Exactly solvable chaos in an electromechanical oscillator

Abstract

A novel electromechanical chaotic oscillator is described that admits an exact analytic solution. The oscillator is a hybrid dynamical system with governing equations that include a linear second order ordinary differential equation with negative damping and a discrete switching condition that controls the oscillatory fixed point. The system produces provably chaotic oscillations with a topological structure similar to either the Lorenz butterfly or Rössler's folded-band oscillator depending on the configuration. Exact solutions are written as a linear convolution of a fixed basis pulse and a sequence of discrete symbols. We find close agreement between the exact analytical solutions and the physical oscillations. Waveform return maps for both configurations show equivalence to either a shift map or tent map, proving the chaotic nature of the oscillations.

Easy encoding and low bit‐error‐rate chaos communication system based on reverse‐time chaotic oscillator

“…As shown in (10) to (12), if the input signal is δ(t), then the output of the matched filter defined by (9) is ξ( − t). Therefore, the filter proposed here is the matched filter of the basis function ξ(t).…”

“…Chaotic signals have the features of irregularity, broad band, aperiodicity, and easy to be generated. Coding information in the symbolic dynamics of such signals is an intriguing method and its application for communication becomes more and more prevalent [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Chaos-based communication systems, with coding information embedded in the chaotic waveforms, provide a large channel capacity, low probability of detection [16], enhanced data security [17] and a very simple hardware implementation [18][19][20].…”

Chaos in optimal communication waveforms

“…is an exact analytic solution to a physically realizable chaotic oscillator [17][18][19][20][21][22][23]. Remarkably, the autonomous oscillation of this system generates a waveform constructed of acausal basis functions.…”

Li–Yorke Chaos in Hybrid Systems on a Time Scale

“…Both concepts have many applications in various disciplines such as mechanics, electronics, neural networks, population models and economics. See, for instance, [14,16,22,23,39,41,45,46,48] and the references therein.Dynamic equations on time scales (DETS) have been extensively investigated in the literature [16,31].However, to the best of our knowledge, the presence of chaos has never been achieved in DETS. Motivated by the deficiency of mathematical methods for the investigation of chaos in such equations, we suggest the results of the present study.…”

“…Both concepts have many applications in various disciplines such as mechanics, electronics, neural networks, population models and economics. See, for instance, [14,16,22,23,39,41,45,46,48] and the references therein.…”