In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-or...In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.展开更多
Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kau...Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps,breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.展开更多
This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincare section, t...This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincare section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.展开更多
We propose a theoretical model for spatial variations of the temperature varianceσ2(z,r)(z is the dis-tance from the sample bottom and r the radial coordinate)in turbulent Rayleigh-Bénard convection(RBC).Adaptin...We propose a theoretical model for spatial variations of the temperature varianceσ2(z,r)(z is the dis-tance from the sample bottom and r the radial coordinate)in turbulent Rayleigh-Bénard convection(RBC).Adapting the“attached-eddy”modelofshearflowtothe plumesofRBC,wederivedanequationforσ2 which is based on the universal scaling of the normalized RBC temperature spectra.This equation in-cludes both logarithmic and power-law dependences on z/λth,whereλth is the thermal boundary layer thickness.The equation parameters depend on r and the Prandtl number Pr,but have only an extremelyweak dependence on the Rayleigh number Ra Thus our model provides a near-universal equation for thetemperature variance profile in turbulent RBC.展开更多
This paper proposes a new chaotic system and its fractional-order chaotic system. The necessary condition for the existence of chaotic attractors in this new fractional-order system is obtained. It finds that this new...This paper proposes a new chaotic system and its fractional-order chaotic system. The necessary condition for the existence of chaotic attractors in this new fractional-order system is obtained. It finds that this new fractional-order system is chaotic for q 〉 0.783 if the system parameter m=6. The chaotic attractors for q=0.8, and q=0.9 are obtained. A circuit is designed to realize its fractional-order chaos system for q=0.9 by electronic workbench.展开更多
In this paper,using scalar feedback controller and stability theory of fractional-order systems,a gener-alized synchronization method for different fractional-order chaotic systems is established.Simulation results sh...In this paper,using scalar feedback controller and stability theory of fractional-order systems,a gener-alized synchronization method for different fractional-order chaotic systems is established.Simulation results show theeffectiveness of the theoretical results.展开更多
The permanence of a nonlinear higher order discrete time system from macroeconomics is studied, and a sufficient condition is proposed for the permanence of the system described by 11(,...,)nnnnkxrxfxx---=+ where :kfR...The permanence of a nonlinear higher order discrete time system from macroeconomics is studied, and a sufficient condition is proposed for the permanence of the system described by 11(,...,)nnnnkxrxfxx---=+ where :kfRR, the initial values 01,,kxx-are real numbers and [0,1)r is constant after exploring the relationship between this equation and 1(,...,)nnnkxfxx--= for certain classes of function f. As an application a short proof is given to a known result in a simpler way than ever reported.展开更多
In this paper, a very simple generalized synchronization method between different chaotic systems is presented. Only a scalar controller is used in this method. The method of obtaining the scalar controller from chaot...In this paper, a very simple generalized synchronization method between different chaotic systems is presented. Only a scalar controller is used in this method. The method of obtaining the scalar controller from chaotic systems is established. The sufficient and necessary condition of generalized synchronization is obtained from a rigorous theory, and the sufficient and necessary condition of generalized synchronization is irrelative to chaotic system itself. Theoretical analyses and simulation results show that the method established in this paper is effective.展开更多
The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differe...The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.展开更多
This study deals with the analytical investigation of oscillatory squeeze film flow through a Brinkman viscoelastic Oldroyd-B fluid-saturated porous layer subject to two vertically harmonically oscillatory disks.The v...This study deals with the analytical investigation of oscillatory squeeze film flow through a Brinkman viscoelastic Oldroyd-B fluid-saturated porous layer subject to two vertically harmonically oscillatory disks.The validity of the present proposed analytical solutions is first demonstrated for the Newtonian fluids when bothΛ_(1)andΛ_(2)tend to zero by comparison with the previous literature.Results demonstrate that an increase in the elasticity parameterΛ_(1)correlates with a rise in axial velocities,indicating that the relaxation timeΛ_(1)facilitates enhanced squeeze flow.In the case of squeeze film flow in porous layers,low oscillating frequencies exert minimal effects on axial velocities,independent of variations in the viscoelasticity parameterΛ_(1).However,at higher oscillating frequencies,axial velocities escalate with increasing the viscoelasticity parameterΛ_(1).Furthermore,the retardation timeΛ_(2)of the viscoelastic fluid shows no significant effect on the axial velocity,regardless of oscillating frequency changes in both pure fluids and porous layers.展开更多
