Periodic Dynamical Systems Research Articles

Exponential convergence of the weighted Birkhoff average

In this paper, we consider the polynomial and exponential convergence rates of the weighted Birkhoff averages of irrational rotations on tori. It is shown that these can be achieved for finite and infinite dimensional tori which correspond to the quasiperiodic and almost periodic dynamical systems respectively, under certain balance between the nonresonant condition and the decay rate of the Fourier coefficients. Diophantine rotations with finite and infinite dimensions are provided as examples. For the first time, we prove the universality of exponential convergence and arbitrary polynomial convergence in the quasiperiodic case and almost periodic case under analyticity respectively.

InVAErt networks: A data-driven framework for model synthesis and identifiability analysis

Use of generative models and deep learning for physics-based systems is currently dominated by the task of emulation. However, the remarkable flexibility offered by data-driven architectures would suggest to extend this representation to other aspects of system analysis including model inversion and identifiability. We introduce InVAErt (pronounced invert) networks, a comprehensive framework for data-driven analysis and synthesis of parametric physical systems which uses a deterministic encoder and decoder to represent the forward and inverse solution maps, a normalizing flow to capture the probabilistic distribution of system outputs, and a variational encoder designed to learn a compact latent representation for the lack of bijectivity between inputs and outputs. We formally analyze how changes in the penalty coefficients affect the stationarity condition of the loss function, the phenomenon of posterior collapse, and propose strategies for latent space sampling, since we find that all these aspects significantly affect both training and testing performance. We verify our framework through extensive numerical examples, including simple linear, nonlinear, and periodic maps, dynamical systems, and spatio-temporal PDEs.

New Stability Results for Periodic Solutions of Generalized Van der Pol Oscillator via Second Bogolyubov’s Theorem

A certain class of nonlinear differential equations representing a generalized Van der Pol oscillator is proposed in which we study the behavior of the existing solution. After using the appropriate variables, the first Levinson’s change converts the equations into a system with two equations, and the second converts these systems into a Lipschitzian system. Our main result is obtained by applying the Second Bogolubov’s Theorem. We established some integrals, which are used to compute the average function of this system and arrive at a new general condition for the existence of an asymptotically stable unique periodic solution. One of the well-known results regarding asymptotic stability appears, owing to the Second Bogolubov’s Theorem, and the advantage of this method is that it can be applied not only in the periodic dynamical systems, but also in non-almost periodic dynamical systems.

Stochastic paleoclimatology: Modeling the EPICA ice core climate records.

We analyze and model the stochastic behavior of paleoclimate time series and assess the implications for the coupling of climate variables during the Pleistocene glacial cycles. We examine 800 kiloyears of carbon dioxide, methane, nitrous oxide, and temperature proxy data from the European Project for Ice Coring in Antarctica (EPICA) Dome-C ice core, which are characterized by 100 ky glacial cycles overlain by fluctuations across a wide range of timescales. We quantify this behavior through multifractal time-weighted detrended fluctuation analysis, which distinguishes near-red-noise and white-noise behavior below and above the 100 ky glacial cycle, respectively, in all records. This allows us to model each time series as a one-dimensional periodic nonautonomous stochastic dynamical system, and assess the stability of physical processes and the fidelity of model-simulated time series. We extend this approach to a four-variable model with intervariable coupling terms, which we interpret in terms of possible interrelationships among the four time series. Within the framework of our coupling coefficients, we find that carbon dioxide and temperature act to stabilize each other and methane and nitrous oxide, whereas the latter two destabilize each other and carbon dioxide and temperature. We also compute the response function for each pair of variables to assess the model performance by comparison to the data and confirm the model predictions regarding stability amongst variables. Taken together, our results are consistent with glacial pacing dominated by carbon dioxide and temperature that is modulated by terrestrial biosphere feedbacks associated with methane and nitrous oxide emissions.

The vicinity of Earth–Moon [formula omitted] and [formula omitted] in the Hill restricted 4-body problem

The Hill restricted 4-body problem (HR4BP) is a periodic time-dependent Hamiltonian dynamical system that is both a generalization of the circular restricted 3-body problem (CR3BP) and Hill’s problem. In this work, the HR4BP is used to model the motion of an infinitesimal particle under the mutual gravitation of the Sun, Earth, and Moon. In the HR4BP, the equilibria L1 and L2 of the CR3BP are replaced by π-periodic orbits, the so-called EM L1,2 dynamical equivalents. We compute center manifold reductions about EM L1,2 to obtain a local overview of the dynamics in their vicinity. In this way, we show the existence of families of 2- and 3-dimensional quasi-periodic invariant tori around EM L1,2. Additionally, we exploit the first integral constructed in the center manifold reduction to compute hyperbolic invariant manifolds. We draw comparisons to the bicircular and quasi-bicircular restricted 4-body problems.

On some dense sets in the space of dynamical systems

Abstract The natural consequence of the existence of different kinds of chaos is the study of their mutual dependence and the relationship between these concepts and the entropy of systems. This observation also applies to the local approach to this issue. In this article, we will focus on this problem in the context of “points focusing chaos.” We aim to show their mutual independence by considering the sets of appropriate periodic dynamical systems in the space of discrete dynamical systems.

