Communications in Nonlinear Science and Numerical Simulation

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Using a combination of Riccati differential equation approach, Lyapunov-Krasovskii functional, inequality techniques, some new sufficient conditions for exponentially stability of the error system are formulated in form of a solution to the standard Riccati differential equation. The designed controller ensures that the synchronization of non-autonomous chaotic systems are proposed via delayed feedback control. Numerical simulations are presented to illustrate the effectiveness of these synchronization criteria.

In this paper, we consider the (n − 1, 1)-type integral boundary value problem of nonlinear fractional differential equationwhere are parameter and 0<μ<α, α∈(n-1,n] is a real number and n⩾3, is the Riemann–Liouville’s fractional derivative, are continuous and semipositone. We gave the corresponding Green’s function for the boundary value problem and its some properties. Moreover, we derive an interval of λ such that any λ lying in this interval, the semipositone boundary value problem has multiple positive solutions.Highlights► We study the integral boundary value problems for coupled systems of nonlinear semipositone fractional differential equations. ► To obtain the solution of the integral boundary value problems, derive the corresponding Green’s function for the boundary value problem and its some properties. ► Moreover, we give some sufficient conditions for the existence of positive solution of the semipositone equations.

The traditional Grey Model is easy to understand and simple to calculate, with satisfactory accuracy, but it is also lack of flexibility to adjust the model to acquire higher forecasting precision. This research studies feasibility and effectiveness of a novel Grey model together with the concept of the Bernoulli differential equation in ordinary differential equation. In this research, the author names this newly proposed model as Nonlinear Grey Bernoulli Model (NGBM). The NGBM is nonlinear differential equation with power index n. By controlling n, the curvature of the solution curve could be adjusted to fit the result of one time accumulated generating operation (1-AGO) of raw data. One extreme case from Grey system textbook is studied by NGBM, and two published articles are chosen for practical tests of NGBM. The results prove the novel NGBM is feasible and efficient. Finally, NGBM is used to forecast 2005 foreign exchange rates of twelve Taiwan major trading partners, including Taiwan.

This paper considers the classical problem of hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid. It is well known that similarity solutions of the energy equation are possible for the boundary conditions of constant surface temperature and constant heat flux. However, no such solution has been attempted for the convective surface boundary condition. The paper demonstrates that a similarity solution is possible if the convective heat transfer associated with the hot fluid on the lower surface of the plate is proportional to x−1/2. Numerical solutions of the resulting similarity energy equation are provided for representative Prandtl numbers of 0.1, 0.72, and 10 and a range of values of the parameter characterizing the hot fluid convection process. For the case of constant heat transfer coefficient, the same data provide local similarity solutions.

In this paper, a differential transform method (DTM) is used to find the numerical solution of a special 12th-order boundary value problems with two point boundary conditions. The analysis is accompanied by testing differential transform method both on linear and nonlinear problems from the literature [Wazwaz AM. Approximate solutions to boundary value problems of higher-order by the modified decomposition method. Comput Math Appl 2000:40;679–91; Siddiqi SS, Ghazala Akram. Solutions of 12th order boundary value problems using non-polynomial spline technique. Appl Math Comput 2007. doi:10.1016/j.amc.2007.10.015; Siddiqi SS, Twizell EH. Spline solutions of linear 12th-order boundary value problems. J Comput Appl Math 1997;78:371–90]. Numerical experiments and comparison with existing methods are performed to demonstrate reliability and efficiency of the proposed method.

