An air gun generates acoustic signals for seismic exploration by releasing a high-pressure gas.A large error is always gradually introduced into the ideal-gas model when the pressure in the air-gun chamber exceeds 100...An air gun generates acoustic signals for seismic exploration by releasing a high-pressure gas.A large error is always gradually introduced into the ideal-gas model when the pressure in the air-gun chamber exceeds 100 atm.In the van der Waals non-ideal-gas theory,the gas in the air gun can be regarded as an actual gas,and the error is less than 2%.The van der Waals model is established in combination with the quasi-static open thermodynamic system and bubble-motion equation by considering the bubble rise,bubble interaction,and throttling eff ect.The mismatch between the van der Waals and ideal-gas models is related to the pressure.Theoretically,under high-pressure conditions,the van der Waals air-gun model yields results that are closer to the measured results.Marine vertical cables are extended to the seafl oor using steel cables that connect the cement blocks,but the corresponding hydrophones are suspended in the seawater.Thus,noise associated with ships,ocean surges,and coupling problems is avoided,and the signal-to-noise ratio and resolution of marine seismic data are improved.This acquisition method satisfies the conditions of recording air-gun far-fi eld wavelets.According to an actual vertical-cable observation system,the van der Waals air-gun model is used to model the wavelet of different azimuth and take-off angles.The characteristics of the experimental and simulated data demonstrate good agreement,which indicates that the van der Waals method is accurate and reliable.The accuracy of the model is directly related to the resolution,thus aff ecting the resolution ability of the stratum.展开更多
An air-gun source is the most commonly used excitation method in off shore seismic exploration.The excitation characteristics of an air-gun source aff ect seismic data quality.Far-field wavelet simulation is an import...An air-gun source is the most commonly used excitation method in off shore seismic exploration.The excitation characteristics of an air-gun source aff ect seismic data quality.Far-field wavelet simulation is an important approach to study these characteristics.Compared to the measured wavelet,far-field wavelet simulation based on a traditional bubble-motion equation and ideal gas wavelet model has some disadvantages,such as a greater amplitude and smaller pulse attenuation velocity.Here,we start from the linear acoustic wave equation in the spherical coordinate system to deduce an improved,simpler bubble-motion equation and develop a Van der Waals gas wavelet model based on this equation.Unlike the existing methods,our method considers the high-pressure environment during actual excitation,heat exchange between the bubble and outside water,and change in the air fl ow at the muzzle.The results show that the far-fi eld wavelet simulated using this model is closer to the measured wavelet than that of the ideal gas wavelet model.At the same time,our method has a more succinct equation and a higher calculation effi ciency.展开更多
The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic cha...The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic chaotic vibration by means of the total variation theory. Some results are derived on the exponential growth of total variation of the snapshots on the spatial interval in the long-time horizon when the map and the initial condition satisfy some conditions.展开更多
In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitatio...In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.展开更多
