}(x)\leq -\alpha\cdot (V(x)-\gamma) \] with quatifier elimination.

This approach is illustrated on the Lorenz system [1], the Lorenz-Haken system [2], the Lorenz family [3] as well as the Duffing system and the nonlinear pendulum [4]. In addition we considered the Duffing-Ueada system [5]. The source code used in [1] is available on Github.



  1. Röbenack, K.; Voßwinkel, R.; Richter, H.: Automatic generation of bounds for polynomial systems with application to the Lorenz system.
    Chaos, Solitons & Fractals 113 (2018): 25-30.
  2. Röbenack, K.; Voßwinkel, R; Richter, H.: Calculating positive invariant sets: A quantifier elimination approach.
    Journal of Computational and Nonlinear Dynamics 14.7 (2019): 074502.
  3. Röbenack, K.: Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination.
    PAMM 20.1 (2021): e202000161.
  4. Röbenack, K.; Gerbet, D.: Computation of Positively Invariant Sets of the Duffing System and the Nonlinear Pendulum under Bounded Excitation.
    2020 7th International Conference on Control, Decision and Information Technologies (CoDIT). Vol. 1. IEEE, 2020.
  5. Natkowski, L., Gerbet, D., & Röbenack, K.: On the Systematic Construction of Lyapunov Functions for Polynomial Systems. PAMM, 23(1) 2023, e202200197.