In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)...In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)(Ω).In the framework of time-dependent spaces,we verify the existence of absorbing sets and the asymptotic compactness of the process,then we obtain the existence of the time-dependent global attractor A={A_t}t∈Rin Mt.Furthermore,we achieve the regularity of A,that is,A_(t) is bounded in M_(t)^(1) with a bound independent of t.展开更多
In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi...In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.展开更多
For the Cauchy problem to Keller–Segel system,we show well-posedness and time-decay estimates in the critical scaling-invariant Besov spaces by using Littlewood–Paley analysis together with the decay estimates of he...For the Cauchy problem to Keller–Segel system,we show well-posedness and time-decay estimates in the critical scaling-invariant Besov spaces by using Littlewood–Paley analysis together with the decay estimates of heat kernels.展开更多
We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation.What is more,each soliton molecule can be transformed into an asymmetr...We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation.What is more,each soliton molecule can be transformed into an asymmetric soliton by changing the parameterφ.In addition,the collision between soliton molecules(or asymmetric soliton)and several soliton solutions is observed.Finally,some related pictures are presented.展开更多
Laser induced periodic surface structures(LIPSS)represent a kind of top down approach to produce highly reproducible nano/microstructures without going for any sophisticated process of lithography.This method is much ...Laser induced periodic surface structures(LIPSS)represent a kind of top down approach to produce highly reproducible nano/microstructures without going for any sophisticated process of lithography.This method is much simpler and cost effective.In this work,LIPSS on Si surfaces were generated using femtosecond laser pulses of 800 nm wavelength.Photocatalytic substrates were prepared by depositing TiO2 thin films on top of the structured and unstructured Si wafer.The coatings were produced by sputtering from a Ti target in two different types of oxygen atmospheres.In first case,the oxygen pressure within the sputtering chamber was chosen to be high(3×10^–2 mbar)whereas it was one order of magnitude lower in second case(2.1×10^–3 mbar).In photocatalytic dye decomposition study of Methylene blue dye it was found that in the presence of LIPSS the activity can be enhanced by 2.1 and 3.3 times with high pressure and low pressure grown TiO2 thin films,respectively.The increase in photocatalytic activity is attributed to the enlargement of effective surface area.In comparative study,the dye decomposition rates of TiO2 thin films grown on LIPSS are found to be much higher than the value for standard reference thin film material Pilkington Activ^TM.展开更多
The design of observers for a class of practical physical chaotic systems is discussed.By using only one state variable and its time derivatives,a control law is constructed to achieve the synchronization between the ...The design of observers for a class of practical physical chaotic systems is discussed.By using only one state variable and its time derivatives,a control law is constructed to achieve the synchronization between the investigated chaotic systems and their observers,and the results are proved theoretically.Several observers of chaotic systems are designed by using this method.展开更多
The generation of an attosecond pulse in the ultraviolet range is described in the terms of the catastrophe theory. A simple criterion of tunneling is proposed. The criterion allows constructing the quasiclassical mod...The generation of an attosecond pulse in the ultraviolet range is described in the terms of the catastrophe theory. A simple criterion of tunneling is proposed. The criterion allows constructing the quasiclassical model of the generator of attosecond laser pulses based on the interaction of an electric field of extremely powerful femtosecond pulse with the valence electron in the potential well of the gas atom.展开更多
