Continuous Dynamical Systems Research Articles

Formulations of the elastodynamic equations in anisotropic and multiphasic porous media from the principle of energy conservation

Abstract Elastodynamic equations have been formulated with Newton's second law of motion, Lagrange's equation, or Hamilton's principle for over 150 years. In this work, contrary to classical continuum mechanics, a novel strategic methodology is proposed for formulating general mechanical equations using the principle of energy conservation. First, based on Hamilton's principle, Hamilton's equations, Lagrange's equation, and the elastodynamic equation of motion are derived in arbitrarily anisotropic and multiphasic porous elastic media, for the first time. Secondly, these equations are all formulated using the principle of energy conservation for the related media. Both formulation results using the two kinds of principles are compared and validated by each other. The advantages of our methodology lie in that the elastodynamic equation of motion, Lagrange's equation, and Hamilton's equations in continuum mechanics are directly formulated using a simple constraint of energy conservation without introducing variational concepts. It is easy to understand and has clear physical meanings. Our methodology unlocks the physics essences of Hamilton's principle in continuum mechanics, which is a consequence of the principle of energy conservation. Although the linear stress–strain constitutive relation is considered, our methodology can still be used in a nonlinear dynamical system. The methodology also paves an alternative way of treating other complex continuous dynamical systems in a broad sense. In addition, as an application, the continuity conditions at various medium interfaces are also revisited and extended using our proposed approach, which explains the law of reflections and refractions.

Impulsive time series modeling with application to luteinizing hormone data.

This work considers the estimation of impulsive time series pertaining to biomedical systems and, in particular, to endocrine ones. We assume a signal model in the form of the output of a continuous linear time-invariant system driven by a sequence of instantaneous impulses, which concept is utilized here, in particular, for modeling of the male reproductive hormone axis. An estimation method to identify the impulsive sequence and the continuous system dynamics from sampled measurements of the output is proposed. Hinging on thorough mathematical analysis, the method improves upon a previously developed least-squares algorithm by resolving the trade-off between model fit and input sparsity, thus removing the need for manual tuning of user-defined estimation algorithm parameters. Experiments with synthetic data and Markov chain Monte-Carlo estimation demonstrate the viability of the proposed method, but also indicate that measurement noise renders the estimation problem ill-posed, as multiple estimates along a curve in the parameter space yield similar fits to data. The method is furthermore applied to clinical luteinizing hormone data collected from healthy males and, for comparability, one female, with similar results. Comparison between the estimated and theoretical elimination rates, as well as simulation of the estimated models, demonstrate the efficacy of the method. The sensitivity of the impulse distribution to the estimated elimination rates is investigated on a subject-specific data subset, revealing that the input sequence and elimination rate estimates can be interdependent. The dose-dependent effect of a selective gonadotropin releasing hormone receptor antagonist on the frequency and weights of the estimated impulses is also analyzed; a significant impact of the medication on the impulse weights is confirmed. To demonstrate the feasibility of the estimation approach for other hormones with pulsatile secretion, the modeling of cortisol data sets collected from three female adolescents was performed.

Global dynamics visualisation from Lagrangian Descriptors. Applications to discrete and continuous systems

This paper introduces a new global dynamics and chaos indicator based on the method of Lagrangian Descriptor apt for discriminating ordered and deterministic chaotic motions in multidimensional systems. The selected implementation of this method requires only the knowledge of orbits on finite time windows and is free of the computation of the tangent vector dynamics (i.e.,variational equations are not needed). To demonstrate its ability in visualising different dynamical behaviours, in particular for highlighting chaotic regions, several stability maps of classical systems, obtained with different phase space methods, are reproduced. The benchmark examples are rooted in discrete and continuous nearly-integrable dynamical systems, with prominent features played by resonances. These include the Chirikov standard map, higher dimensional symplectic and volume preserving maps, fundamental models of resonances, and a 3 degrees-of-freedom nearly-integrable Hamiltonian system with a dense web of resonances. The indicator thus appears to be relevant for understanding phase space transport mediated by resonances in nearly-integrable system, as ubiquitous in celestial mechanics or astrodynamics.

