In this paper,we define some non-elementary amplitude functions that are giving solutions to some second-order nonlinear ODEs with forcing term and systems of ODEs with chaotic behavior,such as the chaotic cases of th...In this paper,we define some non-elementary amplitude functions that are giving solutions to some second-order nonlinear ODEs with forcing term and systems of ODEs with chaotic behavior,such as the chaotic cases of the Lorenz system.For this purpose,we will introduce a special function,that is a function of the dependent variable ϕ and the independent variable t,and place it into the solution-function.These solutions are equal to the amplitude,or upper limit of integration in a non-elementary integral that can be arbitrary.The first derivative to these amplitude functions contains one or two integrals that disappear at the second derivation or at the third derivation.We are giving the solutions a name,a symbol and putting them into a group of functions and into the context of other functions.Using these integral amplitude functions,we can define solutions to some well-known second-order ODEs and systems of ODEs exhibiting chaotic behavior.展开更多
文摘In this paper,we define some non-elementary amplitude functions that are giving solutions to some second-order nonlinear ODEs with forcing term and systems of ODEs with chaotic behavior,such as the chaotic cases of the Lorenz system.For this purpose,we will introduce a special function,that is a function of the dependent variable ϕ and the independent variable t,and place it into the solution-function.These solutions are equal to the amplitude,or upper limit of integration in a non-elementary integral that can be arbitrary.The first derivative to these amplitude functions contains one or two integrals that disappear at the second derivation or at the third derivation.We are giving the solutions a name,a symbol and putting them into a group of functions and into the context of other functions.Using these integral amplitude functions,we can define solutions to some well-known second-order ODEs and systems of ODEs exhibiting chaotic behavior.
