Differential Dynamical Systems Research Articles

Municipal Solid Waste(MSW) Management in Urbanization Process Based on Dynamical Model

It was announced by Chinese government at the Copenhagen Conference that there would be 40-45% mitigation of the 2005 level intensity of carbon by 2020. Every industry in China is preparing to realize this national reduction target. A lot of work have been tried to achieve low-carbon development in different industries, but relatively little work has linked lowcarbon development to municipal solid waste management(MSWM). This article focuses on how to develop low-carbon MSWM using a quantitative approach. Firstly, the MSWM system including some mutual influence factors is investigated and some historical data are given in support for the research of their quantitative relationship. Secondly, a differential dynamic system model with fuzzy coefficients is proposed to predict waste amount, disposal cost, energy consumption and carbon intensity. Finally, an application to Chengdu in southwestern China, as a representative of a developed city, is presented to show the trend of MSWM in a low-carbon economy and prove the effectiveness of the proposed model.

Preface

As we all know, many biological and physical systems, such as neuronal systems and disease systems, are featured by certain nonlinear and complex patterns in their elements and networks. These phenomena carry significant biological and physical information and regulate down-stream mechanism in many instances. This issue of Discrete and Continuous Dynamical Systems, Series S, comprises a collection of recent works in the general area of nonlinear differential equations and dynamical systems, and related applications in mathematical biology and engineering. The common themes of this issue include theoretical analysis, mathematical models, computational and statistical methods on dynamical systems and differential equations, as well as applications in fields of neurodynamics, biology, and engineering etc. <br>&nbsp; &nbsp Research articles contributed to this issue explore a large variety of topics and present many of the advances in the field of differential equations, dynamical systems and mathematical modeling, with emphasis on newly developed theory and techniques on analysis of nonlinear systems, as well as applications in natural science and engineering. These contributions not only present valuable new results, ideas and techniques in nonlinear systems, but also formulate a few open questions which may stimulate further study in this area. We would like to thank the authors for their excellent contributions, the referees for their tireless efforts in reviewing the manuscripts and making suggestions, and the chief editors of DCDS-S for making this issue possible. We hope that these works will help the readers and researchers to understand and make future progress in the field of nonlinear analysis and mathematical modeling.

On Nonnegative Solutions of Fractionalq-Linear Time-Varying Dynamic Systems with Delayed Dynamics

This paper is devoted to the investigation of nonnegative solutions and the stability and asymptotic properties of the solutions of fractional differential dynamic linear time-varying systems involving delayed dynamics with delays. The dynamic systems are described based onq-calculus and Caputo fractional derivatives on any order.

Applied Mathematical Theory for Monetary-Fiscal Interaction in a Supranational Monetary Union

I utilize a differentiable dynamical system a la Lotka-Voletrra and explain monetary and fiscal interaction in a supranational monetary union. The paper demonstrates an applied mathematical approach that provides useful insights about the interaction mechanisms in theoretical economics in general and a monetary union in particular. I find that a common central bank is necessary but not sufficient to tackle the new interaction problems in a supranational monetary union, such as the free-riding behaviour of fiscal policies. Moreover, I show that supranational institutions, rules or laws are essential to mitigate violations of decentralized fiscal policies.

Stability and boundedness of differential equations relative to initial time difference by employing vector Lyapunov functions

The existing stability results relative to initial time difference are not easy to be applied into practice since it is too hard to find a Lyapunov function satisfying all the conditions. This paper presents several quasi-stability criteria for nonlinear differential dynamical systems relative to initial time difference by employing vector Lyapunov functions. These new results might be more fruitful since it can satisfy less rigid requirements. As an application, the results are applied into a nonlinear dynamical system to get the quasi-stability properties.

GENERIC DIFFEOMORPHISMS WITH ROBUSTLY TRANSITIVE SETS

Let <TEX>${\Lambda}$</TEX> be a robustly transitive set of a diffeomorphism <TEX>$f$</TEX> on a closed <TEX>$C^{\infty}$</TEX> manifold. In this paper, we characterize hyperbolicity of <TEX>${\Lambda}$</TEX> in <TEX>$C^1$</TEX>-generic sense.