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U(1)-action. When the limiting rotation is non-resonant, these maps admit formal U(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbed U(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

On Bochner's almost-periodicity criterion

We give an extension of Bochner's criterion for the almost periodic functions. By using our main result, we extend two results of A. Haraux. The first is a generalization of Bochner's criterion which is useful for periodic dynamical systems. The second is a characterization of periodic functions in term of Bochner's criterion.

In-process impulse response of milling to identify stability properties by signal processing

In this work we present a quantitative measurement-based method to describe the stability behaviour of a periodic dynamical system. In-process response for impulse during milling is analysed to provide operational stability prediction. The proposed method combines chatter detection techniques with the theory of time-periodic delay differential equations applied for general milling operations. Signal processing based on dynamic modal decomposition approach not only presents qualitative properties (like stability/instability) but also quantifies a measure of stability through the determination of the Floquet multipliers. The corresponding monodromy operator of the milling process is approximated from the measured system’s response during machining operation. By changing the technological parameters, the variation of the modulus of the Floquet multipliers can be monitored. The stability limit can precisely be interpolated with the unitary multiplier; furthermore, the stability limit can be extrapolated while the manufacturing parameters remain in the chatter-free region. The presented approach is validated by numerical results and laboratory tests.

Almost periodic fuzzy multidimensional dynamic systems and applications on time scales

In this paper, we introduce a new division of fuzzy vectors depending on a determinant algorithm and develop a theory of almost periodic fuzzy multidimensional dynamic systems on time scales. Moreover, several applications are provided. In particular, a new type of fuzzy dynamic systems called fuzzy q-dynamic systems (i.e., fuzzy quantum dynamic systems) is proposed and studied. Our results are not only effective on classical fuzzy dynamic systems including their continuous and discrete situations (i.e., T=Z or R) but are also valid for other fuzzy multidimensional dynamic systems on various hybrid domains like qZ¯,±N12, ∪k=1+∞[2k,2k+1] etc.

On the structure of the Levinson center for monotone non-autonomous dynamical systems with a first integral

In this paper we give a description of the structure of compact global attractor (Levinson center) for monotone Bohr/Levitan almost periodic dynamical system $x'=f(t,x)$ (*) with the strictly monotone first integral. It is shown that Levinson center of equation (*) consists of the Bohr/Levitan almost periodic (respectively, almost automorphic, recurrent or Poisson stable) solutions. We establish the main results in the framework of general non-autonomous (cocycle) dynamical systems. We also give some applications of theses results to different classes of differential/difference equations.

Normal stability of slow manifolds in nearly periodic Hamiltonian systems

Kruskal [J. Math. Phys. 3, 806 (1962)] showed that each nearly periodic dynamical system admits a formal U(1) symmetry, generated by the so-called roto-rate. We prove that such systems also admit nearly invariant manifolds of each order, near which rapid oscillations are suppressed. We study the nonlinear normal stability of these slow manifolds for nearly periodic Hamiltonian systems on barely symplectic manifolds—manifolds equipped with closed, non-degenerate 2-forms that may be degenerate to leading order. In particular, we establish a sufficient condition for long-term normal stability based on second derivatives of the well-known adiabatic invariant. We use these results to investigate the problem of embedding guiding center dynamics of a magnetized charged particle as a slow manifold in a nearly periodic system. We prove that one previous embedding and two new embeddings enjoy long-term normal stability and thereby strengthen the theoretical justification for these models.

Optimization of pressure swing adsorption via a trust-region filter algorithm and equilibrium theory

Optimization of pressure swing adsorption (PSA) remains a challenging task, as these are periodic dynamic systems governed by nonlinear PDE systems. This study develops an optimization strategy that incorporates reduced PSA models and also samples information from a high-fidelity ’truth’ model. The reduced model is based on equilibrium theory and can be applied to optimize a large variety of cyclic adsorption processes. The optimization is performed via a trust-region filter (TRF) method. The TRF method uses reduced models to minimize the information required from the high-fidelity adsorption model during mathematical optimization. Moreover, this method guarantees convergence to the optimum of the high-fidelity model. This approach is applied to optimize a 4 column pressure swing adsorption process, modeled by partial differential algebraic equations, for the separation of a CO2/CH4 mixture. The results show that the reduced model significantly reduces the computational time of the method’s trust-region step compared to previous studies.

Exact asymptotic behavior of pulsating traveling waves for a periodic monostable lattice dynamical system

This paper is concerned with the pulsating traveling waves of a periodic lattice dynamical system with monostable nonlinearity. We first establish the exponential upper and lower bounds of the pulsating wave profiles at minus infinity. Then we prove the uniqueness result and derive the asymptotic behavior of all non-critical monostable pulsating traveling waves. This might be the first time to obtain the exact asymptotic behavior of the pulsating traveling waves for periodic discrete systems towards the unstable steady state.

