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
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 paper limit cycles of cubic Van der Pci equation x- (b01 +b21 x2 + b02x+b12xx)x + x(1 -b20x - b30x2) = 0 is discussed, where . The existenceof limit cycle for such an equation depends on its coefficients b01 a...In this paper limit cycles of cubic Van der Pci equation x- (b01 +b21 x2 + b02x+b12xx)x + x(1 -b20x - b30x2) = 0 is discussed, where . The existenceof limit cycle for such an equation depends on its coefficients b01 and b21 and the convergence of the function F(x) =展开更多
One of the basic problems in bifurcation theory is to understand the way in whichhorseshoes are created. In this paper, we study the bifurcation behavior exhibited by the toral Vander Pol equation subject to periodic ...One of the basic problems in bifurcation theory is to understand the way in whichhorseshoes are created. In this paper, we study the bifurcation behavior exhibited by the toral Vander Pol equation subject to periodic forcing. Our attention is focased on routes relevant to horseshoestype chaos.展开更多
In this paper we analyze the qualitative behaviour of the equation ε+q(X) +εX=bp(t), where e is a small parameter.We divide the interval of parameter b into four sets of subintervals,A, B,C and D.For bA,B or D,we di...In this paper we analyze the qualitative behaviour of the equation ε+q(X) +εX=bp(t), where e is a small parameter.We divide the interval of parameter b into four sets of subintervals,A, B,C and D.For bA,B or D,we discuss the different structures of the attractors of the equation and their stabilities.When chaotic phenomena appear,we also estimate the entropy.For bC,the set of bifurcation intervals,we analyze the bifurcating type and get a series of consequences from the results of Newhouse and Palis.展开更多
The relation between the Lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the ...The relation between the Lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the definition of the Lyapunov dimension D-L((A)) of the autonomous system, the definition of the Lyapunov dimension D-L of the non-autonomous dynamical system is also given, and the difference between them is the integer 1, namely, D-L((A)) - D-L = 1. For a quasi-periodically excited dynamical system, similar conclusions are formed.展开更多
Investigating local dynamics of equilibrium points of nonlinear systems plays an important role in studying the behavior of dynamical systems. There are many different definitions for stable and unstable solutions in ...Investigating local dynamics of equilibrium points of nonlinear systems plays an important role in studying the behavior of dynamical systems. There are many different definitions for stable and unstable solutions in the literature. The main goal to develop stability definitions is exploring the responses or output of a system to perturbation as time approaches infinity. Due to the wide range of application of local dynamical system theory in physics, biology, economics and social science, it still attracts many researchers to play with its definitions to find out the answers for their questions. In this paper, we start with a brief review over continuous time dynamical systems modeling and then we bring useful examples to the playground. We study the local dynamics of some interesting systems and we show the local stable behavior of the system around its critical points. Moreover, we look at local dynamical behavior of famous dynamical systems, Hénon-Heiles system, Duffing oscillator and Van der Pol equation and analyze them. Finally, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples.展开更多
The Wayland algorithm has been improved in order to evaluate the degree of visible determinism for dynamical systems that generate time series. The objective of this study is to show that the Double-Wayland algorithm ...The Wayland algorithm has been improved in order to evaluate the degree of visible determinism for dynamical systems that generate time series. The objective of this study is to show that the Double-Wayland algorithm can distinguish between time series generated by a deterministic process and those generated by a stochastic process. The authors conducted numerical analysis of the van der Pol equation and a stochastic differential equation as a deterministic process and a Ganssian stochastic process, respectively. In case of large S/N ratios, the noise term did not affect the translation error derived from time series data, but affected that from the temporal differences of time series. In case of larger noise amplitudes, the translation error from the differences was calculated to be approximately 1 using the Double-Wayland algorithm, and it did not vary in magnitude. Furthermore, the translation error derived from the differenced sequences was considered stable against noise. This novel algorithm was applied to the detection of anomalous signals in some fields of engineering, such as the analysis of railway systems and bio-signals.展开更多
This article proposes a new wake oscillator model for vortex induced vibrations of an elastically supported rigid circular cylinder in a uniform current. The near wake dynamics related with the fluctuating nature of v...This article proposes a new wake oscillator model for vortex induced vibrations of an elastically supported rigid circular cylinder in a uniform current. The near wake dynamics related with the fluctuating nature of vortex shedding is modeled based on the classical van der Pol equation, combined with the equation for the oscillatory motion of the body. An appropriate approach is developed to estimate the empirical parameters in the wake oscillator model. The present predicted results are compared to the experimental data and previous wake oscillator model results. Good agreement with experimental results is found.展开更多
In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form u...In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.展开更多
The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields.The intrinsic complexity of their dynamics defies many existing tools based on individual orbits,whil...The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields.The intrinsic complexity of their dynamics defies many existing tools based on individual orbits,while the Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits,which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator.However,it is difficult to identify and represent the most relevant eigenfunctions in practice.Here,combined with the Koopman analysis,a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems.By invoking the error minimization,a fundamental set of Koopman eigenfunctions are derived,which may reproduce the input dynamics through a nonlinear transformation provided by the neural network.The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.展开更多
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.
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 paper limit cycles of cubic Van der Pci equation x- (b01 +b21 x2 + b02x+b12xx)x + x(1 -b20x - b30x2) = 0 is discussed, where . The existenceof limit cycle for such an equation depends on its coefficients b01 and b21 and the convergence of the function F(x) =
基金The project supported by National Natural Science Foundation of China
文摘One of the basic problems in bifurcation theory is to understand the way in whichhorseshoes are created. In this paper, we study the bifurcation behavior exhibited by the toral Vander Pol equation subject to periodic forcing. Our attention is focased on routes relevant to horseshoestype chaos.