A Direct Method for Searching Power System Second Order Resonance Point

The dynamic characteristics of the power system will be a sharp deterioration near the second order resonance point. In-depth study of the Power system dynamic characteristics near the second order resonance point, and then how to cause the dynamic characteristics of the system worse at the second order resonance point near the stable boundary, need to quickly search system of second order resonance point. Based on the principle of minimal expansion system method, this paper presents the direct method for search the second order resonance point in power system. For the N dimensions differential dynamic system was established in the 4N+4 dimension expansion equations. The power system second order resonance point search is changed into expanding equation fixed point solution. Significantly reduce computational load, improve the search accuracy for solving second order resonance point calculation. For in-depth study of power system dynamic characteristics near the second order resonance point have a certain value.

Qualitative analysis of the chemostat model with variable yield and a time delay

In this paper, we consider the chemostat model with inhibitory exponential substrate, variable yield and a time delay. A detailed qualitative analysis about existence and boundedness of its solutions and the local asymptotic stability of its equilibria are carried out. The Hopf bifurcation of solutions to the system is studied. Using Lyapunov–LaSalle invariance principle, we show that the washout equilibrium is global asymptotic stability for any time delay. Based on some known techniques on limit sets of differential dynamical systems, we show that, for any time delay, the chemostat model is permanent if and only if only one positive equilibrium exits.

Neurodynamic System Theory and Applications

Neurodynamical systems have gradually become a popular research topic owing to their broad applications in such fields as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision. In view of some inevitable factors, there have been formed various neurodynamical systems including delayed neural networks, stochastic neural networks, impulsive neural networks, reaction-diffusion neural networks, and fuzzy neural networks. Over the last few decades, considerable attention has been devoted to this research area not only for enriching the theory of differential equations and dynamical systems but also for deeply understanding the dynamic states of neural networks for better modelling the brain. The current special issue puts its emphasis on the study of neurodynamical system theory and applications. Call for papers has been carefully prepared by the guest editors and posted on the journal’s web page, which has received many attentions followed by some submissions among wide topics such as delayed neural systems, stochastic neural systems, impulsive neural systems, reaction-diffusion neural systems, fuzzy neural systems, evolutionary neural systems, mathematical modeling of neural systems, computational neuroscience, neurodynamical optimization and adaptive dynamic programming, cognitive models, pattern recognition, and neural network applications. All manuscripts submitted to this special issue went through a thorough peer-refereeing process. Based on the reviewers’ reports, eleven original research articles are finally accepted. The contents embrace the synchronization of coupled neural networks, the numerical analysis of stochastic delayed partial differential equations, and the stability analysis of delayed impulsive reaction-diffusion neural networks and switched neural networks. It is certainly impossible to provide in this short editorial a more comprehensive description for all articles in this special issue. However, the team of the guest editors believes that the results included reflect some recent trends in research and outline new ideas for future studies of neurodynamical system theory and applications.

Dynamical Aspects of Initial/Boundary Value Problems for Ordinary Differential Equations

Dynamical aspects of initial/boundary value problems for ordinary differential equations have become a rapidly growing area of research in the theory of differential equations and dynamical systems and have gathered substantial research interests during the last decades. The attractiveness of this field not only is derived from theoretical interests but also is motivated by the insights that such dynamical aspects could reveal in several phenomena observed in applied sciences. The current special issue places its emphasis on the study of the dynamical aspects of initial/boundary value problems for ordinary differential equations. Call for papers has been carefully prepared by the guest editors and posted on the journals web page, which has attracted many researchers to submit their contribution on wide topics such as oscillation theory, delay differential equation, impulsive differential equation, multipoint boundary value problems, stochastic mutualism system, chaotic system, homoclinic solutions, Hamiltonian systems, stability and bifurcation, exponential extinction, singular elliptic problem, nonuniform exponential contraction and dichotomy. All manuscripts submitted to this special issue went through a thorough peer-refereeing process. Based on the reviewers’ reports, we collect twenty-five original research articles by more than fifty active international researchers in differential equations and from different countries such as Korea, China, Malaysia, Singapore, Czech Republic, Turkey, Slovenia, India, and USA. Besides, one survey on recent results for the existence of singular periodic problems is also contained. It is certainly impossible to provide in this short editorial note a more comprehensive description for all articles in this special issue. However, the team of the guest editors believes that the results included reflect some recent trends in research and outline new ideas for future studies of dynamical aspects of initial/boundary value problems for ordinary differential equations.

