Time-periodic Systems Research Articles

Invariant Surfaces of Periodic Systems with Conservative Cubic First Approximation

Two classes of time-periodic systems of ordinary differential equations with a small parameter e ≥ 0, those with “fast” and “slow” time, are studied. The corresponding conservative unperturbed systems $${{\dot {x}}_{i}}$$ = $$ - {{\gamma }_{i}}{{y}_{i}}{{\varepsilon }^{\nu }}$$ , $${{\dot {y}}_{i}}$$ = γi( $$x_{i}^{3}$$ – $${{\eta }_{i}}{{x}_{i}}$$ )eν (i = $$\overline {1,n} $$ , ν = 0, 1) have 1 to 3n singular points. The following results are obtained in explicit form: (1) conditions on perturbations independent of the parameter under which the initial systems have a certain number of invariant surfaces of dimension n + 1 homeomorphic to the torus for all sufficiently small parameter values; (2) formulas for these surfaces and their asymptotic expansions; (3) a description of families of systems with six invariant surfaces.

On higher-order averaging of time-periodic systems: reconciliation of two averaging techniques

In this paper we show how higher-order averaging can be used to remedy serious technical issues with the direct application of the averaging theorem. While doing so, we reconcile two higher-order averaging methodologies that were developed independently using different tools and within different communities: (i) perturbation theory using a near-identity transformation and (ii) chronological calculus using Lie algebraic tools. We provide the underpinning concepts behind each averaging approach and provide a mathematical proof for their equivalence up to the fourth order. Moreover, we provide a higher-order averaging study and analysis for two applications: the classical problem of the Kapitza pendulum and the modern application of flapping flight dynamics of micro-air-vehicles and/or insects.

Frequency-Domain Subspace Identification of Linear Time-Periodic (LTP) Systems

This paper proposes a new methodology for subs-pace-based state-space identification for linear time-periodic (LTP) systems. Since LTP systems can be lifted to equivalent linear time-invariant (LTI) systems, we first lift input–output data from an unknown LTP system as if they were collected from an equivalent LTI system. Then, we use frequency-domain subspace identification methods to find the LTI system estimate. Subsequently, we propose a novel method to obtain a time-periodic realization for the estimated lifted LTI system by exploiting the specific parametric structure of Fourier series coefficients of the frequency-domain lifting method. Our method can be used to obtain state-space estimates for unknown LTP systems as well as to obtain Floquet transforms for known LTP systems.

Flow Equation Approach to Periodically Driven Quantum Systems

We present a theoretical method to generate a highly accurate {\em time-independent} Hamiltonian governing the finite-time behavior of a time-periodic system. The method exploits infinitesimal unitary transformation steps, from which renormalization group-like flow equations are derived to produce the effective Hamiltonian. Our tractable method has a range of validity reaching into frequency regimes that are usually inaccessible via high frequency $\omega$ expansions in the parameter $h/\omega$, where $h$ is the upper limit for the strength of local interactions. We demonstrate our approach on both interacting and non-interacting many-body Hamiltonians where it offers an improvement over the more well-known Magnus expansion and other high frequency expansions. For the interacting models, we compare our approximate results to those found via exact diagonalization. While the approximation generally performs better globally than other high frequency approximations, the improvement is especially pronounced in the regime of lower frequencies and strong external driving. This regime is of special interest because of its proximity to the resonant regime where the effect of a periodic drive is the most dramatic. Our results open a new route towards identifying novel non-equilibrium regimes and behaviors in driven quantum many-particle systems.

Optimal transition of flapping wing micro-air vehicles from hovering to forward flight

