The time-delayed fractal Van der Pol–Helmholtz–Duffing(VPHD)oscillator is the subject of this paper,which explores its mechanisms and highlights its stability analysis.While time-delayed technologies are currently g...The time-delayed fractal Van der Pol–Helmholtz–Duffing(VPHD)oscillator is the subject of this paper,which explores its mechanisms and highlights its stability analysis.While time-delayed technologies are currently garnering significant attention,the focus of this research remains crucially relevant.A non-perturbative approach is employed to refine and set the stage for the system under scrutiny.The innovative methodologies introduced yield an equivalent linear differential equation,mirroring the inherent nonlinearities of the system.Notably,the incorporation of quadratic nonlinearity into the frequency formula represents a cutting-edge advancement.The analytical solution's validity is corroborated using a numerical approach.Stability conditions are ascertained through the residual Galerkin method.Intriguingly,it is observed that the delay parameter,in the context of the fractal system,reverses its stabilizing influence,impacting both the amplitude of delayed velocity and the position.The analytical solution's precision is underscored by its close alignment with numerical results.Furthermore,the study reveals that fractal characteristics emulate damping behaviors.Given its applicability across diverse nonlinear dynamical systems,this non-perturbative approach emerges as a promising avenue for future research.展开更多
The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative approach.Using a new analytical technique,name...The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative approach.Using a new analytical technique,namely,the modification of a He’s fractal derivative that converts the fractal derivative to the traditional derivative in continuous space,this study provides an effective and easy-to-apply procedure that is dependent on the He’s fractal derivative approach.The analytic approximate solution has a significant match with the results of the numerical simulation as the fractal parameter is very closer to unity,which proves the reliability of the method.Stability behavior is discussed and illustrated graphically.These new powerful analytical tools are developed in an attempt to obtain effective analytical tools,valid for any fractal nonlinear problems.展开更多
The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator,involving fractional-order by the simplest method.An effective procedure has been introduced for executing the ...The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator,involving fractional-order by the simplest method.An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach.The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order.A detailed solving process is given for the obtained oscillator with the traditional system.展开更多
文摘The time-delayed fractal Van der Pol–Helmholtz–Duffing(VPHD)oscillator is the subject of this paper,which explores its mechanisms and highlights its stability analysis.While time-delayed technologies are currently garnering significant attention,the focus of this research remains crucially relevant.A non-perturbative approach is employed to refine and set the stage for the system under scrutiny.The innovative methodologies introduced yield an equivalent linear differential equation,mirroring the inherent nonlinearities of the system.Notably,the incorporation of quadratic nonlinearity into the frequency formula represents a cutting-edge advancement.The analytical solution's validity is corroborated using a numerical approach.Stability conditions are ascertained through the residual Galerkin method.Intriguingly,it is observed that the delay parameter,in the context of the fractal system,reverses its stabilizing influence,impacting both the amplitude of delayed velocity and the position.The analytical solution's precision is underscored by its close alignment with numerical results.Furthermore,the study reveals that fractal characteristics emulate damping behaviors.Given its applicability across diverse nonlinear dynamical systems,this non-perturbative approach emerges as a promising avenue for future research.
基金Princess Nourah bint Abdulrahman University Researchers Supporting Project number(PNURSP2023R17)
文摘The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative approach.Using a new analytical technique,namely,the modification of a He’s fractal derivative that converts the fractal derivative to the traditional derivative in continuous space,this study provides an effective and easy-to-apply procedure that is dependent on the He’s fractal derivative approach.The analytic approximate solution has a significant match with the results of the numerical simulation as the fractal parameter is very closer to unity,which proves the reliability of the method.Stability behavior is discussed and illustrated graphically.These new powerful analytical tools are developed in an attempt to obtain effective analytical tools,valid for any fractal nonlinear problems.
文摘The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator,involving fractional-order by the simplest method.An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach.The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order.A detailed solving process is given for the obtained oscillator with the traditional system.
