Multidimensional Dynamical Systems Research Articles

4400 TEU cargo ship dynamic analysis by Gaidai reliability method

Modern cargo vessel transport constitutes an important part of global economy; hence it is of paramount importance to develop novel, more efficient reliability methods for cargo ships, especially if onboard recorded data is available. Classic reliability methods, dealing with timeseries, do not have the advantage of dealing efficiently with system high dimensionality and cross-correlation between different dimensions. This study validates novel structural reliability method suitable for multi-dimensional structural systems versus a well-established bivariate statistical method. An example of this reliability study was a chosen container ship subjected to large deck panel stresses during sailing. Risk of losing containers, due to extreme motions is the primary concern for ship cargo transport. Due to non-stationarity and complicated nonlinearities of both waves and ship motions, it is challenging to model such a phenomenon. In the case of extreme motions, the role of nonlinearities dramatically increases, activating effects of second and higher order. Moreover, laboratory tests may also be questioned. Therefore, data measured on actual ships during their voyages in harsh weather provides a unique insight into statistics of ship motions. This study aimed at benchmarking and validation of the state-of-the-art method, which enables extraction of the necessary information about the extreme system dynamics from onboard measured time histories. The method proposed in this study opens up broad possibilities of predicting simply, yet efficiently potential failure or structural damage risks for the nonlinear multi-dimensional cargo vessel dynamic systems as a whole. Note that advocated novel reliability method can be used for a wide range of complex engineering systems, thus not limited to cargo ship only.

Sloppy model analysis identifies bifurcation parameters without normal form analysis.

Bifurcation phenomena are common in multidimensional multiparameter dynamical systems. Normal form theory suggests that bifurcations are driven by relatively few combinations of parameters. Models of complex systems, however, rarely appear in normal form, and bifurcations are controlled by nonlinear combinations of the bare parameters of differential equations. Discovering reparameterizations to transform complex equationsinto a normal form is often very difficult, and the reparameterization may not even exist in a closed form. Here we show that information geometry and sloppy model analysis using the Fisher information matrix can be used to identify the combination of parameters that control bifurcations. By considering observations on increasingly long timescales, we find those parameters that rapidly characterize the system's topological inhomogeneities, whether the system is in normal form or not. We anticipate that this novel analytical method, which we call time-widening information geometry (TWIG), will be useful in applied network analysis.

Convolutional LSTM-Attention Based Encoder–Decoder Neural Network for Prediction of Chaotic Vibrations of Multi-Dimensional Dynamic Systems

The present research proposes a convolutional long-short term memory (ConvLSTM) with an attention mechanism (AM) model, termed as ConvLSTM-AM, to conduct prediction of chaotic vibrations of multi-dimensional dynamic systems. The proposed data-driven model is based on an encoder–decoder architecture where the lengths of inputs and outputs are variable. Different from other conventional benchmarks which consider the temporal correlation solely to deal with the chaotic sequences, this research work takes the spatial information into account. In this sense, the ConvLSTM is adopted as an encoder to acquire useful chaotic spatiotemporal patterns and retain long-term successive dependencies. LSTM and AM in this research are taken as the main structures of the procedure in decoding, in which the LSTM is used as the further temporal processor and AM is stacked on the top to exploit more salient information of the historical data. Among that, a residual connection between the outputs of LSTM and the information of attention is considered in AM to prevent gradient vanishment. Two datasets of chaotic vibrations of multi-dimensional systems are employed to adequately illustrate the effectiveness and feasibility of the proposed model. Besides, five conventional benchmarks are built to demonstrate the advantages of the proposed model in terms of both training and generalization performance. As found in the research, the training time is reduced with lower testing loss in comparing with the other five counterparts, as the spatial information introduced expedites the training convergence. The present research provides a useful guidance for predicting and analysing chaotic vibrations of multi-dimensional dynamic systems.