A 135-sector inventory and embodiment analysis for carbon emissions and resources use by Chinese economy 2007 is presented in this paper by an ecological input–output modeling based on the physical entry scheme. Included emissions and resources belong to six categories as: (1) greenhouse gas (GHG) in terms of CO2, CH4, and N2O; (2) energy in terms of coal, crude oil, natural gas, hydropower, nuclear power, and firewood; (3) water in terms of freshwater; (4) exergy in terms of coal, crude oil, natural gas, grain, bean, tuber, cotton, peanut, rapeseed, sesame, jute, sugarcane, sugar beet, tobacco, silkworm feed, tea, fruits, vegetables, wood, bamboo, pulp, meat, egg, milk, wool, aquatic products, iron ore, copper ore, bauxite, lead ore, zinc ore, pyrite, phosphorite, gypsum, cement, nuclear fuel, and hydropower; (5) and (6) solar and cosmic emergies in terms of sunlight, wind power, deep earth heat, chemical power of rain, geopotential power of rain, chemical power of stream, geopotential power of stream, wave power, geothermal power, tide power, topsoil loss, coal, crude oil, natural gas, ferrous metal ore, non-ferrous metal ore, non-metal ore, cement, and nuclear fuel. Accounted based on the embodied intensities are carbon emissions and resources use embodied in the final use as rural consumption, urban consumption, government consumption, gross fixed capital formation, change in inventories, and export, as well as in the international trade balance. The resulted database is basic to environmental account of carbon emissions and resources use at various levels.

This paper describes the security weakness of a recently proposed image encryption algorithm based on a logistic-like new chaotic map. We show that the chaotic map’s distribution is far from ideal, thus making it a bad candidate as a pseudo-random stream generator. As a consequence, the images encrypted with this algorithm are shown to be breakable through different attacks of variable complexity.

Peristaltic transport in a two-dimensional non-uniform tube filled with Herschel–Bulkley fluid is studied under the assumptions of long wavelength and low Reynold number. The fluid flow is investigated in the wave frame of reference moving with the velocity of the peristaltic wave. Exact solution for the velocity field, the temperature profile, the stream functions and the pressure gradient are obtained. The physical behavior of τ, n, α and ϕ on the pressure rise versus flow rate are discussed through graphs. At the end of the article trapping phenomena for Herschel–Bulkley and also for Newtonian, Bingham and power law (which are the special cases of Herschel–Bulkley fluid) fluid are discussed.

In this paper, we adopt the homotopy analysis method (HAM) to obtain solutions of linear and nonlinear fractional diffusion and wave equation. The fractional derivative is described in the Caputo sense. Some illustrative examples are presented.

In this work, stability analysis of the fractional-order modified Autonomous Van der Pol–Duffing (MAVPD) circuit is studied using the fractional Routh–Hurwitz criteria. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than 3. Furthermore, the fractional Routh–Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria. Based on the fractional Routh–Hurwitz conditions and using specific choice of linear controllers, it is shown that the fractional-order MAVPD system is controlled to its equilibrium points; however, its integer-order counterpart is not controlled. Moreover, chaos synchronization of MAVPD system is found only in the fractional-order case when using a specific choice of nonlinear control functions. This shows the effect of fractional order on chaos control and synchronization. Synchronization is also achieved using the unidirectional linear error feedback coupling approach. Numerical results show the effectiveness of the theoretical analysis.

This paper is concerned with the practical complexity of the symbolic computation of limit cycles associated with Hilbert’s 16th problem. In particular, in determining the number of small-amplitude limit cycles of a non-linear dynamical system, one often faces computing the focus values of Hopf-type critical points and solving lengthy coupled polynomial equations. These computations must be carried out through symbolic computation with the aid of a computer algebra system such as Maple or Mathematica, and thus usually gives rise to very large algebraic expressions. In this paper, efficient computations for the focus values and polynomial equations are discussed, showing how to deal with the complexity in the computation of non-linear dynamical systems.

The embodiment of natural resources and greenhouse gas emissions for the urban economy of Beijing economy 2002 by a physical balance modeling is carried out based on an extension of the economic input–output table into an ecological one integrating the economy with its various environmental driving forces. Included resources and greenhouse gas emissions belong to six categories as energy resources in terms of primary energy and secondary energy; water resource; emissions of CO2, CH4, and N2O; exergy in terms of energy sources, biological resources and minerals; and solar emergy and cosmic emergy in terms of climate resources, soil, energy sources, and minerals.