Characteristics of Stability Boundary of Nonlinear Continuous Dynamical Systems

The theory of differential equations has been widely known and developed in recent years. One of the issues that many authors give their undivided attention to is the stability boundary of nonlinear dynamical systems. In this work, we first review several properties of equilibrium points on the stability boundary. We next extend the characteristics of the stability boundary for a fairly large class of nonlinear dynamical systems. These characteristics are the key to completely determine the stability boundary of nonlinear dynamical systems.

On the induced measure-theoretic entropy for random dynamical systems

In this paper, we introduce the concepts of induced topological entropy and induced measure-theoretic entropy for random dynamical systems. By providing the topological version of induced measure-theoretic entropy, we establish the Katok entropy formula for a continuous bundle random dynamical system over an invertible ergodic system. We also consider the induced version with lower and upper pointwise dimensions, and establish a measure-theoretic entropy formula for lower and upper induced pointwise dimensions for a continuous bundle random dynamical system over an invertible ergodic system.

Transition to hyperchaos: Sudden expansion of attractor and intermittent large-amplitude events in dynamical systems.

Hyperchaos is distinguished from chaos by the presence of at least two positive Lyapunov exponents instead of just one in dynamical systems. A general scenario is presented here that shows emergence of hyperchaos with a sudden large expansion of the attractor of continuous dynamical systems at a critical parameter when the temporal dynamics shows intermittent large-amplitude spiking or bursting events. The distribution of local maxima of the temporal dynamics is non-Gaussian with a tail, confirming a rare occurrence of the large-amplitude events. We exemplify our results on the sudden emergence of hyperchaos in three paradigmatic models, namely, a coupled Hindmarsh-Rose model, three coupled Duffing oscillators, and a hyperchaotic model.

DYNAMIC ANALYSIS OF A CHAOTIC 3D QUADRATIC SYSTEM USING PLANAR PROJECTION.

The theory of dynamical systems is one of the most important theorems of scientific research because it relies heavily on most of the major fields of applied mathematics to give a sufficiently broad view of reality, but it still poses some problems, especially with regard to the modeling of certain physical phenomena. Since most of these systems are designed as continuous or discrete dynamic systems with large dimensions and multiple bifurcation parameters, researchers face major problems in qualitative study. In this paper, we propose a method to study bifurcations of continuous three-dimensional dynamic systems in general and chaotic systems in particular, which contains many bifurcation parameters. This method is mainly based on the projection on the plane and on the appropriate bifurcation parameter.

Nilpotent Center in a Continuous Piecewise Quadratic Polynomial Hamiltonian Vector Field

In this paper, we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line [Formula: see text], where these kinds of systems have a nilpotent center at [Formula: see text], which comes from the combination of two cusps of both Hamiltonian systems. By the Poincaré compactification we classify the global phase portraits of these systems. We must mention that it is extremely rare to find works studying the center-focus problem in piecewise smooth systems with nonelementary singular points as we did here.

Диагностирование непрерывных динамических систем с использованием структурных функций чувствительности

The paper considers an algorithm for searching for single defects of structural blocks of a diagnostic object in the form of an arbitrary change in the parameters of one dynamic block as part of a continuous dynamic system. This approach corresponds to the real picture of the manifestation of a specific structural defect in the unit and its influence on the dynamics of the system. The use of known algorithms of this kind is complicated by the need to use models with trial deviations of parameters, or by the need to analyze the signs of signal transmissions. The algorithm based on trial deviations of model parameters uses the specification of these changes in the model, which is a laborious task. The algorithm for analyzing transmission signs using a normalized diagnostic feature, as well as using a binary diagnostic feature, requires additional calculations of the signs of signal transmissions from block outputs to checkpoints. An algorithm for searching for single defects in the form of changes in the parameters of dynamic blocks is considered, based on the use of a model of the structural sensitivity of integral estimates of the output signals of an object to changes in the dynamic properties of a block. The algorithm allows you to move from the analysis of time functions to the analysis of the numerical values of their integral estimates. This approach allows you to identify a faulty dynamic unit without determining the actual parameters of its transfer function. The procedure for obtaining normalized diagnostic features and calculating a quantitative measure of the distinguishability of defects has been determined. The results of the implementation of the algorithm for the gas pressure stabilization system at a straight-tube furnace for heating oil products of an oil refinery are presented. The implementation of the algorithm shows that the considered algorithm gives acceptable values of the distinguishability of all single defects.