Lifetime assessment for riser systems

As marine renewable energy technologies developing, there is a growing need for energy transportation systems. During offshore operations, deep sea risers can be subjected to excessive environmental loadings, causing operational risks. In this study, hydrodynamic loads, caused by in situ sea currents, acting on a riser under real-world sea conditions were modelled and examined, with experimental data being used as a calibration tool. Major safety problems for various offshore energy systems being an accurate assessment of excessive riser external loads, under influence of local sea currents, and hence resulting vortex induced vibrations (VIV).The method outlined in this study may be applied to complex sustainable energy systems, that are exposed to environmental loads, throughout the whole period of their intended service life. Approach advocated in this study offers practical way to estimate failure risks for nonlinear multidimensional dynamic offshore riser systems in an easy yet accurate manner.With regard to defense technology, risers and umbilicals play an important role for modern submarine operations.

Lifetime assessment for container vessels

This study presents novel reliability-based technique, being especially useful for lifetime assessment of multi-dimensional dynamic systems, consisting of cross-correlated components, that have been either measured, or numerically simulated over representative time span. This study has potential applications within wide range of marine structural systems design, where reliability analysis is required. As demonstrated in this study, it is possible to accurately forecast risks of a system failure due to excessive structural responses. Furthermore, time series-based traditional reliability techniques do not always manage system's high-dimensionality, along with complex cross-correlations between different system components. When sailing in harsh weather, cargo ships are exposed to structural damage risks, this study used operating cargo vessel as an underlying example. Another serious safety concern for modern ship trans-Atlantic navigation is the potential loss of cargo, due to the ship's violent movements. Simulating such a situation is quite challenging, due to waves and ship motions being both non-stationary and highly nYar. Validity of laboratory tests may also be questioned, hence, on-board data, gathered from actual cargo ships during their trans-Atlantic voyages, under realistic storms gives a unique research opportunity.Novel Monte Carlo based multi-dimensional reliability method, based on-board measurements is presented in this study. Advocated methodology can be applied to a large variety of complex sustainable marine systems, that are designed to endure environmental loadings for the entire length of their planned service life. This study's approach offers the opportunity to robustly and accurately evaluate operational risks of an entire nonlinear multidimensional dynamic system.

ENRM: An alternative tool for studying dynamical systems

In recent years, the recurrence plot technique has been increasingly used in nonlinear complex time series, to analyse the periodicity, chaos and non-stationarity. As far, some important methods have arisen based on the recurrence plot technique, among which the recurrence quantification analysis (RQA) is the most famous one, defined based on the recurrence matrix. As a tool of the RQA, the entropy based on recurrence microstates (ENRM), has been proposed in Corsoet al. (2018), to measure the chaotic and mixing behaviours of the underlying time series. In this paper, we first develop some techniques for more efficiently computing the ENRM. Another important contribution of this paper is to extend the concept of the ENRM to study multi-dimensional dynamical systems, where we regard each particle trajectory as a time series. This allows us to compare the ENRM to the finite time Lyapunov exponent (FTLE), which is another commonly used quantity for measuring chaotic behaviours of multi-dimensional dynamical systems. Examples will show different behaviours of particle trajectories with different magnitudes of ENRM values, and also the difference between the ENRM and the FTLE. Especially, by investigating a real data set, we show that the ENRM is a nice tool for measuring the complex behaviours of dynamical systems in the occasion where only sparse particle trajectories are provided.

Обобщение формулы Аккермана для некоторого класса многомерных динамических систем с векторным входом

A compact analytical formula is obtained that determines the entire set of solutions of the modal control problem for a wide class of multidimensional dynamical systems with vector input, where the number of states is divisible by the number of control inputs, and the controllability index is equal to the quotient of this division. This formula generalizes to systems with the vector input the Ackermann formula applied to multidimensional systems with scalar input. The basis to obtaining the generalized Ackermann formula lies in the original concepts of the Luenberger generalized canonical form and operations of the matrices block transposition. For the most convenient calculation of controller, the original system with vector input is reduced to the generalized Luenberger canonical form using the two successive similarity transformations. A lemma is proved that demonstrates the compact analytical form of the inverse transformation matrix. Transition equivalence makes it possible to obtain a complete countably infinite parametrized set of solutions to the modal control problem under consideration. Its parametrization is provided by selecting block coefficients of the matrix polynomial, which determinant corresponds to the given scalar characteristic polynomial. In cases, where the matrix polynomial involved in parametrization is not reduced to the multipliers, the generalized Ackermann formula contains solutions to the modal control problem that could not be obtained using the existing decomposition method. Examples are presented demonstrating both suitability of the proposed formula for analytical synthesis of modal controllers by state in systems with vector input and its advantages in comparison with the decomposition method