This paper provides an integrated study on the ecological account for the Chinese economy in 2004 based on emergy synthesis theory. The detailed flows of the Chinese economy is diagramed, accounted and analyzed in categories using the biophysically based ecological accounting. Through calculating environmental and economic inputs within and outside the Chinese economy, this paper discusses the Chinese international exchange, describes the resource structure, and assesses its sustainability as a whole. Also, the comparison of systematic indicators, such as emergy/dollar ratio, environmental load ratio, and emergy self-support ratio, with those of the other countries is tabled and explored to illustrate the general status of the Chinese economy in the world. Take, for example, the environmental load ratio, which was 9.29 in China 2004, it reveals that the Chinese economy put high pressure on the local environment compared with those of the environment-benign countries, such as Brazil (0.75), Australia (0.86) and New Zealand (0.81). In addition, in this paper, the accounting method of tourism is adjusted based on the previous researches.

For the embodiment of natural resources and environmental emissions in Chinese economy 2005, a biophysical balance modeling is carried out based on an extension of the economic input–output table into an ecological one integrating the economy with its various environmental driving forces. Included resource flows into the primary resource sectors and environmental emission flows from the primary emission sectors belong to seven categories as energy resources in terms of fossil fuels, hydropower and nuclear energy, biomass, and other sources; freshwater resources; greenhouse gas emissions in terms of CO2, CH4, and N2O; industrial wastes in terms of waste water, waste gas, and waste solid; exergy in terms of fossil fuel resources, biological resources, mineral resources, and environmental resources; solar emergy and cosmic emergy in terms of climate resources, soil, fossil fuels, and minerals. The resulted database for embodiment intensity and sectoral embodiment of natural resources and environmental emissions is of essential implications in context of systems ecology and ecological economics in general and of global climate change in particular.

New exact solutions including bright soliton solutions, breather and periodic types of chirped soliton solutions, kink-wave and homoclinic wave solutions for the 2D Ginzburg–Landau equation are obtained using the special envelope transform and the auxiliary function method. It is shown that the specially envelope transform and the auxiliary function method provide a powerful mathematical tool for solving nonlinear equations arising in mathematical physics.

This paper presents the vortical and self-similar solutions for 2D compressible Euler equations using the separation method. These solutions complement Makino's solutions in radial symmetry without rotation. The rotational solutions provide new information that furthers our understanding of ocean vortices and reference examples for numerical methods. In addition, the corresponding blowup, time-periodic or global existence conditions are classified through an analysis of the new Emden equation. A conjecture regarding rotational solutions in 3D is also made.

This paper presents the application of coherent vortex simulation (CVS) filtering, based on an orthogonal wavelet decomposition of vorticity, to study mixing in 2D homogeneous isotropic turbulent flows. The Eulerian and Lagrangian dynamics of the flow are studied by comparing the evolution of a passive scalar and of particles advected by the coherent and incoherent velocity fields, respectively. The former is responsible for strong mixing and produces the same anomalous diffusion as the total flow, due to transport by the coherent vortices, while mixing in the latter is much weaker and corresponds to classical diffusion.

The use of active feedback control strategy is a common way to stabilize and control dangerous vibrations in vibrating systems and structures, such as bridges, highways, buildings, space and aircrafts. These structures are distributed-parameter systems. Unfortunately, the existing vibrations control techniques, even for these simplified models, are fraught with numerical difficulties and engineering limitations. In this paper, a negative velocity feedback is added to the dynamical system of twin-tail aircraft, which is represented by two coupled second-order nonlinear differential equations having both quadratic and cubic nonlinearities. The system describes the vibration of an aircraft tail subjected to multi-parametric excitation forces. The method of multiple time scale perturbation is applied to solve the nonlinear differential equations and obtain approximate solutions up to the third order approximations. The stability of the system is investigated applying frequency response equations. The effects of the different parameters are studied numerically. Some different resonance cases are investigated. A comparison is made with the available published work.