In this work, we formulate a minimum-time optimal control problem to steer a FWMAV dynamical system from a hovering condition to forward flight with a prescribed forward speed using time-periodic and averaged dynamics formulations. For the averaged dynamics representation, we optimize the back and forth flapping angle and the up and down-stroke angles of attack of the wing. We represent the flapping angle via a generic periodic function with some parameters that determine the waveform of the flapping angle over the cycle. We formulate the optimal control problem such that the cost functional is the final time, and the slowly time-varying parameters of the flapping angle wave form and the up and down-stroke angles of attack are considered inputs to the averaged dynamics. On the other hand, the instantaneous the flapping speed and wing pitching angle are considered direct inputs to the time periodic system. The problem is then to steer the averaged dynamics from the hovering fixed point (origin) to a prescribed average forward speed, and the time periodic dynamics from the hovering periodic orbit to the orbit of the forward flight condition. We show that the averaging is not suitable for the steering between hovering and forward flight and that time-periodic dynamics are required for the controller to achieve proper transition. Also, we investigated the effect of using the time-averaged stability derivatives obtained using a computational fluid dynamics simulation versus using the time-varying hovering derivatives.

Reformulating Phase-Shifting Property in Dynamic Harmonic Domain: Time-Periodic Case

The phase-shifting property has been formulated in the dynamic harmonic domain in a recently published paper. The main idea is to apply a phase-shifting matrix to the output when phase-shifting the inputs by an angle ${\alpha }$ . This letter demonstrates that the phase-shifting property has to be reformulated when applied to time-periodic systems involving switching functions. The proposed reformulated property assumes proper commutation of the phase-shifting matrix and achieves zero error regardless the periodic system under study.

Global stability effects of parametric excitation

Stability investigations of general non-conservative parametrically excited systems with asynchronous excitation are presented. Focusing on the global stability effects outside of the traditional resonance areas, systems with two degrees of freedom are considered featuring displacement- and/or velocity-proportional parametric excitation with variable phase relations. In particular, facing the lack of studies on this subject, special attention is paid to time-periodic systems containing gyroscopic and circulatory terms. Through the application of the semi-analytical method of normal forms, general conditions for the appearance of possible global effects are derived. Apart from the “total instability” – presently the only known global effect – new stabilizing and destabilizing effects affecting the stability over the whole range of excitation frequencies are discovered. The derived conditions show, that such global effects are expected to be rather common in complex mechanical system, especially those, featuring circulatory terms. The qualitative analytical results are also confirmed by numerical stability analysis based on Floquet theory. As a mechanical example a minimal model of a squealing disk brake is examined. It is shown that this complex model is indeed subject to parametric excitation leading to global stability effects. These findings may contribute to a better understanding of the squealing phenomenon. Further, the newly obtained knowledge may as well be utilized for extended vibration suppression in mechanical systems in the context of parametric anti-resonance.

Examining the effect of land-use function complementarity on intra-urban spatial interactions using metro smart card records

Spatial interaction is an important phenomenon that reflects the human–land relationship and has long been a core topic in multiple fields, such as urban planning, transportation planning, commodity trade, and epidemic prevention. However, as an underlying cause of spatial interaction, function complementarity has been ignored by existing research for a long time. At the same time, the increase in Big Data of travel behavior provides an opportunity to model spatial interactions in detail. In this paper, we proposed three types of land-use function complementarity indices according to the spatiotemporal characteristics of human mobility. These complementarity indices are introduced to spatial interaction to improve the gravity model. We also examined the effects of land function complementarity on intra-urban spatial interaction using smart card records of metro system for different time periods and directions. The results showed that all models could be improved by introducing the land-use function complementarity indices, but the models with a single travel pattern and clear direction were explained more by the complementary indices. The indices we propose in this paper could be used for predicting spatial flow and trip distribution, and also could be considered as factors in researches about transportation and land-use planning.

Floquet group theory and its application to selection rules in harmonic generation

Symmetry is one of the most generic and useful concepts in science, often leading to conservation laws and selection rules. Here we formulate a general group theory for dynamical symmetries (DSs) in time-periodic Floquet systems, and derive their correspondence to observable selection rules. We apply the theory to harmonic generation, deriving closed-form tables linking DSs of the driving laser and medium (gas, liquid, or solid) in (2+1)D and (3+1)D geometries to the allowed and forbidden harmonic orders and their polarizations. We identify symmetries, including time-reversal-based, reflection-based, and elliptical-based DSs, which lead to selection rules that are not explained by currently known conservation laws. We expect the theory to be useful for ultrafast high harmonic symmetry-breaking spectroscopy, as well as in various other systems such as Floquet topological insulators.