In this research work, Homotopy perturbation method (HPM) is applied to find the approximate solution of the Van der Pol Differential equation (VDPDE), which is a well-known nonlinear ODE. Firstly, the approximate sol...In this research work, Homotopy perturbation method (HPM) is applied to find the approximate solution of the Van der Pol Differential equation (VDPDE), which is a well-known nonlinear ODE. Firstly, the approximate solution of Van Der Pol equation is developed using Dirichlet boundary conditions. Then a comparison between the present results and previously published results is presented and a good agreement is observed. Finally, HPM method is applied to find the approximate solution of VDPDE with Robin and Neumann boundary conditions.展开更多
In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a...In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.展开更多
本文针对含有自激励,参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析,发现其准周期运动的频谱中含有均匀边频带这一新的特性.首先,采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mat...本文针对含有自激励,参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析,发现其准周期运动的频谱中含有均匀边频带这一新的特性.首先,采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应,得到了其非线性频率响应曲线;再利用Floquet理论对周期解进行稳定性分析,得到了两种类型的分岔及它们的位置.然后,基于van der Pol-Mathieu方程准周期运动的频谱中边频带相邻频率之间是等距的且含有两个不可约的基频的特性(其中一个基频是已知的,另一个基频事先是未知的),推导了相应的两时间尺度IHB法,精确计算出van der Pol-Mathieu方程的准周期运动的另一个未知基频和所有的频率成份及其对应的幅值,尤其在临界点附近处的准周期运动响应.得到的准周期运动结果和利用四阶龙格-库塔(RK)数值法得到的结果高度吻合.最后,研究发现了含外激励van der Pol-Mathieu方程在不同激励频率时的一些丰富而有趣的非线性动力学现象.展开更多
以含分数阶微分项的van der Pol-Mathieu方程为对象,研究了谐波激励作用下主共振的动力学行为和稳定性。采用平均法得到了方程近似解析解,通过数值方法验证了解析结果的准确性。建立了系统稳态响应的幅频方程,利用Lyapunov第一方法得到...以含分数阶微分项的van der Pol-Mathieu方程为对象,研究了谐波激励作用下主共振的动力学行为和稳定性。采用平均法得到了方程近似解析解,通过数值方法验证了解析结果的准确性。建立了系统稳态响应的幅频方程,利用Lyapunov第一方法得到定常解的稳定条件,确定解的稳定性。在此基础上,分析了参激项、自激项以及分数阶微分项参数对系统幅频特性的影响。结果表明:改变参激项系数主要影响系统的响应幅值和共振频率范围;改变自激项系数主要影响系统响应幅值和多值性;改变分数阶微分项系数和阶次对系统的动力学行为具有双重调节的作用。展开更多
回顾了对MOS LC差分振荡器的认识现状。通过简单的推导和Van der Pol方程现有结论,得到了交叉耦合MOS特性对振荡器性能的影响。这些结论包括:1)起振条件;2)输出幅度与参数间的解析表达式;3)振荡器输出频率与LC谐振回路和交叉耦合MOS管...回顾了对MOS LC差分振荡器的认识现状。通过简单的推导和Van der Pol方程现有结论,得到了交叉耦合MOS特性对振荡器性能的影响。这些结论包括:1)起振条件;2)输出幅度与参数间的解析表达式;3)振荡器输出频率与LC谐振回路和交叉耦合MOS管非线性特性影响的关系;4)过渡过程的时间常数;5)振荡器输出的谐波特性。这些结论揭示了MOS LC差分振荡器新的现象,对设计者了解振荡器的工作状态和优化设计有一定的参考意义。展开更多
研究了Van der Pol-Duffing振子的混沌动力学行为,应用直接微扰法构造了系统的通解,由该通解获得了预测混沌出现的Melnikov判据.在非微扰情形,相图和相应Poincaré截面的演化结果表明:系统阻尼和外驱动力的变化都可以导致系统由倍...研究了Van der Pol-Duffing振子的混沌动力学行为,应用直接微扰法构造了系统的通解,由该通解获得了预测混沌出现的Melnikov判据.在非微扰情形,相图和相应Poincaré截面的演化结果表明:系统阻尼和外驱动力的变化都可以导致系统由倍周期分叉进入混沌状态,当频率参数取相同值时,系统混沌被完全抑制.展开更多
基于9阶van der Pol方程的分岔结果,设计了1个平衡点和2个极限环共存的三稳态电路.利用平均法分析了9阶van der Pol方程的分岔性质,设计了能够实现三稳态现象的无量纲方程的系统参数.根据基尔霍夫电路定理,利用运算放大器和模拟乘法器...基于9阶van der Pol方程的分岔结果,设计了1个平衡点和2个极限环共存的三稳态电路.利用平均法分析了9阶van der Pol方程的分岔性质,设计了能够实现三稳态现象的无量纲方程的系统参数.根据基尔霍夫电路定理,利用运算放大器和模拟乘法器等元件,构建了9阶van der Pol方程的电路原理图,并通过PSpice仿真和硬件电路试验验证了该电路的可行性和可靠性.试验结果表明,该电路系统中有1个稳定平衡点与2个稳定极限环共存的现象,为研究确定性激励以及随机激励下三稳态系统的动力学行为奠定了基础.展开更多
研究了环面上非线性 van der Pol方程的图形建模及可视化计算的问题 .在图形环境下 ,对 van der Pol环面方程从建模、实验到结果分析的全过程进行了可视化建模和计算 ,并建立了一个对系统运动轨迹进行全面自动分析试验的可视化仿真框架 ...研究了环面上非线性 van der Pol方程的图形建模及可视化计算的问题 .在图形环境下 ,对 van der Pol环面方程从建模、实验到结果分析的全过程进行了可视化建模和计算 ,并建立了一个对系统运动轨迹进行全面自动分析试验的可视化仿真框架 .该方法不但可以不用传统程序代码对模型及算法编程 。展开更多
研究了修正的等熵Van der Waals气体动力学Euler方程Riemann问题及其基本波的相互作用.利用Maxwell提出的等面积法则,将Van der Waals气体状态方程修正为与实际相符,从而守恒律方程组从混合型转化为双曲型.利用广义特征线分析法,构造性...研究了修正的等熵Van der Waals气体动力学Euler方程Riemann问题及其基本波的相互作用.利用Maxwell提出的等面积法则,将Van der Waals气体状态方程修正为与实际相符,从而守恒律方程组从混合型转化为双曲型.利用广义特征线分析法,构造性地得到了Riemann问题的解是存在的.进一步,得到了基本波相互作用.展开更多
基金This work has been supported by the following:the National Natural Science Foundation of China(No.91958206,91858215)the National Key Research and Development Program Pilot Project(No.2018YFC1405901,2017YFC0307401)+1 种基金the Fundamental Research Funds for the Central Universities(No.201964016)the Marine Geological Survey Program of China Geological Survey(No.DD20190819).
