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Dynamical systems that evolve continuously over time are ubiquitous throughout science and engineering. Machine learning (ML) provides data-driven approaches to model and predict the dynamics of such systems. A core issue with this approach is that ML models are typically trained on discrete data, using ML methodologies that are not aware of underlying continuity properties. This results in models that often do not capture any underlying continuous dynamics—either of the system of interest, or indeed of any related system. To address this challenge, we develop a convergence test based on numerical analysis theory. Our test verifies whether a model has learned a function that accurately approximates an underlying continuous dynamics. Models that fail this test fail to capture relevant dynamics, rendering them of limited utility for many scientific prediction tasks; while models that pass this test enable both better interpolation and better extrapolation in multiple ways. Our results illustrate how principled numerical analysis methods can be coupled with existing ML training/testing methodologies to validate models for science and engineering applications. Many challenging problems in science and engineering rely on the study of dynamical systems that evolve continuously in time, and yet this feature proves difficult to be captured reliably using modern machine learning (ML) models. This paper develops a convergence test based on numerical analysis and illustrates how this methodology can be combined with existing ML techniques to validate models for science and engineering applications. [ABSTRACT FROM AUTHOR]

In terms of the first- and second-order sensitivities of the complex mode for a symmetric damped linear discrete dynamic system with respect to design parameters, the algebraic method, direct method, full-mode method and truncated modal method corresponding to full-mode method are proposed. These four algorithms insist on using the expression forms in N-dimensional vector space, which provide the convenience in engineering application. In the numerical examples, the correctness and validity of all the proposed algorithms are demonstrated for the single-frequency, closed-frequency and multiple-frequency systems. The algebraic method, direct method and full-mode method are accurate algorithms. No statement is made that one method is absolutely superior to the other, but this paper provides the chance to compare their efficiency and accuracies of all the three accurate algorithms. Truncated modal method proposed here is an approximate algorithm. The truncation criterion is simple and efficient, which is a more competitive alternative to the full-mode method, and it is shown that the truncated modal method has high precision and good convergence speed for sensitivity analysis by numerical examples. [ABSTRACT FROM AUTHOR]

The present paper addresses the dynamical motion of two degrees-of-freedom (DOF) auto-parametric system consisting of a connected rolling cylinder with a damped spring. This motion has been considered under the action of an excitation force. Lagrange's equations from second kind are utilized to obtain the governing system of motion. The uniform approximate solutions of this system are acquired up to higher order of approximation using the technique of multiple scales in view of the abolition of emerging secular terms. All resonance cases are characterized, and the primary and internal resonances are examined simultaneously to set up the corresponding modulation equations and the solvability conditions. The time histories of the amplitudes, modified phases, and the obtained solutions are graphed to illustrate the system's motion at any given time. The nonlinear stability approach of Routh–Hurwitz is used to examine the stability of the system, and the different zones of stability and instability are drawn and discussed. The characteristics of the nonlinear amplitude for the modulation equations are investigated and described, as well as their stabilities. The gained results can be considered novel and original, where the methodology was applied to a specific dynamical system. [ABSTRACT FROM AUTHOR]

In this paper, the Rayleigh beam system with two dynamical boundary controls is treated. Theoretically, the well-posedness of the weak solution is obtained. Later, we discretize the system by using the Implicit Euler scheme in time and the P 3 Hermite finite element in space. In addition, we show the decay of the discrete energy and we establish some a priori error estimates. Finally, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

In this paper we investigate the global dynamics in an HIV virus infection model with saturated incidence. The model includes two viral strains, one is wild-type (i.e. drug sensitive) and another is drug-resistant. The wild-type strain can mutate and become drug-resistant during the process of reverse transcription. The nonnegativity and boundedness of solutions are established. The basic reproduction numbers of two strains and the existence of equilibria are also obtained. The threshold criteria on the local and global stability of equilibria and the uniform persistence of the model are established by using the linearization method, constructing suitable Lyapunov functions and the theory of persistence in dynamical systems. Moreover, the mathematical analysis and numerical examples show that model may have a positive equilibrium which is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper, a novel four-dimensional continuous chaotic system is constructed from the simplified Lorenz-like system. The novel system has three equilibria, strange attractors, coexisting attractors, and performs Hopf bifurcation with the variation of system parameters. The coexisting attractors, which are the most remarkable dynamic features of the system, are numerically studied. The coexisting attractors show that the system coexists as a pair of point, periodic, and chaotic attractors. Some basic dynamic behaviours are studied as well. The synchronisation control problem of the system is analysed. The theoretical and numerical analyses demonstrate that the system can easily achieve synchronisation by using the passive control technique. [ABSTRACT FROM AUTHOR]

Yaw damper, widely employed in high-speed railway vehicles, has played an important role in improving the hunting stability. This paper mainly makes a comprehensive analysis on the effect of yaw damper with its series stiffness value on the stability and bifurcation type of the railway bogie. With the prerequisite of the linear critical speed calculated by mathematical method, Center Manifold Theorem is adopted to reduce the dimension of the model to a planar dynamical system. And the symbolic expression associated with yaw damper and its series stiffness to determine bifurcation type at the critical speed is obtained by the method of Normal Form. As a result, the influence of the variation tendency of the yaw damper and series stiffness on the bifurcation type of the bogie is given qualitatively in contrast to different couples. Finally, numerical analysis of corresponding bifurcation diagrams is given to verify the accuracy of the conclusion. [ABSTRACT FROM AUTHOR]

This paper is concerned with the dissipativity of Runge-Kutta methods for a class of nonlinear functional-integro-differential equations (FIDEs). The dissipativity results of Runge-Kutta methods for the FIDEs are given. It is shown under a suitable condition that an algebraically stable Runge-Kutta method is dissipative when applied to the FIDEs. Numerical examples are given to illustrate the correctness of our theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents the results of the numerical and analytical study of natural and quasinatural bending-gravitational oscillations of a circular elastic ice plate floating on the fluid surface and frozen onto a cylindrical vertical support. In the framework of the shallow water theory for bounded and unbounded reservoirs, the dependence of natural and quasinatural frequencies of geometrical parameters of the oscillation region is studied. [ABSTRACT FROM AUTHOR]

The present paper solves numerically the problem of vibrations of infinite structures under a moving load. A velocity formulation of the space-time finite element method was applied. In the case of simplex shaped space-time finite elements, the 'steady state' dynamic behaviour of the system was obtained. A properly performed discretization allowed of propagating information in a given direction at a limited velocity. The solutions were obtained under the assumption that the deformation is quasi-stationary, i.e., stationary in the coordinate system that moves with the load. The unbounded Timoshenko beam subjected to a distributed moving load was used as a test example. The dynamical system is placed on an elastic foundation. The matrices describing an infinite dynamical system subjected to a moving load are derived and the stability of the numerical scheme is analysed. The numerical results are compared with the analytical solutions in the literature and the classical numerical method. [ABSTRACT FROM AUTHOR]