Sampling and replication operators are used for a description of the mode-locking radiation. Such description allows taking into account the influence of the shape of the gain curve of the active medium of the mode-lo...Sampling and replication operators are used for a description of the mode-locking radiation. Such description allows taking into account the influence of the shape of the gain curve of the active medium of the mode-locking laser on the form of the pulses generated by it.展开更多
Femtosecond laser ablation-driven periodic surface structuring offers a promising method for large-scale and high-throughput nanolithography technique.However,the self-organized periodic structures typically manifest ...Femtosecond laser ablation-driven periodic surface structuring offers a promising method for large-scale and high-throughput nanolithography technique.However,the self-organized periodic structures typically manifest constraints in terms of tunable period and depth,as well as suboptimal regularity,which restricts their broader application potential.Here,in terms of a rarely explored laser-induced photochemical mechanism for nonablative structuring,we demonstrate manufacturing of sub-wavelength oxidative grating structures on silicon films with active structural modulation.In this scenario,the plasmonic field plays a pivotal role in dragging oxygen ions from surface into the silicon,greatly speeding up oxidation rates.While high oxygen doping levels can already be achieved with single-pulse exposure,far superior results are obtained with the application of 40-MHz burst mode pulse trains,mitigating the formation of excessively large nanocrystallites.Furthermore,it is revealed that the periodicity and modulation depth of laser-writing nanograting are both dependent on the number of pulse per burst.This offers a convenient scheme for actively controlling laser plasmonic lithography.展开更多
文摘In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.
基金supported by the National Natural Science Foundation of China (project Nos. 11371086,11671258,11975145)the Fund of Science and Technology Commission of Shanghai Municipality (project No. 13ZR1400100)the Fund of Donghua University,Institute for Nonlinear Sciences and the Fundamental Research Funds for the Central Universities。
文摘Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps,breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10926072 and 10972082)Chongqing Municipal Education Commission(Grant No.KJ080515)Natural Science Foundation Project of CQ CSTC,China(GrantNo.2008BB2409)
文摘This paper studies chaotic dynamics in a fractional-order unified system by means of topological horseshoe theory and numerical computation. First it finds four quadrilaterals in a carefully-chosen Poincare section, then shows that the corresponding map is semiconjugate to a shift map with four symbols. By estimating the topological entropy of the map and the original time-continuous system, it provides a computer assisted verification on existence of chaos in this system, which is much more convincible than the common method of Lyapunov exponents. This new method can potentially be used in rigorous studies of chaos in such a kind of system. This paper may be a start for proving a given fractional-order differential equation to be chaotic.
基金the National Natural Science Foundation of China(Grants 11772111 and91952101)the Max Planck Partner Group.
文摘We propose a theoretical model for spatial variations of the temperature varianceσ2(z,r)(z is the dis-tance from the sample bottom and r the radial coordinate)in turbulent Rayleigh-Bénard convection(RBC).Adapting the“attached-eddy”modelofshearflowtothe plumesofRBC,wederivedanequationforσ2 which is based on the universal scaling of the normalized RBC temperature spectra.This equation in-cludes both logarithmic and power-law dependences on z/λth,whereλth is the thermal boundary layer thickness.The equation parameters depend on r and the Prandtl number Pr,but have only an extremelyweak dependence on the Rayleigh number Ra Thus our model provides a near-universal equation for thetemperature variance profile in turbulent RBC.
基金supported by Chongqing Education Committee of China (Grant No KJ070502)
文摘This paper proposes a new chaotic system and its fractional-order chaotic system. The necessary condition for the existence of chaotic attractors in this new fractional-order system is obtained. It finds that this new fractional-order system is chaotic for q 〉 0.783 if the system parameter m=6. The chaotic attractors for q=0.8, and q=0.9 are obtained. A circuit is designed to realize its fractional-order chaos system for q=0.9 by electronic workbench.
基金the Foundation of Chongqing Education Committee under Grant No.J070502
文摘In this paper,using scalar feedback controller and stability theory of fractional-order systems,a gener-alized synchronization method for different fractional-order chaotic systems is established.Simulation results show theeffectiveness of the theoretical results.
