Noise-enhanced Response of Nonlinear Oscillators

AbstractAlthough often considered to be undesirable, noise can produce beneficial effects in a system. Here, the authors discuss two representative nonlinear systems and the influence of noise on the responses of these systems. One of these systems is a set of coupled monostable Duffing oscillators, while the second of these systems is a Rayleigh-Duffing system that has been considered in honor of Dr. Y. Ueda. For the coupled oscillators, it is shown that an appropriately chosen noise addition can be used to localize energy as well as shift energy localization locations. In the case of the Rayleigh-Duffing system, the authors illustrate how the addition of noise to a deterministic input can push the system from a periodic attractor in the case without noise to a “broken-egg attractor” in the case with noise. These representative examples serve to illustrate a range of possible noise-influenced responses, and it is expected that similar as well as a wider range of responses can be expected in other nonlinear systems

NOISE-INFLUENCED DYNAMICS OF NONLINEAR OSCILLATORS

Noise is usually considered detrimental to the performance of a system and the effects of noise are usually mitigated through design and/or control. In this dissertation, noise-influenced phenomena and qualitative changes in responses of nonlinear systems with noise are explored. Here, the author considers a range of nonlinear dynamical systems, including an array of nonlinear, coupled oscillators, a vertically excited pendulum, the Duffing oscillator, and a Rayleigh-Duffing mixed type oscillator. These systems are studied analytically and numerically via stochastic direct numerical integration, and analytically via the Fokker-Planck equation. The array of nonlinear, coupled oscillators is also experimentally studied. The topics covered in this dissertation are as follows: i) the destruction and formation of energy localizations in an array of oscillators, ii) a technique to stabilize an inverted pendulum by using noise, iii) a noise-utilizing control scheme, iv) the effects of noise on the response of a nonlinear system that exhibits chaotic behavior, v) and the effects of phase lag on the information rate of a Duffing oscillator. The understanding gained through this dissertation efforts can be of benefit to a variety of nonlinear systems, including structural systems at the macro-scale, micro-scale, and nano-scale

<Contributed Talk 7>NOISE-ENHANCED RESPONSE OF NONLINEAR OSCILLATORS

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPA

The van der Pol physical reservoir computer

The van der Pol oscillator has historical and practical significance to spiking neural networks. It was proposed as one of the first models for heart oscillations, and it has been used as the building block for spiking neural networks. Furthermore, the van der Pol oscillator is also readily implemented as an electronic circuit. For these reasons, we chose to implement the van der Pol oscillator as a physical reservoir computer (PRC) to highlight its computational ability, even when it is not in an array. The van der Pol PRC is explored using various logical tasks with numerical simulations, and a field-programmable analog array circuit for the van der Pol system is constructed to verify its use as a reservoir computer. As the van der Pol oscillator can be easily constructed with commercial-off-the-shelf circuit components, this PRC could be a viable option for computing on edge devices. We believe this is the first time that the van der Pol oscillator has been demonstrated as a PRC

A four-state adaptive Hopf oscillator.

Adaptive oscillators (AOs) are nonlinear oscillators with plastic states that encode information. Here, an analog implementation of a four-state adaptive oscillator, including design, fabrication, and verification through hardware measurement, is presented. The result is an oscillator that can learn the frequency and amplitude of an external stimulus over a large range. Notably, the adaptive oscillator learns parameters of external stimuli through its ability to completely synchronize without using any pre- or post-processing methods. Previously, Hopf oscillators have been built as two-state (a regular Hopf oscillator) and three-state (a Hopf oscillator with adaptive frequency) systems via VLSI and FPGA designs. Building on these important implementations, a continuous-time, analog circuit implementation of a Hopf oscillator with adaptive frequency and amplitude is achieved. The hardware measurements and SPICE simulation show good agreement. To demonstrate some of its functionality, the circuit's response to several complex waveforms, including the response of a square wave, a sawtooth wave, strain gauge data of an impact of a nonlinear beam, and audio data of a noisy microphone recording, are reported. By learning both the frequency and amplitude, this circuit could be used to enhance applications of AOs for robotic gait, clock oscillators, analog frequency analyzers, and energy harvesting