Active synthesis of a gyroscopic-nonreciprocal acoustic metamaterial

A class of active nonreciprocal metamaterials is developed to control the flow and distribution of energy along periodic dynamical systems. Such a development constitutes a radical departure from the currently available approaches where the non-reciprocities are generated either by utilizing various physical sources of passive nonlinearities, gyroscopic circulators, spatiotemporal modulation, or active control of nonlinear systems. The proposed active Nonreciprocal Gyroscopic Meta-Material (NGMM) cell consists of a one-dimensional acoustic duct provided with linear dynamic control capabilities that virtually synthesize a gyroscopic control action that generates non-reciprocal characteristics of tunable magnitude and direction. The controller is designed in order to enable the spatial control and redistribution of the wave propagation energy flow along the acoustic duct. During this entire process, the system behaves in a linear fashion. Numerical examples are presented to demonstrate the basic features, non-reciprocal behavior, as well as the energy flow characteristics. The presented concept and controller design of the NGMM can be extended to various critical structures to achieve realistic acoustic diode configurations in a simple and programmable manner.

Lyapunov Quantities in the Problem about Local Bifurcations of Non-Autonomous Periodic Dynamical Systems

The article discusses the problems about the main scenarios of dynamical systems’ local bifurcations described by non-autonomous differential equations with periodic right-hand side. New formulae for calculating Lyapunov quantities are proposed. The proposed formulae are obtained on the basis of the general operator method for studying local bifurcations and do not require a transition to normal forms and the usage of the theorems about the central manifold. The indicated method allowed for obtaining new bifurcation formulae for studying the main scenarios of local bifurcations. The paper shows how these bifurcation formulae lead to new formulae for calculating Lyapunov quantities in the problems about bifurcations of forced oscillations and Andronov–Hopf bifurcations.

Stochastic semidiscretization method: Second moment stability analysis of linear stochastic periodic dynamical systems with delays

In this paper, an efficient numerical approach is presented, which allows the analysis of the moment dynamics, stability, and stationary behavior of linear periodic stochastic delay differential equations. The method leads to a high dimensional stochastic mapping with periodic statistical properties, from which the periodic first and second moment mappings are derived. The application of the method is demonstrated first through the analysis of the stochastic delay Mathieu equation. Then a practical case study, where the effect of spindle speed variation on the stability and the resulting surface quality of turning operations is investigated.

Modeling periodic and non-periodic response of dynamical systems using an efficient Chebyshev-based time-spectral approach

A Chebyshev-based time-spectral method (C-TSM) is developed to model periodic and non-periodic nonlinear dynamical systems. It is shown that the proposed technique can accurately model such problems eliminating the need to use expensive classical dual-timestepping time-accurate integration. Furthermore, for autonomous dynamical systems subjected to single or multiple fundamental frequencies, the C-TSM analysis can be used without the prior knowledge of those frequencies. This offers an apparent advantage over the Fourier-based time-spectral methods. In addition, the current approach lends itself as a very useful tool for transient adjoint-based sensitivity analysis since it greatly reduces the memory requirements by solving and storing time-dependent response at a handful of collocation points instead of storing the entire time-history of the primal solution. The efficacy of the present technique is demonstrated by directly comparing the results with Fourier-based time-spectral, as well as time-accurate methods.

Active nonreciprocal metamaterial using an eigen-structure assignment control strategy.

A class of active nonreciprocal metamaterials (ANMMs) is developed to control the flow and distribution of energy along periodic dynamical systems. Such a development constitutes a radical departure from the currently available approaches where the non-reciprocities are generated either by utilizing various sources of passive nonlinearities, gyroscopic circulators, spatiotemporal modulation, or active control of nonlinear systems. The proposed ANMM cell consists of a one-dimensional acoustic duct provided with linear active control capabilities. The controller is designed by simultaneous assignment of both the eigenvalues and eigenvectors, i.e., the entire eigen-structure, of the closed-loop system. Conventionally, the placement of the eigenvalues has been considered to improve the damping and system's response. However, in this study, the emphasis is placed also on tailoring the eigenvectors in order to enable the spatial control and redistribution of the wave propagation energy flow along the acoustic duct in such a manner that it can introduce non-reciprocity as well as control its direction and reversal. During this entire process, the system remains behaving in a linear fashion. Numerical examples are presented to demonstrate the basic features and non-reciprocal behavior, as well as the energy flow characteristics.

Fast Construction of Forward Flow Maps using Eulerian Based Interpolation Schemes

We propose a modification to a recently developed Eulerian interpolation scheme for constructing the flow map for autonomous, periodic and aperiodic dynamical systems. We show that the proposed methods significantly improve the computational efficiency when a large number of flow maps are needed, yet retain second-order accuracy as in the original approach. The idea is to pre-compute and to store a carefully selected set of intermediate flow maps. When the initial and the final time of a required flow map are known, our proposed methods can simply load these pre-processed short-time flow maps for flow map construction. Numerical examples are included to validate our theoretical prediction, and demonstrate the effectiveness of these proposed Eulerian interpolation schemes as a simple numerical tool for other applications such as the finite-time Lyapunov exponent in determining the Lagrangian coherent structure of dynamical systems and the geometrical optics problems.