基金the Technology Research Foundation of the State Ministry of Education (No. 02130)
文摘The permanence of a nonlinear higher order discrete time system from macroeconomics is studied, and a sufficient condition is proposed for the permanence of the system described by 11(,...,)nnnnkxrxfxx---=+ where :kfRR, the initial values 01,,kxx-are real numbers and [0,1)r is constant after exploring the relationship between this equation and 1(,...,)nnnkxfxx--= for certain classes of function f. As an application a short proof is given to a known result in a simpler way than ever reported.
文摘In this paper, a very simple generalized synchronization method between different chaotic systems is presented. Only a scalar controller is used in this method. The method of obtaining the scalar controller from chaotic systems is established. The sufficient and necessary condition of generalized synchronization is obtained from a rigorous theory, and the sufficient and necessary condition of generalized synchronization is irrelative to chaotic system itself. Theoretical analyses and simulation results show that the method established in this paper is effective.
文摘The finite element method has established itself as an efficient numerical procedure for the solution of arbitrary-shaped field problems in space. Basically, the finite element method transforms the underlying differential equation into a system of algebraic equations by application of the method of weighted residuals in conjunction with a finite element ansatz. However, this procedure is restricted to even-ordered differential equations and leads to symmetric system matrices as a key property of the finite element method. This paper aims in a generalization of the finite element method towards the solution of first-order differential equations. This is achieved by an approach which replaces the first-order derivative by fractional powers of operators making use of the square root of a Sturm-Liouville operator. The resulting procedure incorporates a finite element formulation and leads to a symmetric but dense system matrix. Finally, the scheme is applied to the barometric equation where the results are compared with the analytical solution and other numerical approaches. It turns out that the resulting numerical scheme shows excellent convergence properties.
基金Project supported by the National Natural Science Foundation of China(No.12262026)the Natural Science Foundation of the Inner Mongolia Autonomous Region of China(No.2021MS01007)+1 种基金the Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region of China(No.NMGIRT2323)the Fundamental Research Funds for the Central Universities(Nos.2232022G-13,2232023G-13,and 2232024G-13)。
文摘This study deals with the analytical investigation of oscillatory squeeze film flow through a Brinkman viscoelastic Oldroyd-B fluid-saturated porous layer subject to two vertically harmonically oscillatory disks.The validity of the present proposed analytical solutions is first demonstrated for the Newtonian fluids when bothΛ_(1)andΛ_(2)tend to zero by comparison with the previous literature.Results demonstrate that an increase in the elasticity parameterΛ_(1)correlates with a rise in axial velocities,indicating that the relaxation timeΛ_(1)facilitates enhanced squeeze flow.In the case of squeeze film flow in porous layers,low oscillating frequencies exert minimal effects on axial velocities,independent of variations in the viscoelasticity parameterΛ_(1).However,at higher oscillating frequencies,axial velocities escalate with increasing the viscoelasticity parameterΛ_(1).Furthermore,the retardation timeΛ_(2)of the viscoelastic fluid shows no significant effect on the axial velocity,regardless of oscillating frequency changes in both pure fluids and porous layers.
基金supported by the National Natural Science Foundation of China(No.12171082)the Fundamental Research Funds for the Central Universities(No.2232023G-13).
文摘In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)(Ω).In the framework of time-dependent spaces,we verify the existence of absorbing sets and the asymptotic compactness of the process,then we obtain the existence of the time-dependent global attractor A={A_t}t∈Rin Mt.Furthermore,we achieve the regularity of A,that is,A_(t) is bounded in M_(t)^(1) with a bound independent of t.
基金Supported by the Science and Technology Commission of Shanghai Municipality (Grant No.23ZR1402100)the Fundamental Research Funds for the Central Universities (Grant Nos. 2232022G-13 and 2232023G-13)
文摘In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.