The purpose of this paper is to investigate the use of the 2× and 3× super-harmonic frequency components for detecting the presence of a single transverse breathing crack in a non-linear rotor system. This procedure is based on the detection of the super-harmonic frequency components of the non-linear dynamical behaviour at the associated sub-critical resonant peaks.The non-linear behaviour of the rotor system with a breathing crack is briefly analysed numerically: it will be illustrated that the effects of the crack size and location induce the variation of non-linear responses and the emerging of new resonance – antiresonance peaks of the cracked rotor at second, third and fourth harmonic frequency components. Then, the influence of the crack–unbalance interactions and more particularly the relative orientation between the front crack and the unbalance are also undertaken with considerations of various crack depths, and unbalance magnitudes. It is demonstrated that for a given crack depth, the unbalance does not only affect the vibration amplitude of the 1× amplitudes, but also the and sub-critical resonant peaks. Finally, it is illustrated that the emerging of super-harmonic frequency components provides useful information on the presence of cracks and may be used on an on-line crack monitoring rotor system. Using this methodology, the detection of small levels of damage may be easily undertaken.

This paper proposes the hyperchaotic system of 6th-order cellular neural network (CNN), realizes its synchronization based on state observer. In addition, a multi-ary number communication system based on this hyperchaotic system is given in this paper. This communication system has the features of large capacity of signals transmission and high security.

In this paper, the generalized Darboux transformation is established to the AB system, which mainly describes marginally unstable baroclinic wave packets in geophysical fluids and ultra-short pulses in nonlinear optics. We find a unified formula of Nth-order rogue wave solution for the AB system by the direct iterative rule. In particular, rogue waves possessing several free parameters from first to second order are calculated. The dynamics and some interesting structures of the rogue waves are illustrated through some figures.

We identify a contact transformation that linearizes the given equation in the Riccati and Abel chains of nonlinear scalar and coupled ordinary differential equations to the same order. The contact transformation is not of Cole-Hopf type and is \emph {new} to the literature. The linearization of Abel chain of equations is also demonstrated explicitly for the first time. The contact transformations can be utilized to derive dynamical symmetries of the associated nonlinear ODEs. The wider applicability of identifying this type of contact transformations and the method of deriving dynamical symmetries by using them is illustrated through a two dimensional generalization of the Riccati and Abel chains as well.

Flow induced in a viscoelastic fluid by a linearly stretched sheet is investigated assuming that the fluid is Maxwellian and the sheet is subjected to a transverse magnetic field. The objective is to investigate the effects of parameters such as elasticity number, magnetic number, radiative heat transfer, Prandtl number, and Eckert number on the temperature field above the sheet. To do this, boundary layer theory will be used to simplify energy and momentum equations assuming that fluid physical/rheological properties remain constant. A suitable similarity transformation will be used to transform boundary layer equations from PDEs into ODEs. Homotopy analysis method (HAM) will be invoked to find an analytical solution for the temperature field above the sheet knowing the velocity profiles (see Alizadeh-Pahlavan et al. [Alizadeh-Pahlavan A, Aliakbar V, Vakili-Farahani F, Sadeghy K. MHD flows of UCM fluids above porous stretching sheets using two-auxiliary parameter homotopy analysis method. Commun. Nonlinear Sci Numer Simulat, in press]). The importance of manipulating the transverse velocity component, v, will be discussed on the temperature field above the sheet.

In the present work, unsteady MHD flow of a Maxwellian fluid above an impulsively stretched sheet is studied under the assumption that boundary layer approximation is applicable. The objective is to find an analytical solution which can be used to check the performance of computational codes in cases where such an analytical solution does not exist. A convenient similarity transformation has been found to reduce the equations into a single highly nonlinear PDE. Homotopy analysis method (HAM) will be used to find an explicit analytical solution for the PDE so obtained. The effects of magnetic parameter, elasticity number, and the time elapsed are studied on the flow characteristics.

