The Rössler attractor model is an important model that provides valuable insights into the behavior of chaotic systems in real life and is applicable in understanding weather patterns,biological systems,and secur...The Rössler attractor model is an important model that provides valuable insights into the behavior of chaotic systems in real life and is applicable in understanding weather patterns,biological systems,and secure communications.So,this work aims to present the numerical performances of the nonlinear fractional Rössler attractor system under Caputo derivatives by designing the numerical framework based on Ultraspherical wavelets.The Caputo fractional Rössler attractor model is simulated into two categories,(i)Asymmetric and(ii)Symmetric.The Ultraspherical wavelets basis with suitable collocation grids is implemented for comprehensive error analysis in the solutions of the Caputo fractional Rössler attractor model,depicting each computation in graphs and tables to analyze how fractional order affects the model’s dynamics.Approximate solutions obtained through the proposed scheme for integer order are well comparable with the fourth-order Runge-Kutta method.Also,the stability analyses of the considered model are discussed for different equilibrium points.Various fractional orders are considered while performing numerical simulations for the Caputo fractional Rössler attractor model by using Mathematica.The suggested approach can solve another non-linear fractional model due to its straightforward implementation.展开更多
In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-di...In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.展开更多
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
基金"La derivada fraccional generalizada,nuevos resultados y aplicaciones a desigualdades integrales"Cod UIO-077-2024supported via funding from Prince Sattam bin Abdulaziz University project number(PSAU/2025/R/1446).
文摘The Rössler attractor model is an important model that provides valuable insights into the behavior of chaotic systems in real life and is applicable in understanding weather patterns,biological systems,and secure communications.So,this work aims to present the numerical performances of the nonlinear fractional Rössler attractor system under Caputo derivatives by designing the numerical framework based on Ultraspherical wavelets.The Caputo fractional Rössler attractor model is simulated into two categories,(i)Asymmetric and(ii)Symmetric.The Ultraspherical wavelets basis with suitable collocation grids is implemented for comprehensive error analysis in the solutions of the Caputo fractional Rössler attractor model,depicting each computation in graphs and tables to analyze how fractional order affects the model’s dynamics.Approximate solutions obtained through the proposed scheme for integer order are well comparable with the fourth-order Runge-Kutta method.Also,the stability analyses of the considered model are discussed for different equilibrium points.Various fractional orders are considered while performing numerical simulations for the Caputo fractional Rössler attractor model by using Mathematica.The suggested approach can solve another non-linear fractional model due to its straightforward implementation.
文摘In this case-study, we examine the effects of linear control on continuous dynamical systems that exhibit chaotic behavior using the symbolic computer algebra system Mathematica. Stabilizing (or controlling) higher-dimensional chaotic dynamical systems is generally a difficult problem, Musielak and Musielak, [1]. We numerically illustrate that sometimes elementary approaches can yield the desired numerical results with two different continuous higher order dynamical systems that exhibit chaotic behavior, the Lorenz equations and the Rössler attractor.
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.