文摘An air gun generates acoustic signals for seismic exploration by releasing a high-pressure gas.A large error is always gradually introduced into the ideal-gas model when the pressure in the air-gun chamber exceeds 100 atm.In the van der Waals non-ideal-gas theory,the gas in the air gun can be regarded as an actual gas,and the error is less than 2%.The van der Waals model is established in combination with the quasi-static open thermodynamic system and bubble-motion equation by considering the bubble rise,bubble interaction,and throttling eff ect.The mismatch between the van der Waals and ideal-gas models is related to the pressure.Theoretically,under high-pressure conditions,the van der Waals air-gun model yields results that are closer to the measured results.Marine vertical cables are extended to the seafl oor using steel cables that connect the cement blocks,but the corresponding hydrophones are suspended in the seawater.Thus,noise associated with ships,ocean surges,and coupling problems is avoided,and the signal-to-noise ratio and resolution of marine seismic data are improved.This acquisition method satisfies the conditions of recording air-gun far-fi eld wavelets.According to an actual vertical-cable observation system,the van der Waals air-gun model is used to model the wavelet of different azimuth and take-off angles.The characteristics of the experimental and simulated data demonstrate good agreement,which indicates that the van der Waals method is accurate and reliable.The accuracy of the model is directly related to the resolution,thus aff ecting the resolution ability of the stratum.
基金supported by National Natural Science Foundation of China (No. 41674118)
文摘An air-gun source is the most commonly used excitation method in off shore seismic exploration.The excitation characteristics of an air-gun source aff ect seismic data quality.Far-field wavelet simulation is an important approach to study these characteristics.Compared to the measured wavelet,far-field wavelet simulation based on a traditional bubble-motion equation and ideal gas wavelet model has some disadvantages,such as a greater amplitude and smaller pulse attenuation velocity.Here,we start from the linear acoustic wave equation in the spherical coordinate system to deduce an improved,simpler bubble-motion equation and develop a Van der Waals gas wavelet model based on this equation.Unlike the existing methods,our method considers the high-pressure environment during actual excitation,heat exchange between the bubble and outside water,and change in the air fl ow at the muzzle.The results show that the far-fi eld wavelet simulated using this model is closer to the measured wavelet than that of the ideal gas wavelet model.At the same time,our method has a more succinct equation and a higher calculation effi ciency.
基金It was supported in part by the National Natural Foundation of China (No. 10371136) and the Guangdong Natural Science Foundation of Guangdong Province (No.021765,031603)
文摘The one-dimensional linear wave equation with a van der Pol nonlinear boundary condition is one of the simplest models that may cause isotropic or nonisotropic chaotic vibrations. It characterizes the nonisotropic chaotic vibration by means of the total variation theory. Some results are derived on the exponential growth of total variation of the snapshots on the spatial interval in the long-time horizon when the map and the initial condition satisfy some conditions.
基金Supported by the National Natural Science Foundation of China（11201118）
文摘In this paper a modified L-P method and multiple scale method are used to solve sub-harmonic resonance solutions of strong and nonlinear resonance of general Van der Pol equation with parametric and external excitations by parametric transformation. Bifurcation response equation and transition sets of sub-harmonic resonance with strong nonlinearity of general Van der Pol equation with parametric and external excitation are worked out.Besides, transition sets and bifurcation graphs are drawn to help to analysis the problems theoretically. Conclusions show that the transition sets of general and nonlinear Van der Pol equation with parametric and external excitations are more complex than those of general and nonlinear Van der Pol equation only with parametric excitation, which is helpful for the qualitative and quantitative reference for engineering and science applications.
文摘In this research work, Homotopy perturbation method (HPM) is applied to find the approximate solution of the Van der Pol Differential equation (VDPDE), which is a well-known nonlinear ODE. Firstly, the approximate solution of Van Der Pol equation is developed using Dirichlet boundary conditions. Then a comparison between the present results and previously published results is presented and a good agreement is observed. Finally, HPM method is applied to find the approximate solution of VDPDE with Robin and Neumann boundary conditions.
文摘In this paper, we define some non-elementary amplitude functions that are giving solutions to some well-known second-order nonlinear ODEs and the Lorenz equations, but not the chaos case. We are giving the solutions a name, a symbol and putting them into a group of functions and into the context of other functions. These solutions are equal to the amplitude, or upper limit of integration in a non-elementary integral that can be arbitrary. In order to define solutions to some short second-order nonlinear ODEs, we will make an extension to the general amplitude function. The only disadvantage is that the first derivative to these solutions contains an integral that disappear at the second derivation. We will also do a second extension: the two-integral amplitude function. With this extension we have the solution to a system of ODEs having a very strange behavior. Using the extended amplitude functions, we can define solutions to many short second-order nonlinear ODEs.