Analytical solutions for heat and mass transfer by laminar flow of a Newtonian, viscous, electrically conducting and heat generation/absorbing fluid on a continuously vertical permeable surface in the presence of a radiation, a first-order homogeneous chemical reaction and the mass flux are reported. The plate is assumed to move with a constant velocity in the direction of fluid flow. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a suction velocity varying with time. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. Graphical results for velocity, temperature and concentration profiles of both phases based on the analytical solutions are presented and discussed.

The effects of two photon absorption (TPA) and gain dispersion on soliton propagation in amplified medium are investigated. For finite gain bandwidth, the effect of gain dispersion becomes significant along with TPA and is treated as perturbation in fundamental soliton propagation. Including these perturbing effects an analytical expression of integrated intensity is formulated applying a completely new methodology by adopting Rayleigh’s dissipation function in the framework of variational approach. With classical analogy, the Euler–Lagrange equation in non-conservative system is used to solve the problem analytically. In order to justify the analytical prediction a numerical verification is made by split-step beam propagation method following Ginzburg–Landau equation.

This paper presents a new control strategy for a three phase PWM converter, which consists of applying an adaptive nonlinear control. The input–output feedback linearization approach is based on the exact cancellation of the nonlinearity, for this reason, this technique is not efficient, because system parameters can vary. First a nonlinear system modelling is derived with state variables of the input current and the output voltage by using power balance of the input and output, the nonlinear adaptive backstepping control can compensate the nonlinearities in the nominal system and the uncertainties. Simulation results are obtained using Matlab/Simulink. These results show how the adaptive backstepping law updates the system parameters and provide an efficient control design both for tracking and regulation in order to improve the power factor.

In order to reduce the computational amount and improve the computational precision for parameter optimization of Muskingum model, a new algorithm, Gray-encoded accelerating genetic algorithm (GAGA) is proposed. With the shrinking of searching range, the method gradually directs to an optimal result with the excellent individuals obtained by Gray genetic algorithm (GGA). The global convergence is analyzed for the new genetic algorithm. Its efficiency is verified by application of Muskingum model. Compared with the nonlinear programming methods, least residual square method and the test method, GAGA has higher precision. And compared with GGA and BGA (binary-encoded genetic algorithm), GAGA has rapider convergent speed.

The large-eddy simulation is used to study fluid mixing due to Rayleigh–Taylor instability in a time-dependent acceleration field. The numerical results show that the two fluids in the mixing layer move back towards their initial position if the acceleration has a sudden change in the direction during evolution process. If the direction inverse happens at the earlier stage while the amplitude of perturbation remains quite small, the two fluids may roughly resume their initial state. However, if the inverse happens at a rather late time and the instability has already evolved into the nonlinear stage, the two fluids can no longer be separated from each other, and there is no distinct interface between them.

We show that, in spatially periodic Hamiltonian systems driven by a time-periodic coordinate-independent (AC) force, the upper energy of the chaotic layer grows unlimitedly as the frequency of the force goes to zero. This remarkable effect is absent in any other physically significant systems. It gives rise to the divergence of the rate of the spatial chaotic transport. We also generalize this phenomenon for the presence of a weak noise and weak dissipation. We demonstrate for the latter case that the adiabatic AC force may greatly accelerate the spatial diffusion and the reset rate at a given threshold.

The effect of inner cylinder acceleration in the transition from circular Couette flow to Taylor vortex regime was numerically investigated. The solution of the eigenvalue problem when the Couette flow was perturbed allowed the determination of the relationship between the inner cylinder acceleration rate and the acceleration time duration expressed in terms of viscous diffusion time. The analysis of the numerical results also allowed the concept of quasi-steady acceleration in the Couette system to be qualitatively described.

