In this paper a new Rabinovitch-Fabrikant (R-F) four dimensional (4D) continuous time dynamical s... more In this paper a new Rabinovitch-Fabrikant (R-F) four dimensional (4D) continuous time dynamical system was generated from three dimensional (3D) Rabinovitch-Fabrikant dynamical system using the state augmentation technique by adding new state variables u. The system employs thirteen terms includes five cross-product terms and one irreversible function. The dynamical behaviors of the system were investigated which include equilibrium points, stability analysis, wave form analysis, phase space analysis, multistability, Hopf-bifurcation, the Lyapunov exponent and Lyapunov dimension. The values of Lyapunov exponents are:L1 = 14.025946, L2 = 0.295151, L3 = −2.854401, L4 = −13.736833. and Lyapunov dimension is (3.83474), so the system is unstable and hyperchaotic with coexistence attractors. Chaos was handled in two ways: adaptive control and adaptive synchronization, it was found that the new system is stable and achieved good results.
Date of publication (dd/mm/yyyy): 05/04/2021 Abstract – In this paper, a three dimensional non li... more Date of publication (dd/mm/yyyy): 05/04/2021 Abstract – In this paper, a three dimensional non linear discrete-time dynamical system was introduced, the numerical solution was carried by Newton's Raphson method. The basic properties was investigated by mean of it's fixed points, bifurcation and numerical diagrams. Stability analysis measured by eigenvalues of the characteristic equation roots ‘Lyapunov function and Jury's test which all show that the system is unstable. Chaos diagnose was calculated by Lyapunov exponent and binary test, the maximum value of Lyapunov exponent is obtain as ( 3.8), Lyapunov dimension is obtain as ( 2.0872) and binary test (0-1) is obtain as (k = 0.9942 1), which all shows the system is highly chaotic. Finally, the system was controlled effectively by designed feedback and adaptive controllers, the results after controls were very good system trajectories are stable and regular.
Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2021
In this paper, we will use the differential equations of the SIR model as a non-linear system, by... more In this paper, we will use the differential equations of the SIR model as a non-linear system, by using the Runge-Kutta numerical method to calculate simulated values for known epidemiological diseases related to the time series including the epidemic disease COVID-19, to obtain hypothetical results and compare them with the dailyreal statisticals of the disease for counties of the world and to know the behavior of this disease through mathematical applications, in terms of stability as well as chaos in many applied methods. The simulated data was obtained by using Matlab programms, and compared between real data and simulated datd were well compatible and with a degree of closeness. we took the data for Italy as an application. The results shows that this disease is unstable, dissipative and chaotic, and the Kcorr of it equal (0.9621), ,also the power spectrum system was used as an indicator to clarify the chaos of the disease, these proves that it is a spread,outbreaks,chaotic an...
In this paper, we used four feedback control method to suppress a modified hyperchaotic Pan syste... more In this paper, we used four feedback control method to suppress a modified hyperchaotic Pan system to unstable equilibrium, and we found that the critical value for each method based on the Routh-Hurwitz theorem, we study the relationship between this value and asymptotically stable, unstable and Hopf Bifurcation. Finally, we found that the least complexity and cost of method depended only on the system's constants of critical value and do not depended on the method itself. Theoretical analysis, numerical simulation, illustrative examples and comparison are given to demonstrate the effectiveness of the proposed controllers.
In this paper a new Rabinovitch-Fabrikant (R-F) four dimensional (4D) continuous time dynamical s... more In this paper a new Rabinovitch-Fabrikant (R-F) four dimensional (4D) continuous time dynamical system was generated from three dimensional (3D) Rabinovitch-Fabrikant dynamical system using the state augmentation technique by adding new state variables u. The system employs thirteen terms includes five cross-product terms and one irreversible function. The dynamical behaviors of the system were investigated which include equilibrium points, stability analysis, wave form analysis, phase space analysis, multistability, Hopf-bifurcation, the Lyapunov exponent and Lyapunov dimension. The values of Lyapunov exponents are:L1 = 14.025946, L2 = 0.295151, L3 = −2.854401, L4 = −13.736833. and Lyapunov dimension is (3.83474), so the system is unstable and hyperchaotic with coexistence attractors. Chaos was handled in two ways: adaptive control and adaptive synchronization, it was found that the new system is stable and achieved good results.
Date of publication (dd/mm/yyyy): 05/04/2021 Abstract – In this paper, a three dimensional non li... more Date of publication (dd/mm/yyyy): 05/04/2021 Abstract – In this paper, a three dimensional non linear discrete-time dynamical system was introduced, the numerical solution was carried by Newton's Raphson method. The basic properties was investigated by mean of it's fixed points, bifurcation and numerical diagrams. Stability analysis measured by eigenvalues of the characteristic equation roots ‘Lyapunov function and Jury's test which all show that the system is unstable. Chaos diagnose was calculated by Lyapunov exponent and binary test, the maximum value of Lyapunov exponent is obtain as ( 3.8), Lyapunov dimension is obtain as ( 2.0872) and binary test (0-1) is obtain as (k = 0.9942 1), which all shows the system is highly chaotic. Finally, the system was controlled effectively by designed feedback and adaptive controllers, the results after controls were very good system trajectories are stable and regular.
Turkish Journal of Computer and Mathematics Education (TURCOMAT), 2021
In this paper, we will use the differential equations of the SIR model as a non-linear system, by... more In this paper, we will use the differential equations of the SIR model as a non-linear system, by using the Runge-Kutta numerical method to calculate simulated values for known epidemiological diseases related to the time series including the epidemic disease COVID-19, to obtain hypothetical results and compare them with the dailyreal statisticals of the disease for counties of the world and to know the behavior of this disease through mathematical applications, in terms of stability as well as chaos in many applied methods. The simulated data was obtained by using Matlab programms, and compared between real data and simulated datd were well compatible and with a degree of closeness. we took the data for Italy as an application. The results shows that this disease is unstable, dissipative and chaotic, and the Kcorr of it equal (0.9621), ,also the power spectrum system was used as an indicator to clarify the chaos of the disease, these proves that it is a spread,outbreaks,chaotic an...
In this paper, we used four feedback control method to suppress a modified hyperchaotic Pan syste... more In this paper, we used four feedback control method to suppress a modified hyperchaotic Pan system to unstable equilibrium, and we found that the critical value for each method based on the Routh-Hurwitz theorem, we study the relationship between this value and asymptotically stable, unstable and Hopf Bifurcation. Finally, we found that the least complexity and cost of method depended only on the system's constants of critical value and do not depended on the method itself. Theoretical analysis, numerical simulation, illustrative examples and comparison are given to demonstrate the effectiveness of the proposed controllers.
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