Hybrid modeling of a multidimensional coupled nonlinear system with integration of Hamiltonian mechanics

AbstractThis study concerns hybrid modeling of a multidimensional coupled nonlinear system. The underlying basis for the model is derived from Hamiltonian mechanics capitalizing on the broad utility and efficiency of energy-based reasoning in modeling high-dimensional systems. The hybrid model is essentially an artificial neural network with a computational graph that is modified from conventional neural networks in a few significant ways. The first modification includes incorporating an intermediate scalar function representing the Hamiltonian learned from data. The second modification enhances input/output channels for capturing the multidimensional dynamics of the system. The main goal of such hybrid reasoning is to improve the extrapolation capability of the model by enforcing conformance with some key aspects of the underlying physics in the form of a bias. The results demonstrate that incorporating this physics-based bias into the hybrid model empowers it to produce long-term and physically plausible predictions. The proposed modeling approach also shows high scalability for energy-based modeling of multidimensional dynamic systems in general.

Novel Girsanov correction based Milstein schemes for analysis of nonlinear multi-dimensional stochastic dynamical systems

This work proposes Girsanov corrected explicit, semi-implicit, and implicit Milstein approximations for the solution of nonlinear stochastic differential equations. The solution trajectories provided by the Milstein schemes are corrected by employing the change of measures, aimed at removing the error associated with the diffusion process incurred due to the transformation between two probability measures. The change of measures invoked in the Milstein schemes ensures that the solution from the mapping is measurable with respect to the filtration generated by the error process. The proposed scheme incorporates the error between the approximated mapping and the exact representation as an innovation that is accounted in the Milstein trajectories as an additive term. Numerical demonstrations using parametrically and non-parametrically excited stochastic oscillators, a practical problem involving ring type gyroscope, and a 7-degree-of-freedom nonlinear structural system subjected to both stationary and non-stationary excitation demonstrate the improvement in the solution accuracy for the proposed schemes with much coarser time steps when compared with the classical Milstein approximation with finer time steps.

Rate-induced tipping: thresholds, edge states and connecting orbits

Rate-induced tipping (R-tipping) occurs when time-variation of input parameters of a dynamical system interacts with system timescales to give genuine nonautonomous instabilities. Such instabilities appear as the input varies at some critical rates and cannot, in general, be understood in terms of autonomous bifurcations in the frozen system with a fixed-in-time input. This paper develops an accessible mathematical framework for R-tipping in multidimensional nonautonomous dynamical systems with an autonomous future limit. We focus on R-tipping via loss of tracking of base attractors that are equilibria in the frozen system, due to crossing what we call regular R-tipping thresholds. These thresholds are anchored at infinity by regular R-tipping edge states: compact normally hyperbolic invariant sets of the autonomous future limit system that have one unstable direction, orientable stable manifold, and lie on a basin boundary. We define R-tipping and critical rates for the nonautonomous system in terms of special solutions that limit to a compact invariant set of the autonomous future limit system that is not an attractor. We focus on the case when the limit set is a regular edge state, introduce the concept of edge tails, and rigorously classify R-tipping into reversible, irreversible, and degenerate cases. The central idea is to use the autonomous dynamics of the future limit system to analyse R-tipping in the nonautonomous system. We compactify the original nonautonomous system to include the limiting autonomous dynamics. Considering regular R-tipping edge states that are equilibria allows us to prove two results. First, we give sufficient conditions for the occurrence of R-tipping in terms of easily testable properties of the frozen system and input variation. Second, we give necessary and sufficient conditions for the occurrence of reversible and irreversible R-tipping in terms of computationally verifiable (heteroclinic) connections to regular R-tipping edge states in the autonomous compactified system.