文摘For the Cauchy problem to Keller–Segel system,we show well-posedness and time-decay estimates in the critical scaling-invariant Besov spaces by using Littlewood–Paley analysis together with the decay estimates of heat kernels.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11371086, 11671258, and 11975145)the Fund of Science and Technology Commission of Shanghai Municipality, China (Grant No. 13ZR1400100)+1 种基金the Fund of Institute for Nonlinear Sciences, Donghua Universitythe Fundamental Research Funds for the Central Universities, China (Grant No. 2232021G-13)
文摘We investigate the techniques for velocity resonance and apply them to construct soliton molecules using two solitons of the extended Lax equation.What is more,each soliton molecule can be transformed into an asymmetric soliton by changing the parameterφ.In addition,the collision between soliton molecules(or asymmetric soliton)and several soliton solutions is observed.Finally,some related pictures are presented.
基金Deutsche Forschungsgemeinschaft (DFG), Germany (Grant number GR 1782/12)Science and Engineering Research Board (SERB), India (Grant number EMR/2015/001175)
文摘Laser induced periodic surface structures(LIPSS)represent a kind of top down approach to produce highly reproducible nano/microstructures without going for any sophisticated process of lithography.This method is much simpler and cost effective.In this work,LIPSS on Si surfaces were generated using femtosecond laser pulses of 800 nm wavelength.Photocatalytic substrates were prepared by depositing TiO2 thin films on top of the structured and unstructured Si wafer.The coatings were produced by sputtering from a Ti target in two different types of oxygen atmospheres.In first case,the oxygen pressure within the sputtering chamber was chosen to be high(3×10^–2 mbar)whereas it was one order of magnitude lower in second case(2.1×10^–3 mbar).In photocatalytic dye decomposition study of Methylene blue dye it was found that in the presence of LIPSS the activity can be enhanced by 2.1 and 3.3 times with high pressure and low pressure grown TiO2 thin films,respectively.The increase in photocatalytic activity is attributed to the enlargement of effective surface area.In comparative study,the dye decomposition rates of TiO2 thin films grown on LIPSS are found to be much higher than the value for standard reference thin film material Pilkington Activ^TM.
文摘The design of observers for a class of practical physical chaotic systems is discussed.By using only one state variable and its time derivatives,a control law is constructed to achieve the synchronization between the investigated chaotic systems and their observers,and the results are proved theoretically.Several observers of chaotic systems are designed by using this method.
文摘The generation of an attosecond pulse in the ultraviolet range is described in the terms of the catastrophe theory. A simple criterion of tunneling is proposed. The criterion allows constructing the quasiclassical model of the generator of attosecond laser pulses based on the interaction of an electric field of extremely powerful femtosecond pulse with the valence electron in the potential well of the gas atom.
文摘Sampling and replication operators are used for a description of the mode-locking radiation. Such description allows taking into account the influence of the shape of the gain curve of the active medium of the mode-locking laser on the form of the pulses generated by it.
基金supported by the National Natural Science Foundation of China(12474317 and 62105269).
文摘Femtosecond laser ablation-driven periodic surface structuring offers a promising method for large-scale and high-throughput nanolithography technique.However,the self-organized periodic structures typically manifest constraints in terms of tunable period and depth,as well as suboptimal regularity,which restricts their broader application potential.Here,in terms of a rarely explored laser-induced photochemical mechanism for nonablative structuring,we demonstrate manufacturing of sub-wavelength oxidative grating structures on silicon films with active structural modulation.In this scenario,the plasmonic field plays a pivotal role in dragging oxygen ions from surface into the silicon,greatly speeding up oxidation rates.While high oxygen doping levels can already be achieved with single-pulse exposure,far superior results are obtained with the application of 40-MHz burst mode pulse trains,mitigating the formation of excessively large nanocrystallites.Furthermore,it is revealed that the periodicity and modulation depth of laser-writing nanograting are both dependent on the number of pulse per burst.This offers a convenient scheme for actively controlling laser plasmonic lithography.
