Interpolated Cell Mapping of Dynamical Systems

Abstract

A method is proposed to efficiently determine the basins of attraction of a nonlinear system’s different steady-state solutions. The phase space of the dynamical system is spacially discretized and the continuous problem in time is converted to an iterative mapping. By means of interpolation procedures, an improvement in the system accuracy over the Simple Cell Mapping technique is achieved. Both basins of attraction for a representative nonlinear system and characteristic system trajectories are generated and compared to exact solutions.

Parametric Analysis and Fractal-Like Basins of Attraction by Modified Interpolated Cell Mapping

“…Since ICM [4] cannot distinguish multiple strange attractors or limit cycles, MICM can provide a more complete global analysis of dynamical systems than ICM. By constructing interpolated mapping sequences through iterative N N periods, MICM can locate more complete basins of attraction of periodic attractors than ICM.…”

“…The proposed method is applied to the global analysis of a forced Du$ng's oscillator, that was also studied in references [4,5], governed by the following equation:…”

Non-Linear Dynamic Analysis and Control of Chaos for a Two-Degrees-of-Freedom Rigid Body With Vibrating Support

“…Hence, "nding a method to determine which solution will occur for a given initial condition is the major task. The attractors and corresponding basins of attraction of the system can be found by the modi"ed interpolated cell mapping method (MICM) [11], which improved from interpolated cell mapping method [12]. For three-dimensional system, 303 cells are studied by modi"ed interpolated mapping method, where 303 is the number of the total cells divided in each dimensional region of interest.…”

Non-Linear Dynamics and Chaos Control of a Physical Pendulum With Vibrating and Rotating Support

“…The Lyapunov exponent test is a powerful method to measure the sensitivity of the dynamical system to changes in initial conditions. A new e!ective method, the modi"ed interpolated cell mapping [19,20], is used to obtain the global analysis of dynamic behavior of the non-linear system. The di!erent initial conditions in di!erent basins of attraction may lead to the di!erent attractors.…”