With the advent of deep learning over the last decade, a considerable amount of effort has gone into better understanding and enhancing Stochastic Gradient Descent so as to improve the performance and stability of artificial neural network training. Active research fields in this area include exploiting second order information of the loss landscape and improving the understanding of chaotic dynamics in optimization. This paper exploits the theoretical connection between the curvature of the loss landscape and chaotic dynamics in neural network training to propose a modified SGD ensuring non-chaotic training dynamics to study the importance thereof in NN training. Building on this, we present empirical evidence suggesting that the negative eigenspectrum - and thus directions of local chaos - cannot be removed from SGD without hurting training performance. Extending our empirical analysis to long-term chaos dynamics, we challenge the widespread understanding of convergence against a confined region in parameter space. Our results show that although chaotic network behavior is mostly confined to the initial training phase, models perturbed upon initialization do diverge at a slow pace even after reaching top training performance, and that their divergence can be modelled through a composition of a random walk and a linear divergence. The tools and insights developed as part of our work contribute to improving the understanding of neural network training dynamics and provide a basis for future improvements of optimization methods.