Piecewise linear recurrent neural networks (PLRNNs) form the basis of many successful machine learning applications for time series prediction and dynamical systems identification, but rigorous mathematical analysis of their dynamics and properties is lagging behind. Here, we contribute to this topic by investigating the existence of n-cycles (n ≥ 3) and bordercollision bifurcations in a class of m-dimensional piecewise linear continuous maps which have the general form of a PLRNN. This is particularly important as for one-dimensional maps the existence of 3-cycles implies chaos. It is shown that these n-cycles collide with the switching boundary in a border-collision bifurcation, and parametric regions for the existence of both stable and unstable n-cycles and border-collision bifurcations will be derived theoretically. We then discuss how our results can be extended and applied to PLRNNs. Finally, numerical simulations demonstrate the implementation of our results and are found to be in good agreement with the theoretical derivations. Our findings thus provide a basis for understanding periodic
Abstract. In this paper we study a four dimensional tourism-based social-ecological dynamical system. In fact we analyse tourism profitability, compatibility and sustainability by using bifurcation theory in terms of structural properties of attractors of system. For this purpose first we transformed it into a three dimensional system such that the reduced system is the extended and modified model of the previous three dimensional models suggested for tourism with the same dimension. Then we investigate transcritical, pitchfork and saddle-node bifurcation points of system. And numerically by finding some branches of stable equilibria for system show the profitability of tourism industry. Then by determining the Hopf bifurcation points of system we find a family of stable attractors for that by numerical techniques. Finally we conclude the existence of these stable limit cycles implies profitability and compatibility and then the sustainability of tourism.
In this paper we study a four-dimensional tourism-based social-ecological system including two different types of tourists, i.e., eco-tourists and mass-tourists. First we develop the mathematical model of this system, in order to make it more realistic. In fact, the negative effects of eco-tourists on the environment can not be always ignored. Therefore, we consider a discontinuous function for describing the damage induced by tourist activities to ecosystem quality. By introducing a discontinuous model in this way, we can provide a tool for better investigating the behavior of tourism phenomenon. We analyse the dynamic of the obtained discontinuous system, by using the theory of discontinuous dynamical systems. Moreover, in order to discuss about profitability, compatibility and sustainability of the obtained system, stability regions for that will be found. Furthermore by verifying tangent points of the discontinuous system, more dynamic features of eco and mas-tourists will be shown. In this regard, we determine some regions for existence of the collision of two tangent points for the system. Some numerical simulations are carried out to demonstrate our theoretical results. Here our numerical and theoretical achievements can provide useful information for analysing the tourism industry.
Recurrent neural networks (RNNs) are wide-spread machine learning tools for modeling sequential and time series data. They are notoriously hard to train because their loss gradients backpropagated in time tend to saturate or diverge during training. This is known as the exploding and vanishing gradient problem. Previous solutions to this issue either built on rather complicated, purpose-engineered architectures with gated memory buffers, or -more recently -imposed constraints that ensure convergence to a fixed point or restrict (the eigenspectrum of) the recurrence matrix. Such constraints, however, convey severe limitations on the expressivity of the RNN. Essential intrinsic dynamics such as multistability or chaos are disabled. This is inherently at disaccord with the chaotic nature of many, if not most, time series encountered in nature and society. Here we offer a comprehensive theoretical treatment of this problem by relating the loss gradients during RNN training to the Lyapunov spectrum of RNN-generated orbits. We mathematically prove that RNNs producing stable equilibrium or cyclic behavior have bounded gradients, whereas the gradients of RNNs with chaotic dynamics always diverge. Based on these analyses and insights, we offer an effective yet simple training technique for chaotic data and guidance on how to choose relevant hyperparameters according to the Lyapunov spectrum.
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