Stability analysis and stabilization of linear symmetric matrix-valued continuous, discrete, and impulsive dynamical systems — A unified approach for the stability analysis and the stabilization of linear systems

Matrix-valued dynamical systems are an important class of systems that can describe important processes such as covariance/second-order moment processes, or processes on manifolds and Lie Groups. We address here the case of processes that leave the cone of positive semidefinite matrices invariant, thereby including covariance and second-order moment processes. Both the continuous-time and the discrete-time cases are first considered. In the LTV case, the obtained stability and stabilization conditions are expressed as differential and difference Lyapunov conditions which are equivalent, in the LTI case, to some spectral conditions for the generators of the processes. Convex stabilization conditions are also obtained in both the continuous-time and the discrete-time setting. It is proven that systems with constant delays are stable provided that the systems with zero-delays are stable—which mirrors existing results for linear positive systems. The results are then extended and unified into an impulsive formulation for which similar results are obtained. The proposed framework is very general and can recover and/or extend many of the existing results in the literature on linear systems related to (mean-square) exponential (uniform) stability. Several examples are discussed to illustrate this claim by deriving stability conditions for stochastic systems driven by Brownian motion and Poissonian jumps, Markov jump systems, (stochastic) switched systems, (stochastic) impulsive systems, (stochastic) sampled-data systems, and all their possible combinations.

Fabrication of 3D lignosulfonate composited sponges impregnated by BiVO4/polyaniline/Ag ternary photocatalyst for synergistic adsorption-photodegradation of fluoroquinolones in water

In this work, a 3D lignosulfonate composited poly(vinyl formal) (PVF) sponge impregnated by BiVO4/polyaniline (PANI)/Ag ternary photocatalyst (PLS-BiVO4/PANI/Ag) was designed and fabricated. The synergistic adsorption-photodegradation of five fluoroquinolones (FQs) antibiotics onto PLS-BiVO4/PANI/Ag was investigated. Incorporating lignosulfonate as a carrier on PVF sponge improved its structural properties and enhanced its adsorption performance toward FQs. The BiVO4/PANI/Ag ternary photocatalyst was well-dispersed and tightly bound to the composited sponge, thus it efficiently degraded the adsorbed FQs through surface photodegradation. PLS-BiVO4/PANI/Ag achieved >90% removal efficiency for all examined FQs within 2.0 h in a batch system, while it maintained a steady 80% removal efficiency for 30.0 h in a continuous dynamic treatment system. Photoelectric characterizations and ESR results showed that BiVO4/PANI/Ag was consistent with the type II heterostructure and effectively improved the utilization of visible light and the separation of photogenerated electron-hole pairs. The hole was the dominant active species in the degradation of FQs, while the main degradation pathways included defluorination, decarboxylation and destruction of piperazine ring. The prepared composited sponge is thus characterized by its high performance, convenient operation, eco-friendliness, and practicality in real water treatment systems since it undergoes self-purification and is easily recovered from water without mass loss.

Random dynamical systems with positive entropy imply second type of distributional chaos

In this paper, we investigate the complexity of random dynamical systems. Assume that ϕ is a continuous random dynamical system on Polish space X over an ergodic Polish system (Ω,F,P,θ). Let K be a ϕ-invariant random set with compact ω-section K(ω) such that ϕ(ω)(K(ω))=K(θω). We proved that if htop(ϕ,K)>0, then there exists chaotic phenomenon (strong second type of distributional chaos) in this random dynamical system.