文摘本文针对含有自激励,参数激励和外激励等三种激励联合作用下van der Pol-Mathieu方程的周期响应和准周期运动进行分析,发现其准周期运动的频谱中含有均匀边频带这一新的特性.首先,采用传统的增量谐波平衡法(IHB法)分析了van der Pol-Mathieu方程的周期响应,得到了其非线性频率响应曲线;再利用Floquet理论对周期解进行稳定性分析,得到了两种类型的分岔及它们的位置.然后,基于van der Pol-Mathieu方程准周期运动的频谱中边频带相邻频率之间是等距的且含有两个不可约的基频的特性(其中一个基频是已知的,另一个基频事先是未知的),推导了相应的两时间尺度IHB法,精确计算出van der Pol-Mathieu方程的准周期运动的另一个未知基频和所有的频率成份及其对应的幅值,尤其在临界点附近处的准周期运动响应.得到的准周期运动结果和利用四阶龙格-库塔(RK)数值法得到的结果高度吻合.最后,研究发现了含外激励van der Pol-Mathieu方程在不同激励频率时的一些丰富而有趣的非线性动力学现象.
文摘以含分数阶微分项的van der Pol-Mathieu方程为对象,研究了谐波激励作用下主共振的动力学行为和稳定性。采用平均法得到了方程近似解析解,通过数值方法验证了解析结果的准确性。建立了系统稳态响应的幅频方程,利用Lyapunov第一方法得到定常解的稳定条件,确定解的稳定性。在此基础上,分析了参激项、自激项以及分数阶微分项参数对系统幅频特性的影响。结果表明:改变参激项系数主要影响系统的响应幅值和共振频率范围;改变自激项系数主要影响系统响应幅值和多值性;改变分数阶微分项系数和阶次对系统的动力学行为具有双重调节的作用。
基金Supported by the PhD Programs Foundation of Ministry of Education of China(20070128001)the Innovation Program of Shanghai Municipal Education Commission (09YZ239)the Research Foundation of Shanghai Institute of Technology (YJ2009-12)
文摘本文研究了Adomian分解方法在非线性分数阶微分方程求解中的应用. 利用Riemann-Liouville分数阶导数和Adomian分解方法, 将Duffing方程和Van der Pol方程联合在一个分数阶方程中,并获得了此方程的解析近似解.
文摘回顾了对MOS LC差分振荡器的认识现状。通过简单的推导和Van der Pol方程现有结论,得到了交叉耦合MOS特性对振荡器性能的影响。这些结论包括:1)起振条件;2)输出幅度与参数间的解析表达式;3)振荡器输出频率与LC谐振回路和交叉耦合MOS管非线性特性影响的关系;4)过渡过程的时间常数;5)振荡器输出的谐波特性。这些结论揭示了MOS LC差分振荡器新的现象,对设计者了解振荡器的工作状态和优化设计有一定的参考意义。
文摘研究了Van der Pol-Duffing振子的混沌动力学行为,应用直接微扰法构造了系统的通解,由该通解获得了预测混沌出现的Melnikov判据.在非微扰情形,相图和相应Poincaré截面的演化结果表明:系统阻尼和外驱动力的变化都可以导致系统由倍周期分叉进入混沌状态,当频率参数取相同值时,系统混沌被完全抑制.
文摘基于9阶van der Pol方程的分岔结果,设计了1个平衡点和2个极限环共存的三稳态电路.利用平均法分析了9阶van der Pol方程的分岔性质,设计了能够实现三稳态现象的无量纲方程的系统参数.根据基尔霍夫电路定理,利用运算放大器和模拟乘法器等元件,构建了9阶van der Pol方程的电路原理图,并通过PSpice仿真和硬件电路试验验证了该电路的可行性和可靠性.试验结果表明,该电路系统中有1个稳定平衡点与2个稳定极限环共存的现象,为研究确定性激励以及随机激励下三稳态系统的动力学行为奠定了基础.
文摘研究了环面上非线性 van der Pol方程的图形建模及可视化计算的问题 .在图形环境下 ,对 van der Pol环面方程从建模、实验到结果分析的全过程进行了可视化建模和计算 ,并建立了一个对系统运动轨迹进行全面自动分析试验的可视化仿真框架 .该方法不但可以不用传统程序代码对模型及算法编程 。
文摘研究了修正的等熵Van der Waals气体动力学Euler方程Riemann问题及其基本波的相互作用.利用Maxwell提出的等面积法则,将Van der Waals气体状态方程修正为与实际相符,从而守恒律方程组从混合型转化为双曲型.利用广义特征线分析法,构造性地得到了Riemann问题的解是存在的.进一步,得到了基本波相互作用.
