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Analytical and numerical investigation of two families of Lorenz-like dynamical systems. (English) Zbl 1130.37018

Summary: We investigate two families of Lorenz-like three-dimensional nonlinear dynamical systems (i) the generalized Lorenz system and (ii) the Burke-Shaw system. Analytical investigation of the former system is possible under the assumption (I) which in fact concerns four different systems corresponding to \(\varepsilon = \pm1\), \(m = 0, 1\).
\[ \omega=\varepsilon\,\frac{\sigma}{r}\,,\quad \varepsilon=\pm 1,\;m=1,0\tag{I} \]
The fixed points and stability characteristics of the Lorenz system under the assumption (I) are also classified. Parametric and temporal \((t\to\infty)\) asymptotes are also studied in connection to the memory of both the systems. We calculate the Lyapunov exponents and Lyapunov dimension for the chaotic attractors in order to study the influence of the parameters of the Lorenz system on the attractors obtained not only when the assumption (I) is satisfied but also for other values of the parameters \(\sigma,r,b,\omega\) and \(m\).

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics

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