In real-world networks the interactions between network elements are inherently time-delayed. These timedelays can not only slow the network but can have a destabilizing effect on the network's dynamics leading to poor performance. The same is true in computational networks used for machine learning etc. where time-delays increase the network's memory but can degrade the network's ability to be trained. However, not all networks can be destabilized by time-delays. Previously, it has been shown that if a network or high-dimensional dynamical system is intrinsically stabile, which is a stronger form of the standard notion of global stability, then it maintains its stability when constant time-delays are introduced into the system. Here we show that intrinsically stable systems, including intrinsically stable networks and a broad class of switched systems, i.e. systems whose mapping is time-dependent, remain stable in the presence of any type of time-varying time-delays whether these delays are periodic, stochastic, or otherwise. We apply these results to a number of well-studied systems to demonstrate that the notion of intrinsic stability is both computationally inexpensive, relative to other methods, and can be used to improve on some of the best known stability results. We also show that the asymptotic state of an intrinsically stable switched system is exponentially independent of the system's initial conditions.Here we refer to the emergent behavior of these interacting elements, which is the changing state of these network elements, as the network's dynamics. The network's dynamics can be periodic, as is found in many biological networks [2], synchronizing, which is the desired condition for transmitting power over large distance in power grids [3], and stable or multistable dynamics such as is found in gene-regulatory networks [4], etc.In real-world networks the interactions between network elements are inherently time-delayed. This comes from the fact that network elements are spatially separated, that information and other quantities can only be processed and transmitted at finite speeds, and that these quantities can be slowed by network traffic [5]. These time-delays not only slow the network, leading to poor performance, but can have a destabilizing effect on the network's dynamics, which can lead to network failure [6,7]. The same is true in computational networks used for machine learning etc. where time-delays can be used to increase the network's ability to detect long term temporal dependencies [8] but at the cost of potentially degrading the ability to train the network.Not all networks can be destabilized by time-delays. In [9] the notion of intrinsic stability was introduced, which is a stronger form of the standard notion of stability, i.e. a system in which there is a globally attracting fixed point (see [10] for more details). If a network is intrinsically stable the authors showed that constant-type time-delays, which are delays that do not vary in time, have no effect on the network's stabi...

In many natural and technological systems the rule that governs the system’s dynamics changes over time. In such switched systems the system’s switching can be a significant source of instability. Here we give a simple sufficient criterium to determine if an i.i.d. stochastically switched system is stable in expectation. This method extends recent results for linear switched systems to nonlinear switched systems. It also extends results known for general switched systems giving improved results for systems with i.i.d. stochastic switching. The paper also considers the effects of time-delays on the stability of switched systems. Such time delays, which are intrinsic to any real-world system, can also have a destabilising effect on the system’s dynamics. Previously, it has been shown that if a dynamical system is intrinsically stable, which is a stronger form of global stability, then it maintains its stability even when time-delays are introduced into the system. Here we extend this notion to stochastically switched systems. We refer to this type of stability as patient stability and give a simple sufficient criterium under which such systems are patiently stable, i.e. cannot be destabilised by time delays. Both criteria introduced in this paper side step the need to use Lyapunov, linear matrix inequalities, and semi-definite programming-type methods. Our examples in this paper demonstrate the simplicity of these criteria.