Relativistic Vlasov–Maxwell model was developed to address the particle trapping or particle acceleration in phase space, including situation relevant to Stimulated Raman Scattering and plasma beatwave acceleration. In particular the case of realistically high ratio of driver frequency to plasma wave frequency (ω0/ωp∼30) was also considered using a hybrid version called Vlasov–Hilbert model. Some of the more striking features that have emerged from the Vlasov simulations are discussed in this paper, with particular emphasis on time-dependent matching relations in frequency and wavevector allowing the maintenance of the parametric resonance over a long time, particle dynamics in phase space and action transfer results obtained from the derivation of the integrated Manley–Rowe relations derived for a finite causal system.

We consider a hydrodynamic-type system of balance equations which is closed by the dynamic equation of state taking into account the effects of spatial nonlocality. Symmetry and local conservation laws of this system are studied. A system of ODEs being obtained via the group theory reduction of the initial system of PDEs is investigated. The reduced system is shown to possess a family of the homoclinic solutions. Depending on the values of the parameters, the homoclinic solutions describe the solitary waves of compression or rarefication. The solutions corresponding to the wave of compression are shown to be unstable. More likely, the waves of rarefication are stable. Numerical simulations demonstrate that the solitary waves of rarefication moving toward each other maintain their shape after the interaction.

For decision making in terms of environmental economics for wetland construction, restoration and preservation, net ecosystem services values of constructed, human-interfered and natural wetlands are explored in the present work as a comparative study. The ecosystem services values of a pilot constructed wetland in Beijing, China in different discount rates and time horizons are accounted and compared with those of the natural wetlands all over the world as a mean and of a typical human-interfered wetland in Wenzhou, China. Results show that in both finite and infinite time horizons considered, the constructed wetland has the largest net services value in a reasonable discount rate.

Urban economy is confronted with increasing biophysical limitations derived from the exhaustion of natural resources and the depletion of environmental capacity, and human cultural diversity has been declining during the fast urbanization. The conventional anthropocentric economics, regarding the natural environment as the ‘exterior’ of human economy, is invalid in the scientific evaluation on the contribution of natural resources and environment as well as human culture when facing the current urban crises. The theory of embodied cosmic exergy, as the latest development of ecological economics and ecological thermodynamics, is introduced in this study to construct an ecological evaluation framework of urban economy. The advantage of embodied cosmic exergy dedicated to ecological economics has been discussed in comparison with other ecological evaluation alternatives. Transformities describing hierarchies and manifesting quality are systematically calculated and tabulated. A new framework of embodied cosmic exergy based on network accounting (EmexNA) is sketched out in this study, taking not only diversity flows but also ecological stocks into consideration. The stock based concept of ‘ecological wealth’ and the flow based concept of ‘ecological cost’ as well as related evaluation indicators are developed based on EmexNA. Taking Beijing city as the case, the network accounting and related ecological evaluation of a practical urban economy are carried out in this study in the light of the basic social, economic and environmental data available from 1990 to 2005 of Beijing. The system construction and the ecological mechanism of the development of Beijing economy are correspondingly illuminated and discussed.

The study is concerned with data association of bearings-only multi-target tracking using two stationary observers in a 2-D scenario. In view of each target moving with a constant speed, two objective functions, i.e., distance and slope differences, are proposed and a multi-objective-ant-colony-optimization-based algorithm is then introduced to execute data association by minimizing the two objective functions. Numerical simulations are conducted to evaluate the effectiveness of the proposed algorithm in comparison with the data association results of the joint maximum likelihood (ML) method under different noise levels and track figurations.