Forced vibrations as a mechanism to suppress flutter—An aeroelastic Kapitza’s pendulum

Inspired by Kapitza’s inverted pendulum, forced vibrations are suggested as a mechanism to suppress aeroelastic flutter. We examine high-frequency small-amplitude vibrations (e.g. by an internal oscillating mass) applied on a 2D airfoil, yielding forced periodic excitation in the gyration-radius. Under such excitation, the aeroelastic dynamics involve time-periodic system of two Hill-type ODEs. Harmonic balance is applied, along with Floquet theory approach, in order to find approximated transition curves between stable and unstable regions. The transition curves are obtained from the relevant Hill’s determinants, and are validated by numerical calculations. Structural 3D effects are examined by the aeroelastic strip approach for excitation in a section or the entire wing. The results indicate that rapid small-amplitude oscillations can significantly increase the maximal stable velocity in realistic flight conditions.

Exceptional Points of Degeneracy Induced by Linear Time-Periodic Variation

We present a general theory of exceptional points of degeneracy (EPD) in periodically time-variant systems that do not necessarily require the presence of loss or gain, and we show that even a single resonator with a time-periodic component may develop EPDs. An EPD is a special point in a system parameter space at which two or more eigenmodes coalesce in both their eigenvalues and eigenvectors into a single degenerate eigenmode. We demonstrate the conditions for EPDs to exist in time-periodic systems that are either lossless/gainless or with loss and/or gain and we show that a system with zero time-average loss/gain exhibits EPDs with purely real resonance frequencies, yet the resonator energy grows algebraically in time. We show the occurrence of EPDs in a single LC resonator while the introduced concept is general for any time-periodic system. These findings have significant importance in various electromagnetic/photonic systems and pave the way of applications in areas of sensors, amplifiers and modulators. A potential application of this time varying EPD is highlighted as a highly-sensitive sensor.

Asymptotic spreading for a time-periodic predator-prey system

This paper is concerned with asymptotic spreading for a time-periodic predator-prey system where both species synchronously invade a new habitat. Under two different conditions, we show the bounds of spreading speeds of the predator and the prey, which is proved by the theory of asymptotic spreading of scalar equations, comparison principle and generalized eigenvalue. We show either the predator or the prey has a spreading speed that is determined by the linearized equation at the trivial steady state while the spreading speed of the other also depends on the interspecific nonlinearity. From the viewpoint of population dynamics, our results imply that the predator may play a negative effect on the spreading of the prey while the prey may play a positive role on the spreading of the predator.

Effects of localized spatial variations on the uniform persistence and spreading speeds of time periodic two species competition systems

This paper is concerned with the asymptotic dynamics of two species competition systems of the form $ \begin{equation*} \begin{cases} u_t(t,x) = \mathcal{A} u+u(a_1(t,x)-b_1(t,x)u-c_1(t,x)v),\quad x\in {\mathbb{R}} \cr v_t(t,x) = \mathcal{A} v+ v(a_2(t,x)-b_2(t,x)u-c_2(t,x) v),\quad x\in {\mathbb{R}} \end{cases} \end{equation*} $ where $ (\mathcal{A}u)(t,x) = u_{xx}(t,x) $, or $ (\mathcal{A}u)(t,x) = \int_{ {\mathbb{R}} }\kappa(y-x)u(t,y)dy-u(t,x) $ ($ \kappa(\cdot) $ is a smooth non-negative convolution kernel supported on an interval centered at the origin), $ a_i(t+T,x) = a_i(t,x) $, $ b_i(t+T,x) = b_i(t,x) $, $ c_i(t+T,x) = c_i(t,x) $, and $ a_i $, $ b_i $, and $ c_i $ ($ i = 1,2 $) are spatially homogeneous when $ |x|\gg 1 $, that is, $ a_i(t,x) = a_i^0(t) $, $ b_i(t,x) = b_i^0(t) $, $ c_i(t,x) = c_i^0(t) $ for some $ a_i^0(t) $, $ b_i^0(t) $, $ c_i^0(t) $, and $ |x|\gg 1 $. Such a system can be viewed as a time periodic competition system subject to certain localized spatial variations. In particular, we study the effects of localized spatial variations on the uniform persistence and spreading speeds of the system. Among others, it is proved that any localized spatial variation does not affect the uniform persistence of the system, does not slow down the spreading speeds of the system, and under some linear determinant condition, does not speed up the spreading speeds. We also study a relevant problem, that is, the continuity of the spreading speeds of time periodic two species competition systems with respect to time periodic perturbations, and prove that the spread speeds of such systems are lower semicontinuous with respect to time periodic perturbations.