Stability and bifurcation of epidemic spreading on adaptive network

Adaptive network is characterized by feedback loop between states of nodes and topology of the network. In this paper, for adaptive epidemic spreading model, epidemic spreading dynamics is studied by using a nonlinear differential dynamic system. The local stability and bifurcation behavior of the equilibrium in this network model are investigated and all kinds of bifurcation point formula are obtained by analyzing its corresponding characteristic equation of Jacobian matrix of the nonlinear system. It is shown that, when the epidemic threshold is less than epidemic persistence threshold R00c, the disease always dies out and the disease-free equilibrium is asymptotically locally stable. If R0c01, a backward bifurcation leading to bistability possibly occurs, and there are possibly three equilibria: a stable disease-free equilibrium, a larger stable endemic equilibrium, and a smaller unstable endemic equilibrium. If R01, the disease is uniformly persistent and only one endemic equilibrium is asymptotically locally stable. It is also found that the system has saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Numerical simulations are given to verify the results of theoretical analysis.

Computing covariant Lyapunov vectors, Oseledets vectors, and dichotomy projectors: A comparative numerical study

Covariant Lyapunov vectors or Oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations, nonautonomous differentiable dynamical systems, and random dynamical systems. These vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles. In recent years new computational methods for approximating Oseledets vectors have been developed, motivated by increasing model complexity and greater demands for accuracy. In this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of Ginelli et al. (2007) [36] and Wolfe and Samelson (2007) [37]. We compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry, spectral separation, and dimension. We also investigate which methods perform well with limited data.

Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property

We study weak hyperbolicity of a differentiable dynamical system which is robustly free of non-hyperbolic periodic orbits of Markus type. Let S be a $\mathrm{C}^1$-class vector field on a closed manifold $M^n$, which is free of any singularities. It is of $\mathrm{C}^1$-weak-star in case there exists a $\mathrm{C}^1$-neighborhood $\mathscr{U}$ of S such that for any X$\in\mathscr{U}$, if $P$ is a common periodic orbit of X and S with S$_{\upharpoonright P}=$X$_{\upharpoonright P}$, then $P$ is hyperbolic with respect to X. We show, in the framework of Liao theory, that S possesses the $\mathrm{C}^1$-weak-star property if and only if it has a natural and nonuniformly hyperbolic dominated splitting on the set of periodic points $\mathrm{Per}$(S), for the case $n=3$.

SRB attractors with intermingled basins for non-hyperbolic diffeomorphisms

We investigate a class of non-hyperbolic diffeomorphisms defined onthe product space. By using the Pesin theory combined with thegeneral theory of differentiable dynamical systems, we prove thatthere are exactly two SRB attractors, and their basins cover a fullmeasure subset of the ambient manifold. Furthermore, we prove that the basins ofSRB attractors have the strange intermingled phenomenon, i.e. theyare measure-theoretically dense in each other. The intermingledphenomena have been observed in many physical systems by numericalexperiments, and considered to be important to some fundamentalproblems in physical, biology and computer science etc. Finally, wedescribe a concrete example for application.