The nonlinear theory of electrostatic dust-acoustic (DA) waves in a magnetized dusty plasma consisting of negatively charged mobile dusts, nonthermal fast electrons and trapped ions with vortex-like distribution is revisited. Previous theory in the literature [Phys. Plasmas {\bf 20}, 104505 (2013)] is rectified and put forward to include the effects of the external magnetic field, the adiabatic pressure of charged dusts as well as the obliqueness of propagation to the magnetic field. Using the reductive perturbation technique, a Korteweg-de Vries (KdV)-like equation is derived which governs the dynamics of the small-amplitude solitary waves in a magnetized dusty nonthermal plasma. It is found that due to the dust thermal pressure, there exists a critical value $(\beta_c)$ of the nothermal parameter $\beta (>1)$, denoting the percentage of energetic electrons, below which the DA solitary waves cease to propagate. The soliton solution (travelling wave) of the KdV-like equation is obtained, and is shown to be only of the rarefactive type. The properties of the solitons are analyzed numerically with the system parameters. The effects of the system parameters including the obliqueness of propagation $(l_z)$ and $\sigma$ on the dynamics of the DA solitons are also discussed numerically, and it is found that the soliton structures can withstand perturbations and turbulence during a considerable time. The results should be useful for understanding the nonlinear propagation of DA solitary waves in laboratory and space plasmas (e.g., Earth's magnetosphere, auroral region, heliospheric environments etc.).

The nonlinear dust-acoustic waves in an unmagnetized dusty plasma, including consideration of the dust charge variation, is analytically investigated by using the formally variable separation approach. The exact analytical solutions in the general case are also obtained.

Diffraction of acoustic plane wave through a semi-infinite hard duct which is placed symmetrically inside an infinite soft/hard duct has been analyzed rigorously. Convective flow has been taken into consideration for the analysis. In this paper the applied method of solution is integral transform and Wiener-Hopf technique. The imposition of boundary conditions result in a 2×2 matrix Wiener-Hopf equation associated with a new canonical scattering problem which has been solved explicitly by expansion coefficient method. The graphs are plotted for sundry parameters of interest. Kernel functions are factorized. The results have applications to duct acoustics.

The nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution are analytically investigated by using the formally variable separation approach. New analytical solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma for the first time. We derive exact analytical expressions for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with vortex-like ion distribution.

The nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation is investigated by using the formally variable separation approach. New solutions for the governing equation of this system have been obtained for dust acoustic waves in a dust plasma firsthand. We derive exact mathematical expressions and numerical simulation studies for the general case of the nonlinear dust acoustic waves in two-dimensional dust plasma with dust charge variation.

Wave chaos is demonstrated by studying a wave propagation in a periodically corrugated wave-guide. In the limit of a short wave approximation (SWA) the underlying description is related to the chaotic ray dynamics. In this case the control parameter of the problem is characterized by the corrugation amplitude and the SWA parameter. The considered model is fairly suitable and tractable for the analytical analysis of a wave localization length. The number of eigenmodes characterized the width of the localized wave packet is estimated analytically.

We consider three-dimensional free-boundary problem on the propagation of incompressible, homogeneous and inviscid fluid with zero surface tension confined in a channel of variable depth. Since for large-scale flows the fluid motion is affected by the rotation of the earth, the model is considered in rotating reference frame. Additionally, small atmospheric pressure variations across the channel are taken into account. It is shown that the non-trivial solution to the problem represents three-dimensional solitary wave which is given by the rotation modified Korteweg-de Vries equation (fKdV):b1ξxxx+b2ξξx+b3(f)ξx=0,where x is the down-channel coordinate and the coefficients bi(i=1,2,3) of the resulting fKdV equation depend on the transverse topography of the channel and, additionally, b3 depends on the Coriolis parameter f. It is also shown that if the vertical profile of the channel is symmetric about the vertical axis, the small atmospheric variations will not appear in the resulting fKdV equation. The effects of channel’s cross-sectional geometry on the shape of the resulting three-dimensional wave profile in a longitudinal direction are studied numerically. Additionally, to better understand the effects of the Earth rotation, the above analysis is performed at different latitudes.