Invariant surfaces of periodic systems with conservative cubic first approximation

Two classes of time-periodic systems of ordinary differential equations with a small parameter e ≥ 0, those with “fast” and “slow” time, are studied. The corresponding conservative unperturbed systems $${{\dot {x}}_{i}}$$ = $$ - {{\gamma }_{i}}{{y}_{i}}{{\varepsilon }^{\nu }}$$ , $${{\dot {y}}_{i}}$$ = γi( $$x_{i}^{3}$$ – $${{\eta }_{i}}{{x}_{i}}$$ )eν (i = $$\overline {1,n} $$ , ν = 0, 1) have 1 to 3n singular points. The following results are obtained in explicit form: (1) conditions on perturbations independent of the parameter under which the initial systems have a certain number of invariant surfaces of dimension n + 1 homeomorphic to the torus for all sufficiently small parameter values; (2) formulas for these surfaces and their asymptotic expansions; (3) a description of families of systems with six invariant surfaces.

Multidimensional thermodynamic uncertainty relations

We extend a class of recently derived thermodynamic uncertainty relations to vector-valued observables. In contrast to the scalar-valued observables examined previously, this multidimensional thermodynamic uncertainty relation provides a natural way to study currents in high-dimensional systems and to obtain relations between different observables. Our proof is based on the generalized Crámer–Rao inequality, which we interpret as a relation between physical observables and the Fisher information. This allows us to develop high-dimensional versions of both the original, steady state uncertainty relation and the more recently obtained generalized uncertainty relation for time-periodic systems. We apply the multidimensional uncertainty relation to obtain a new constraint on the performance of steady-state heat engines, which is tighter than previous bounds and reveals the role of heat-work correlations. As a second application, we show that the uncertainty relation is connected to a bound on the differential mobility. As a result of this connection, we find that a necessary condition for equality in the uncertainty relation is that the system obeys the equilibrium fluctuation–dissipation relation.

Frequency domain analysis of regenerative chatter in machine tools with Linear Time Periodic dynamics

Abstract Unstable vibrations (Chatter) during machining operations is caused by the flexibility of various components of the machine tool or the workpiece, which can be stationary and symmetrical or rotating and non-symmetrical. Therefore, the structural dynamics of machining systems are best described by Linear Time Periodic (LTP) systems, but the existing solutions of chatter use Linear Time Invariant (LTI) descriptions of the machine tool dynamics. This paper presents an update to the widely used Multi-Frequency Solution of regenerative chatter by modelling the machine tool dynamics as a LTP system. The stability of the LTP system is then determined by utilizing the concept of Harmonic Transfer Functions (HTF) and Nyquist stability criterion. The presented method provides a unified solution to determine the stability of vibrations in both stationary and rotating coordinate frames, regardless of the stationary or rotating nature of the vibration modes. The accuracy of the presented approach is verified by comparing the predicted stability diagrams with the results of Multi-Frequency Solution, Semi-Discretization Method, and with experimental results.

The principal eigenvalue for periodic nonlocal dispersal systems with time delay

The theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time-periodic nonlocal dispersal cooperative system with time delay. Then we apply it to a Nicholson's blowflies population model and obtain a threshold type result on its global dynamics.