Novel methods for reliability study of multi-dimensional non-linear dynamic systems

This research presents two unique techniques for engineering system reliability analysis of multi-dimensional non-linear dynamic structures. First, the structural reliability technique works best for multi-dimensional structural responses that have been either numerically simulated or measured over a long enough length to produce an ergodic time series. Second, a novel extreme value prediction method that can be used in various engineering applications is proposed. In contrast to those currently used in engineering reliability methodologies, the novel method is easy to use, and even a limited amount of data can still be used to obtain robust system failure estimates. As demonstrated in this work, proposed methods also provide accurate confidence bands for system failure levels in the case of real-life measured structural response. Additionally, traditional reliability approaches that deal with time series do not have the benefit of being able to handle a system's high dimensionality and cross-correlation across several dimensions readily. Container ship that experiences significant deck panel pressures and high roll angles when travelling in bad weather was selected as the example for this study. The main concern for ship transportation is the potential loss of cargo owing to violent movements. Simulating such a situation is difficult since waves and ship motions are non-stationary and complicatedly non-linear. Extreme movements greatly enhance the role of nonlinearities, activating effects of second and higher order. Furthermore, laboratory testing may also be called into doubt due to the scale and the choice of the sea state. Therefore, data collected from actual ships during difficult weather journeys offer a unique perspective on the statistics of ship movements. This work aims to benchmark state-of-the-art methods, making it possible to extract necessary information about the extreme response from available on-board measured time histories. Both suggested methods can be used in combination, making them attractive and ready to use for engineers. Methods proposed in this paper open up possibilities to predict simply yet efficiently system failure probability for non-linear multi-dimensional dynamic structure.

Active Vibration Control of Nonlinear Elastic Wire Rope in Mine Hoisting System

Based on the importance of the safe operation of the mine hoist mechanism, an adaptive control method based on Fuzzy Sliding Mode Control (FSMC) was applied to synchronize the large-amplitude response of geometrically nonlinear elastic wire rope in mine hoist systems. To efficiently and precisely synchronize the lateral vibration of a wire rope under the adaptive synchronizing method, the control equation of the elastic wire rope was given considering the large von Karman geometrical deformation, and it was converted to a six-dimensional nonlinear dynamical system using third-order Galerkin discretization. Based on this dynamic system, the existing FSMC is improved, so that the improved control strategy will be available for the active dynamic control of multidimensional nonlinear dynamical systems. The research shows that the control strategy can effectively synchronize the large-amplitude response of wire rope, thus providing the basis for dynamic design, fault diagnosis, and safe operation of the mine hoist system.

Novel methods for wind speeds prediction across multiple locations

This article provides two unique methodologies that may be coupled to study the dependability of multidimensional nonlinear dynamic systems. First, the structural reliability approach is well suited for multidimensional environmental and structural reactions and is either measured or numerically simulated over sufficient time, yielding lengthy ergodic time series. Second, a unique approach to predicting extreme values has technical and environmental implications. In the event of measurable environmental loads, it is also feasible to calculate the probability of system failure, as shown in this research. In addition, traditional probability approaches for time series cannot cope effectively with the system's high dimensionality and cross-correlation across dimensions. It is common knowledge that wind speeds represent a complex, nonlinear, multidimensional, and cross-correlated dynamic environmental system that is always difficult to analyze. Additionally, global warming is a significant element influencing ocean waves throughout time. This section aims to demonstrate the efficacy of the previously mentioned technique by applying a novel method to the Norwegian offshore data set for the greatest daily wind cast speeds in the vicinity of the Landvik wind station. This study aims to evaluate the state-of-the-art approach for extracting essential information about the extreme reaction from observed time histories. The approach provided in this research enables the simple and efficient prediction of failure probability for the whole nonlinear multidimensional dynamic system.

