Nonlinear Joint Spectral Radius

Abstract

We introduce a nonlinear extension of the joint spectral radius (JSR) for switched discrete-time dynamical systems. The classical JSR characterizes the exponential growth rate of linear switched systems, but its extension to nonlinear systems has remained an open problem. Our approach builds on the framework of nonlinear Perron-Frobenius theory to define a meaningful notion of joint spectral radius for nonlinear switched systems. We establish fundamental properties of this nonlinear JSR, including its relationship to the stability of switched nonlinear systems, and provide computational methods for its approximation. The theoretical framework is illustrated through several examples that demonstrate the utility of our approach for analyzing complex nonlinear switching dynamics.

Please cite this work as:

@article{deidda2025nonlinear,
  title={Nonlinear Joint Spectral Radius},
  author={Deidda, Piero and Guglielmi, Nicola and Tudisco, Francesco},
  journal={arXiv preprint arXiv:2507.11314},
  year={2025}
}

Keywords: Spectral radius Nonlinear dynamics Switched systems Dynamical systems Deep learning neural networks Perron Frobenius Theory