Geometry of transit orbits in the periodically-perturbed restricted three-body problem

In the circular restricted three-body problem, low energy transit orbits are revealed by linearizing the governing differential equations about the collinear Lagrange points. This procedure fails when time-periodic perturbations are considered, such as perturbation due to the sun (i.e., the bicircular problem) or orbital eccentricity of the primaries. For the case of a time-periodic perturbation, the Lagrange point is replaced by a periodic orbit, equivalently viewed as a hyperbolic-elliptic fixed point of a symplectic map (the stroboscopic Poincaré map). Transit and non-transit orbits can be identified in the discrete map about the fixed point, in analogy with the geometric construction of Conley and McGehee about the index-1 saddle equilibrium point in the continuous dynamical system. Furthermore, though the continuous time system does not conserve the Hamiltonian energy (which is time-varying), the linearized map locally conserves a time-independent effective Hamiltonian function. We demonstrate that the phase space geometry of transit and non-transit orbits is preserved in going from the unperturbed to a periodically-perturbed situation, which carries over to the full nonlinear equations.

Persistence of Conley--Morse Graphs in Combinatorial Dynamical Systems

Multivector fields provide an avenue for studying continuous dynamical systems in a combinatorial framework. There are currently two approaches in the literature which use persistent homology to capture changes in combinatorial dynamical systems. The first captures changes in the Conley index, while the second captures changes in the Morse decomposition. However, such approaches have limitations. The former approach only describes how the Conley index changes across a selected isolated invariant set though the dynamics can be much more complicated than the behavior of a single isolated invariant set. Likewise, considering a Morse decomposition omits much information about the individual Morse sets. In this paper, we propose a method to summarize changes in combinatorial dynamical systems by capturing changes in the so-called Conley-Morse graphs. A Conley-Morse graph contains information about both the structure of a selected Morse decomposition and about the Conley index at each Morse set in the decomposition. Hence, our method summarizes the changing structure of a sequence of dynamical systems at a finer granularity than previous approaches.

Dynamics, Periodic Orbit Analysis, and Circuit Implementation of a New Chaotic System with Hidden Attractor

Hidden attractors are associated with multistability phenomena, which have considerable application prospects in engineering. By modifying a simple three-dimensional continuous quadratic dynamical system, this paper reports a new autonomous chaotic system with two stable node-foci that can generate double-wing hidden chaotic attractors. We discuss the rich dynamics of the proposed system, which have some interesting characteristics for different parameters and initial conditions, through the use of dynamic analysis tools such as the phase portrait, Lyapunov exponent spectrum, and bifurcation diagram. The topological classification of the periodic orbits of the system is investigated by a recently devised variational method. Symbolic dynamics of four and six letters are successfully established under two sets of system parameters, including hidden and self-excited chaotic attractors. The system is implemented by a corresponding analog electronic circuit to verify its realizability.

Construction of a gas condensate field development model

Abstract This article has developed and verified a mathematical aggregated approximate model of developing a gas condensate field using a cyclic process. The essence of the cyclic process is to pump the drained gas into the productive formation to reduce the pressure drop into the deposit. This process allows for increased condensate recovery in the future. The model discussed in this article is a continuous dynamic system with control parameters. It is a modification of the dynamic aggregated model of a purely gas field, designed for planning for a sufficiently long period with limited information about the state of the reservoir (the initial flow rate of wells, the initial recoverable gas reserves, the initial reservoir pressure, the dependence of potential condensate content per unit volume of fatty gas on the reservoir pressure). A non-standard approach underlies the model construction. Logical simplifications and a priori assumptions about the processes occurring in the field during its development are at its core. The instruments in the model are the increase in production and injection wells and the proportion of injection wells involved in the production. The purpose of the article is to calculate various variants of the dynamics of the fundamental indicators of the development of a gas condensate field for a sufficiently long-term period at the stage of preliminary design.

