On the correlation dimension of discrete fractional chaotic systems. (English) Zbl 1452.37080
Summary: This paper is mainly devoted to the investigation of discrete-time fractional systems in three aspects. Firstly, the fractional Bogdanov map with memory effect in Riemann-Liouville sense is obtained. Then, via constructing suitable controllers, the fractional Bogdanov map is shown to undergo a transition from regular state to chaotic one. Meanwhile, the positive largest Lyapunov exponent is calculated by the Jacobian matrix algorithm to distinguish the chaotic areas. Finally, the Grassberger-Procaccia algorithm is employed to evaluate the correlation dimension of the controlled fractional Bogdanov system under different parameters. The main results show that the correlation dimension converges to a fixed value as the embedding dimension increases for the controlled fractional Bogdanov map in chaotic state, which also coincides with the conclusion driven by the largest Lyapunov exponent. Moreover, three-dimensional fractional Stefanski map is considered to further verify the effectiveness and generality of the obtained results.
MSC:
| 37M05 | Simulation of dynamical systems |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 37C45 | Dimension theory of smooth dynamical systems |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
| 39A70 | Difference operators |
| 26A33 | Fractional derivatives and integrals |
Keywords:
discrete fractional system; largest Lyapunov exponent; correlation dimension; Grassberger-Procaccia algorithm; chaotificationReferences:
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