Підхід до аналізу поздовжньої стійкості рідинної ракети-носія пакетної схеми компонування

The “core and strap-on boosters” layout of launch vehicle (LV) stages is quite common in heavy LV development. However, POGO oscillations in liquid-propellant LVs with this stage layout have some features. It is shown that the structure of LVs of this type as a dynamic object has a dense spectrum of natural frequencies and complex spatial mode shapes. The longitudinal oscillations of the identical elements of the LV side strap-on boosters may be in phase or in antiphase, while the longitudinal mode shapes of the LV central core and strap-on boosters may differ both in phase and in amplitude. In flight, the thrust of the engines of the side strap-on boosters may also oscillate in phase or in antiphase, as a result of which the interaction of the LV structure with the sustainer propulsion systems of the side strap-on boosters may have both a stabilizing and a destabilizing effect on the POGO stability of a liquid-propellant LV. This paper presents a mathematical model of the “liquid-propellant propulsion systems – LV structure” dynamic system. The model describes the interaction of the longitudinal vibrations of the structure of a two-stage “core and strap-on boosters” LV with the core and strap-on booster propulsion systems. The free longitudinal vibrations of the structure of a ‘core and strap-on boosters’ LV were simulated using computer-aided finite element design tools (CAE systems). The simulation was the first to account for the dissipation of the liquid propellant and LV structure oscillation energy. The paper suggests an approach to analyzing the POGO stability of liquid-propellant “core and strap-on boosters” LVs with the use of the Nyquist criterion generalized to the case of multidimensional dynamic systems. The approach is based on opening the thrust feedback loops of the “liquid-propellant propulsion systems – structure” closed-loop dynamic system and studying the stability of the one-channel systems obtained in this way. Based on the proposed approach, the interaction between the longitudinal vibrations of the “core and strap-on boosters” LV structure and low-frequency processes in the liquid-propellant sustainer propulsion systems of the LV first stage was studied numerically.

Novel reliability method for multidimensional nonlinear dynamic systems

The paper describes a novel structural reliability method, particularly suitable for multi-dimensional structural responses, either measured or numerically simulated over sufficient period of time, resulting in sufficiently long ergodic time series. Unlike other reliability methods the new method does not require to re-start simulation each time system fails, in case of numerical simulation. In case of measured structural response, an accurate prediction of system failure probability is also possible as illustrated in this study. Moreover, classic reliability methods, dealing with time series do not have an advantage of dealing easily with system high dimensionality and cross-correlation between different dimensions.As an example for this reliability study was chosen container ship subjected to large deck panel stresses and extreme roll angles occurring during sailing in harsh weather. Risk of losing containers due to extreme motions is primary concern for ship transport. Due to non-stationarity and complicated nonlinearities of both waves and ship motions, it is a considerable challenge to model such a phenomenon. In case of extreme motions, the role of nonlinearities dramatically increases, activating effects of second and higher order. Moreover, laboratory tests may also be questioned because of the scaling and the sea state choice. Therefore, data measured on actual ships during their voyages in harsh weather provides a unique insight into statistics of ship motions.The aim of this work is to benchmark state of art method, which makes it possible to extract the necessary information about the extreme response from onboard measured time histories. The method proposed in this paper opens up the possibility to predict simply and efficiently failure probability for nonlinear multi-dimensional dynamic system as a whole.

On a study of Sobolev‐type fractional functional evolution equations

Sobolev‐type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate the existence of mild solutions for fractional‐order time‐delay evolution equation of Sobolev type with multiorders in a Banach space with the help of two various methods. First, we study abstract problem by assuming the existence and compactness of E−1 and introduce a closed‐form representation of a mild solution via a newly defined delayed Mittag–Leffler‐type function which is generated by nonpermutable and permutable linear‐bounded operators. Our second method is based on the theory of Sobolev‐type delayed resolvent families generated by the pair (A, E) and a subordination principle, and in this case, any restrictive assumptions are not required on E. Furthermore, we derive an exact analytical representation of solutions for multidimensional fractional functional dynamical systems with nonpermutable and permutable matrices. As an example, the Sobolev‐type delayed wave equation with a damping term is provided to illustrate the applications of the abstract results.