Two-port network equivalent of VSC-HVDC for power flow studies

This paper presents a novel harmonic domain (HD) approach that computes a two-port network equivalent of a switching converter for power flow studies. The equivalent stems from the harmonic domain admittance matrix (HDAM) of a switching converter which accounts for harmonics interaction. Power flow studies, involving switching converters, require modifying the Jacobian matrix via calculation and incorporation of the partial derivative terms corresponding to the switched part within the iterative solution process. The proposed approach is readily applicable to modern power systems involving electronic converters. Also, it avoids calculating partial derivative terms of the switching converter by including it as a simple, yet an effective, equivalent two-port network, i.e., by just modifying the admittance matrix within the iterative power-flow solution process. This paper demonstrates that the procedure to obtain the PI equivalent is general for any linear time-periodic system and does not rely on library components when embedded into a commercial software tool.The proposed approach is validated by two test networks which involve High-Voltage Direct Current (HVDC) systems. The obtained results are compared, in terms of accuracy, with those given by the Power System Computer-Aided Design/Electromagnetic Transients including Direct Current (PSCAD/EMTDC) software tool, which involves the detailed system model, once steady-state is reached. Also, the results of the proposed approach are verified against those obtained from a dedicated power flow software tool, i.e., Power System Simulator for Engineering (PSS/E).

Harmonic transfer functions based controllers for linear time-periodic systems

The analysis, identification and control of periodic systems has gained increasing interest during the last few decades due to the increased use of dynamical systems that exhibit periodic motion. The vast majority of these studies focus on the analysis and control problem for a known state-space formulation of the linear time-periodic (LTP) system. On the other hand, there are also some studies that focus on data-driven identification of LTP systems with unknown state-space formulations. However, most of these methods provide numerical estimates for the harmonic transfer functions (HTFs) of an LTP system that are extremely difficult to work with during controller design. The goal of this paper is to provide a simple controller design methodology for unknown LTP systems by utilizing so-called HTFs estimates. To this end, we first build a mathematical basis of LTP controller design for known LTP systems using the Nyquist diagrams and analytically derived HTFs. We propose a new methodology to design P-, PD- and PID-type controllers for LTP systems using Nyquist diagrams and the eigenlocus of the HTFs. Having established the HTF-based controller design procedure, we extend our methodology to unknown LTP systems by presenting a new sum-of-cosine signal-based data-driven system identification method. We show that the proposed data-driven controller design method allows estimation of the HTFs and it provides simple tools for optimizing certain time-domain performance metrics. We provide numerical examples for both known and unknown LTP system cases to illustrate the performance of the proposed controller design methodology.

Body stiffness and damping depend sensitively on the timing of muscle activation in lampreys.

Unlike most manmade machines, animals move through their world using flexible bodies and appendages, which bend due to internal muscle and body forces, and also due to forces from the environment. Fishes in particular must cope with fluid dynamic forces that not only resist their overall swimming movements but also may have unsteady flow patterns, vortices, and turbulence, many of which occur more rapidly than what the nervous system can process. Has natural selection led to mechanical properties of fish bodies and their component tissues that can respond very quickly to environmental perturbations? Here, we focus on the mechanical properties of isolated muscle tissue and of the entire intact body in the silver lamprey, Ichthyomyzon unicuspis. We developed two modified work loop protocols to determine the effect of small perturbations on the whole body and on isolated segments of muscle as a function of muscle activation and phase within the swimming cycle. First, we examined how the mechanical properties of the whole lamprey body change depending on the timing of muscle activity. Relative to passive muscle, muscle activation can modulate the effective stiffness by about two-fold and modulate the effective damping by >10-fold depending on the activation phase. Next, we performed a standard work loop test on small sections of axial musculature while adding low-amplitude sinusoidal perturbations at specific frequencies. We modeled the data using a new system identification technique based on time-periodic system analysis and harmonic transfer functions (HTFs) and used the resulting models to predict muscle function under novel conditions. We found that the effective stiffness and damping of muscle varies during the swimming cycle, and that the timing of activation can alter both the magnitude and timing of peak stiffness and damping. Moreover, the response of the isolated muscle was highly nonlinear and length dependent, but the body's response was much more linear. We applied the resulting HTFs from our experiments to explore the effect of pairs of antagonistic muscles. The results suggest that when muscles work against each other as antagonists, the combined system has weaker nonlinearities than either muscle segment alone. Together, these results begin to provide an integrative understanding of how activation timing can tune the mechanical response properties of muscles, enabling fish to swim effectively in their complex and unpredictable environment.