Pavel Kuptsov

We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. The theoretical benefit from this approach is that the universal model admits applying common mathematical methods without needing to develop a unique theory for each particular dynamical equations. From the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Rцssler system and also the Hindmarch – Rose model. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. A high similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents.

Amplitude equations are obtained for a system of two coupled van der Pol oscillators that has been recently suggested as a simple system with hyperbolic chaotic attractor allowing physical realization. We demonstrate that an approximate model based on the amplitude equations preserves basic features of a hyperbolic dynamics of the initial system. For two coupled amplitude equations models having the hyperbolic attractors a transition to synchronous chaos is studied. Phenomena typically accompanying this transition, as riddling and bubbling, are shown to manifest themselves in a specific way and can be observed only in a small vicinity of a critical point. Also, a structure of many-dimensional attractor of the system is described in a region below the synchronization point.

While a previously proposed method for estimating inertial manifold dimension, based on explicitly computing angles between pairs of covariant Lyapunov vectors (CLVs), employs efficient algorithms, it remains computationally demanding due to its substantial resource requirements. In this work, we introduce an improved method to determine this dimension by analyzing the angles between tangent subspaces spanned by the CLVs. This approach builds upon a fast numerical technique for assessing chaotic dynamics hyperbolicity. Crucially, the proposed method requires significantly less computational effort and minimizes memory usage by eliminating the need for explicit CLV computation. We test our method on two canonical systems: the complex Ginzburg – Landau equation and a diffusively coupled chain of Lorenz oscillators. For the former, the results confirm the accuracy of the new approach by matching prior dimension estimates. For the latter, the analysis demonstrates the absence of a low-dimensional inertial manifold, highlighting a complex regime that merits further investigation. The presented method offers a practical and efficient tool for characterizing attractors in infinite-dimensional dynamical systems.