文摘In this paper we analyze the qualitative behaviour of the equation ε+q(X) +εX=bp(t), where e is a small parameter.We divide the interval of parameter b into four sets of subintervals,A, B,C and D.For bA,B or D,we discuss the different structures of the attractors of the equation and their stabilities.When chaotic phenomena appear,we also estimate the entropy.For bC,the set of bifurcation intervals,we analyze the bifurcating type and get a series of consequences from the results of Newhouse and Palis.
基金the National Natural Science Foundation of China(No.19772027)the Science Foundation of Shanghai Municipal Commission of Education(99A01)the Science Foundation of Shanghai Municipal Commission of Science and Technology(No.98JC14032)
文摘The relation between the Lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the definition of the Lyapunov dimension D-L((A)) of the autonomous system, the definition of the Lyapunov dimension D-L of the non-autonomous dynamical system is also given, and the difference between them is the integer 1, namely, D-L((A)) - D-L = 1. For a quasi-periodically excited dynamical system, similar conclusions are formed.
文摘Investigating local dynamics of equilibrium points of nonlinear systems plays an important role in studying the behavior of dynamical systems. There are many different definitions for stable and unstable solutions in the literature. The main goal to develop stability definitions is exploring the responses or output of a system to perturbation as time approaches infinity. Due to the wide range of application of local dynamical system theory in physics, biology, economics and social science, it still attracts many researchers to play with its definitions to find out the answers for their questions. In this paper, we start with a brief review over continuous time dynamical systems modeling and then we bring useful examples to the playground. We study the local dynamics of some interesting systems and we show the local stable behavior of the system around its critical points. Moreover, we look at local dynamical behavior of famous dynamical systems, Hénon-Heiles system, Duffing oscillator and Van der Pol equation and analyze them. Finally, we discuss about the chaotic behavior of Hamiltonian systems using two different and new examples.
文摘The Wayland algorithm has been improved in order to evaluate the degree of visible determinism for dynamical systems that generate time series. The objective of this study is to show that the Double-Wayland algorithm can distinguish between time series generated by a deterministic process and those generated by a stochastic process. The authors conducted numerical analysis of the van der Pol equation and a stochastic differential equation as a deterministic process and a Ganssian stochastic process, respectively. In case of large S/N ratios, the noise term did not affect the translation error derived from time series data, but affected that from the temporal differences of time series. In case of larger noise amplitudes, the translation error from the differences was calculated to be approximately 1 using the Double-Wayland algorithm, and it did not vary in magnitude. Furthermore, the translation error derived from the differenced sequences was considered stable against noise. This novel algorithm was applied to the detection of anomalous signals in some fields of engineering, such as the analysis of railway systems and bio-signals.
基金supported by the National High Technology Research and Development Program of China(863 Program,Grant No.2006AA09Z350)the National Natural Science Foundation of China(Grant No.10702073)the Knowledge Innovation Program of Chinese Academy of Sciences(Grant No.KJCX2-YW-L02)
文摘This article proposes a new wake oscillator model for vortex induced vibrations of an elastically supported rigid circular cylinder in a uniform current. The near wake dynamics related with the fluctuating nature of vortex shedding is modeled based on the classical van der Pol equation, combined with the equation for the oscillatory motion of the body. An appropriate approach is developed to estimate the empirical parameters in the wake oscillator model. The present predicted results are compared to the experimental data and previous wake oscillator model results. Good agreement with experimental results is found.
基金“The University of Delhi”under research grant No.Dean(R)/R&D/2010/1311.
文摘In this paper,we propose a new high accuracy discretization based on the ideas given by Chawla and Shivakumar for the solution of two-space dimensional nonlinear hyper-bolic partial differential equation of the form utt=A(x,y,t)uxx+B(x,y,t)uyy+g(x,y,t,u,ux,uy,ut),0<x,y<1,t>0 subject to appropriate initial and Dirichlet boundary conditions.We use only five evaluations of the function g and do not require any fictitious points to discretize the differential equation.The proposed method is directly applicable to wave equation in polar coordinates and when applied to a linear telegraphic hyperbolic equation is shown to be unconditionally stable.Numerical results are provided to illustrate the usefulness of the proposed method.
基金supported by the National Natural Science Foundation of China under Grant No.11775035the Fundamental Research Funds for the Central Universities with contract number 2019XD-A10the Key Program of National Natural Science Foundation of China(No.92067202)
文摘The observation and study of nonlinear dynamical systems has been gaining popularity over years in different fields.The intrinsic complexity of their dynamics defies many existing tools based on individual orbits,while the Koopman operator governs evolution of functions defined in phase space and is thus focused on ensembles of orbits,which provides an alternative approach to investigate global features of system dynamics prescribed by spectral properties of the operator.However,it is difficult to identify and represent the most relevant eigenfunctions in practice.Here,combined with the Koopman analysis,a neural network is designed to achieve the reconstruction and evolution of complex dynamical systems.By invoking the error minimization,a fundamental set of Koopman eigenfunctions are derived,which may reproduce the input dynamics through a nonlinear transformation provided by the neural network.The corresponding eigenvalues are also directly extracted by the specific evolutionary structure built in.