# Convergence Properties for Discrete-time Nonlinear Systems

arXiv: https://arxiv.org/abs/1612.05327 · HTML: https://mlsanity.com/paper/1612.05327/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Dec 2016.

## Built on by (most cited first, 10 of 10)

- [Learning Over Contracting and Lipschitz Closed-Loops for Partially-Observed Nonlinear Systems (Extended Version)](https://mlsanity.com/paper/2304.06193/) (2023)
- [Neural Contraction Metrics with Formal Guarantees for Discrete-Time Nonlinear Dynamical Systems](https://mlsanity.com/paper/2504.17102/) (2025)
- [Predictability and Fairness in Load Aggregation and Operations of Virtual Power Plants](https://mlsanity.com/paper/2110.03001/) (2021)
- [Nonlinear Autoregression with Convergent Dynamics on Novel Computational Platforms](https://mlsanity.com/paper/2108.08001/) (2021)
- [Improving the Performance of Echo State Networks Through State Feedback](https://mlsanity.com/paper/2312.15141/) (2023)
- [Echoes of the Past: A Unified Perspective on Fading memory and Echo States](https://mlsanity.com/paper/2508.19145/) (2025)
- [Bridging Control with Neural Network Verifier alpha-beta-CROWN: A Tutorial](https://mlsanity.com/paper/2605.26577/) (2026)
- [Learning Lipschitz Feedback Policies from Expert Demonstrations: Closed-Loop Guarantees, Generalization and Robustness](https://mlsanity.com/paper/2103.16629/) (2021)
- [Distributed Identification of Contracting and/or Monotone Network Dynamics](https://mlsanity.com/paper/2107.14309/) (2021)
- [A Solvable Molecular Switch Model for Stable Temporal Information Processing](https://mlsanity.com/paper/2508.15451/) (2025)

## Abstract (arXiv)

Three similar convergence notions are considered. Two of them are the long established notions of convergent dynamics and incremental stability. The other is the more recent notion of contraction analysis. All three convergence notions require that all solutions of a system converge to each other. In this paper, we investigate the differences between these convergence properties for discrete-time, time-varying nonlinear systems by comparing the properties in pairs and using examples. We also demonstrate a time-varying smooth Lyapunov function characterization for each of these convergence notions, and, with appropriate assumptions, we provide several sufficient conditions to establish relationships between these properties in terms of Lyapunov functions.