A Modified Quadratic Lorenz Attractor in Geometric Multiplicative Calculus

Abstract

In this study the modified quadratic Lorenz attractor is introduced in geometric multiplicative calculus. The new system is analyzed and discussed for the chaotic behaviour in detail. The equilibria points, the eigenvalues of the multiplicative Jacobian, and the Lyapunov exponents are determined. The numerical simulations are conducted using the Runge-Kutta method in the framework of geometric multiplicative calculus highlighting the chaotic behaviour.

Keywords

• Chaos
• dynamical system
• multiplicative calculus

References

1. Aniszewska, D., Multiplicative runge–kutta methods, Nonlinear Dynamics, 50(1)(2007), 265–272.
2. Aniszewska, D., Rybaczuk, M., Lyapunov type stability and Lyapunov exponent for exemplary multiplicative dynamical systems, Nonlinear Dynamics, 54(4)(2008), 345–354.
3. Chen, G., Ueta, T., Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9(07)(1999), 1465–1466.
4. Eminağa, B., Aktöre, H., Riza, M. A modified quadratic Lorenz attractor, ArXiv e-prints 1508.06840 (Aug. 2015).
5. Holmes, P., Poincar´e, celestial mechanics, dynamical-systems theory and chaos, Physics Reports, 193(3)(1990), 137–163.
6. Lorenz, E.N., Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20(2)(1963), 130–141.
7. Lu, J., Chen, G., A new chaotic attractor coined, International Journal of Bifurcation and Chaos, 12(03)(2002), 659–661.
8. Lu, J.G., Chaotic dynamics of the fractional-order Lu system and its synchronization, Physics Letters A, 354(4)(2006), 305–311.

9. Lyapunov, A.M., The general problem of the stability of motion, International Journal of Control, 55(3)(1992), 531–534.
10. Ott, E., Chaos in Dynamical Systems, Cambridge University Press, 2002.
11. Pehlivan, I., Uyaroglu, Y., A new chaotic attractor from general lorenz system family and its electronic experimental implementation, Turkish Journal of Electrical Engineering and Computer Sciences, 18(2)(2010), 171–184.
12. Poincare, H., Lectures on Celestial Mechanics, 1965.
13. Poincare, H., New Methods of Celestial Mechanics, History of Modern Physics and Astronomy, New York: American Institute of Physics 156 (AIP),— c1993 1, 1993.
14. Riza, M., Aktöre, H., The runge–kutta method in geometric multiplicative calculus, LMS Journal of Computation and Mathematics, 15818(2015), 539–554.
15. Rössler, O., An equation for continuous chaos, Physics Letters A, 57(5)(1976), 397–398.