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.

Kalman Filter Adaptation to Disturbances of the Observer’s Parameters

Currently, one of the most effective algorithms for state estimation of stochastic systems is a Kalman filter. This filter provides an optimal root-mean-square error in state vector estimation only when the parameters of the dynamic system and its observer are precisely known. In real conditions, the observer’s parameters are often inaccurately known; moreover, they change randomly over time. This in turn leads to the divergence of the Kalman estimation process. The problem is currently being solved in a variety of ways. They include the use of interval observers, the use of an extended Kalman filter, the introduction of an additional evaluating observer by nonlinear programming methods, robust scaling of the observer’s transmission coefficient, etc. At the same time, it should be borne in mind that, firstly, all of the above ways are focused on application in specific technical systems and complexes, and secondly, they fundamentally do not allow estimating errors in determining the parameters of the observer themselves in order to compensate them for further improving the accuracy and stability of the filtration process of the state vector. To solve this problem, this paper proposes the use of accurate observations that are irregularly received in a complex measuring system (for example, navigation) for adaptive evaluation of the observer’s true parameters of the stochastic system state vector. The development of the proposed algorithm is based on the analytical dependence of the Kalman estimate variation on the observer’s parameters disturbances obtained using the mathematical apparatus for the study of perturbed multidimensional dynamical systems. The developed algorithm for observer’s parameters adaptive estimation makes it possible to significantly increase the accuracy and stability of the stochastic estimation process as a whole in the time intervals between accurate observations, which is illustrated by the corresponding numerical example.

Instability times in the HD 181433 exoplanetary system

The present work consists of a study of the dynamical stability of a three-body system that takes advantage of the Shannon entropy approach to estimate the diffusivity (DS) in a Delaunay’s action-like phase space. We outline the main features of a numerical computation ofDSfrom the solutions of the equations of motion and, thereupon, we consider how to estimate a macroscopic instability timescale,τinst, (roughly speaking, the lifetime of the system) associated with a given set of initial conditions. Through such estimates, we are able to characterize the system’s space of initial conditions in terms of its orbital stability by applying numerical integrations to the construction of dynamical maps. We compare these measures of chaotic diffusion with other indicators, first in a qualitative fashion and then more quantitatively, by means of long direct integrations. We address an analysis of a particular, near-resonant system, namely HD 181433, and we show that the entropy may provide a complementary analysis with regard to other dynamical indicators. This work is part of a series of studies devoted to presenting the Shannon entropy approach and its possibilities as a numerical tool providing information on chaotic diffusion and the dynamical stability of multidimensional dynamical systems.

The media space of the educational sphere as a logistic system: features of management and personality formation

Development of digitization (including education) leads to difficulties in management and regulation of important multidimensional dynamic social economical systems, where problems of efficient management occur requiring for interdisciplinary approach and active application of information technologies and computers. It is required to develop new methods of risk management (in the fields of national security, personal security), since information nonmilitary action as the most important hazard of hybrid wars is precepted as motion along the road of progress, democracy, tolerance. The research objective implies search and implementation of optimum management of stepwise digitization of education by means of monitoring and reconfiguration of digital platforms as elements of global logistic system (with consideration for consciental risks). Targets include analysis of features of dynamics and management of digitization of educational platforms in relation with maturity of social institutions, COVID-19, ecology of information space, degree of informatization of society, role and composition of mental structures, etc.; detection of factors influencing dynamics of persons and social institutions involved in education. The interrelation between digitization of education and development of educational platforms has been described. The conclusions have been made about the advantages and disadvantages of management formed by analogy with management of dynamic processes running on classical logistic systems. Recommendations have been given on how to reduce the dependence of a human on behavioral stereotypes imposed by various algorithmic energy information structures. In the frames of community, this would provide productive existence in unique personal space, efficient communication with surrounding people together with preservation of mentality and self-consciousness, cultural and national identity.