Local structure of self-affine sets

AbstractThe structure of a self-similar set with the open set condition does not change under magnification. For self-affine sets, the situation is completely different. We consider self-affine Cantor sets $E\subset \mathbb {R}^2$ of the type studied by Bedford, McMullen, Gatzouras and Lalley, for which the projection onto the horizontal axis is an interval. We show that in small square $\varepsilon $-neighborhoods $N$ of almost each point $x$ in $E,$ with respect to many Bernoulli measures on the address space, $E\cap N$ is well approximated by product sets $[0,1]\times C$, where $C$ is a Cantor set. Even though $E$ is totally disconnected, all tangent sets have a product structure with interval fibers, reminiscent of the view of attractors of chaotic differentiable dynamical systems. We also prove that $E$has uniformly scaling scenery in the sense of Furstenberg, Gavish and Hochman: the family of tangent sets is the same at almost all points$x.$

Dynamics of vibration isolation system obeying fractional differentiation

PurposeThe purpose of this paper is to analyze a new, whole‐spacecraft isolator and its performance of vibration isolation, which has been designed to ensure spacecraft safety at the launching stage.Design/methodology/approachThe design is based on the analysis of fractional derivative stress‐strain constitutive relationship of viscoelastic materials. First, the authors study the constitutive relationships for viscoelastic solid of the damping materials, then the authors introduce the results obtained to the equations of motion for the damped isolator.FindingsBy performing a series of transformation, the authors obtain the analytical solution of the equations. It is shown that the results compare favourably to the numerical simulations and experiments. In addition, a saturation phenomenon for the first order damping ratio is also discussed.Practical implicationsIt is found that the constitutive relationships written in terms of the fractional calculus can be applied in the system function of the whole‐spacecraft vibration isolator. Such relationships, developed previously from a model analysis base, have been shown to be useful tools for engineering analyses.Originality/valueSome suggestions are given to improve the design of viscoelastic whole‐spacecraft isolators. The establishment of a theoretical basis for the new fractional differential dynamical system enhances their value, as they may now be used with increased reliability of satellite.

Epidemic spreading model and stability of the networks in mobile environment

In this paper, considering the random motion of nodes, we propose an epidemic spreading model on network of mobile environment based on the mean-filed theory. Using the theory of differential dynamical system, the behaviors of epidemic spreading are further analyzed. It is shown that if R0 1, the epidemic always dies out and the disease-free equilibrium is globally asymptotically stable; and if R0 1, the epidemic is uniformly persistent and the endemic equilibrium is globally asymptotically stable. Numerical simulations are given to validate the results of theoretical analysis.

Solution to time fractional partial differential equations and dynamical systems via integral transforms

In this article, the authors solved certain non-homogeneous time fractional heat equations and certain dynamical systems using joint integral transforms. Transform method is a powerful tool for solving partial fractional differential equations. The result reveals that the transform method is very convenient and effective.

Linear matrix differential dynamical systems with fuzzy matrices

In this paper, we investigate linear first-order fuzzy matrix differential dynamical systems where the coefficients matrix is described by a fuzzy matrix. We show some properties of the matrix differential dynamical systems, and their phase portraits are described by means of examples.

Simulation of low-carbon tourism in world natural and cultural heritage areas: An application to Shizhong District of Leshan City in China

The national goal of 40–45% mitigation of the 2005 level intensity of carbon by 2020 was announced by the Chinese government at the Copenhagen Conference. Every industry in China is preparing to realize this national reduction target. Some attempts have been made to achieve low-carbon development in a few industries, but relatively little work has linked low-carbon development to tourism. This article concentrates on how to develop low-carbon tourism using a quantitative approach. Firstly, the tourism system including some mutual influence factors is investigated and some historical data are given in support for the research of their quantitative relationship. Secondly, a differential dynamic system model with fuzzy coefficients is proposed to predict tourism revenue, energy consumption, waste emissions and the carbon intensity. Finally, an application to Shizhong District of Leshan City in China (LCSD), as a representative of a world natural and cultural heritage area, is presented to show the trend of modern tourism in a low-carbon economy and prove the effectiveness of the proposed model.