Implementasi Metode Beda Hingga Tak Standar untuk Model Penyebaran Campak

The measles distribution model is a system of differential equations that is included in a continuous dynamic system. This research focuses on transforming the continuous form into discrete form by discretization using non-standard finite difference and stability analysis which is then carried out by numerical simulations to prove its stability graphically. Based on the analysis, it is found that the measles distribution model which is assumed to have two fixed points, namely the disease-free fixed point (R_01) and the endemic fixed point (R_01), is stable. The stability of the two fixed points is proven by the Schur-Cohn criteria and is obtained stable with the condition 0ϕ(h)≤5 which meets the value of h0. The results of the numerical simulation show that the measles distribution model is dynamically consistent and tends to the fixed point. In addition, numerical simulations show that the larger the value of h, the more the graph tends to the fixed point.

When machine learning meets fractional-order chaotic signals: detecting dynamical variations

The challenge of classifying multivariate time series generated by discrete and continuous dynamical systems according to their chaotic or non-chaotic behavior has been studied extensively in the literature. The examination of noise or the variation of variables that affect a dynamic system's chaoticity will not be beneficial in analyzing structures employing random number generators (RNG) that are already assured to be chaotic. However, detecting the structural changes and their time intervals in deterministic systems with proven chaoticity can contribute to the literature in encryption applications. Machine Learning algorithms provide flexible possibilities to analyze and predict manipulations that may occur in the dynamics of chaotic and complex systems. This study proposes a deep Long-Short-Term-Memory (LSTM) network with a classification process to predict dynamical changes in a fractional-order chaotic (FOC) system. First, the appropriate system parameters are calculated to satisfy the chaotic behavior in the fractional-order Chen system. The predictive-corrective Adams-Bashforth-Moulton algorithm is used to simulate the FOC Chen system in the time domain. The Lyapunov exponents of the system were obtained according to the Wolf method. Next, three different scenarios have been designed to test and demonstrate the effectiveness of the proposed method. Synthetic FOC signals obtained after sub-sampling and statistical feature extraction processes fed the input of the deep bidirectional LSTM (BiLSTM) network to perform the training and testing process. The classification performance for "q" and "c" classes reaches 100% with the proposed model. The overall average testing accuracy, sensitivity, specificity, precision, F1 score and MCC are 98%, 98%, 99.3%, 98.1%, 98%, and 97.3%, respectively. Our results demonstrate the utility of using a deep BiLSTM network for detecting dynamical variations in nonlinear FOC systems.

Kinematic N-expansive continuous dynamical systems

Expansiveness has been used to study dynamic systems and has been developed for various forms of expansiveness. In this paper, we introduce the concept of kinematic N-expansiveness for flows on a C∞ compact connected manifold M, which is an extension of N-expansive homeomorphisms. We prove that if a vector field X on M is C1 robustly kinematic N-expansive, then it is quasi-Anosov. Furthermore, we consider the divergence-free vector fields and Hamiltonian systems with the kinematic N-expansive property; then, we study their robustness.

Development and Use of a Residence Time Distribution (RTD) Model Control Strategy for a Continuous Manufacturing Drug Product Pharmaceutical Process.

Residence-time-distribution (RTD)-based models are key to understanding the mixing dynamics of continuous manufacturing systems. Such models can allow for material traceability throughout the process and can provide the ability for removal of non-conforming material from the finished product. These models have been implemented in continuous pharmaceutical manufacturing mainly for monitoring purposes, not as an integral part of the control strategy and in-process specifications. This paper discusses the steps taken to develop an RTD model design space and how the model was statistically incorporated into the product’s control strategy. To develop the model, experiments were conducted at a range of blender impeller speeds and total system mass flow rates. RTD parameters were optimized for each condition tested using a tank-in-series-type model with a delay. Using the experimental RTD parameters, an equation was derived relating the mean residence time to the operating conditions (i.e., blender impeller speed and mass flow rate). The RTD parameters were used in combination with real-time upstream process data to predict downstream API concentration, where these predictions allowed validation across the entire operating range of the process by comparison to measured tablet assay. The standard in-process control limits for the product were statistically tightened using the validation acceptance criteria. Ultimately, this model and strategy were accepted by regulatory authorities.