Transport in near-integrable, but partially chaotic, $1 1/2$ degree-of-freedom Hamiltonian systems is blocked by invariant tori and is reduced at \emph{almost}-invariant tori, both associated with the invariant tori of a neighboring integrable system. "Almost invariant" tori with rational rotation number can be defined using continuous families of periodic \emph{pseudo-orbits} to foliate the surfaces, while irrational-rotation-number tori can be defined by nesting with sequences of such rational tori. Three definitions of "pseudo-orbit," \emph{action-gradient--minimizing} (AGMin), \emph{quadratic-flux-minimizing} (QFMin) and \emph{ghost} orbits, based on variants of Hamilton's Principle, use different strategies to extremize the action as closely as possible. Equivalent Lagrangian (configuration-space action) and Hamiltonian (phase-space action) formulations, and a new approach to visualizing action-minimizing and minimax orbits based on AGMin pseudo-orbits, are presented.

Flexible paraboloidal shells, as key components, are increasingly utilized in antennas, reflectors, optical systems, aerospace structures, etc. To explore precise shape and vibration control of the paraboloidal membrane shells, this study focuses on analysis of microscopic control actions of segmented actuator patches laminated on the surface of a free paraboloidal membrane shell. Governing equations of the membrane shell system and modal control forces of distributed actuator patches are presented first, and followed by the analysis of dominating micro-control actions based on various natural modes, actuator locations and geometrical parameters. Finally, according to the parametric analysis, simulation data reveal main factors significantly influencing active control behavior on smart free-floating paraboloidal membrane shell systems, thus providing design guidelines to achieve optimal control of paraboloidal shell systems.

This paper investigates the global asymptotic stability of equilibrium for a class of continuous-time neural networks with delays. Based on suitable Lyapunov functionals and the homeomorphism theory, some sufficient conditions for the existence and uniqueness of the equilibrium point are derived. These results extend the previously works without assuming boundedness and Lipschitz conditions of the activation functions and any symmetry of interconnections. A numerical example is also given to show the improvements of the paper.

In this paper, the multistability is studied for two-dimensional neural networks with multilevel activation functions. And it is showed that the system has n2 isolated equilibrium points which are locally exponentially stable, where the activation function has n segments. Furthermore, evoked by periodic external input, n2 periodic orbits which are locally exponentially attractive, can be found. And these results are extended to k-neuron networks, which is really enlarge the capacity of the associative memories. Examples and simulation results are used to illustrate the theory.

The purpose of this paper is to investigate possible impact of uncertainty on dynamical evolution of an activator–repressor circuit model, for some gene regulatory networks. Escape probability and mean residence time are computed in order to gain insights on the role played by random fluctuations. Some changes or bifurcations in mean residence time are also observed when key model parameters vary.

In this article we analyze the linear stability of nonlinear time-fractional reaction–diffusion systems. As an example, the reaction–subdiffusion model with cubic nonlinearity is considered. By linear stability analysis and computer simulation, it was shown that fractional derivative orders can change substantially an eigenvalue spectrum and significantly enrich nonlinear system dynamics. A overall picture of nonlinear solutions in subdiffusive reaction–diffusion systems is presented.Highlights► Stability of subdiffusive reaction–diffusion system is analyzed. ► Conditions for time and spatial pattern formation are determined. ► It was shown that fractional derivative orders change substantially an eigenvalue spectrum and nonlinear system dynamics. ► A overall picture of nonlinear solutions in subdiffusive regime is presented.

In this paper, a new reliable algorithm based on an adaptation of the standard homotopy analysis method (HAM) is presented, which is the multistage homotopy analysis method (MSHAM). The freedom of choosing the auxiliary linear operator and the auxiliary parameter are still present in the MSHAM. The solutions of the non-chaotic and the chaotic Chen system which is a three-dimensional system of ordinary differential equations with quadratic nonlinearities were obtained by MSHAM. Numerical comparisons between the MSHAM and the classical fourth-order Runge–Kutta (RK4) numerical solutions reveal that the new technique is a promising tool for solving the non-linear chaotic and non-chaotic Chen system.