# Aurelien david thesis

where x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} make up the system state , t {\displaystyle t} is time, and σ {\displaystyle \sigma } , ρ {\displaystyle \rho } , β {\displaystyle \beta } are the system parameters . Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the Rössler equations , which have only one nonlinear term out of seven. Sprott [28] found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel [29] [30] showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved.