A preliminary investigation into the effects of nonlinear response modification within coupled oscillators

This thesis provides an account of an investigation into possible dynamic interactions between two coupled nonlinear sub-systems, each possessing opposing nonlinear overhang characteristics in the frequency domain in terms of positive and negative cubic stiffnesses. This system is a two degree-of-freedom Duffing oscillator coupled in series in which certain nonlinear effects can be advantageously neutralised under specific conditions. This theoretical vehicle has been used as a preliminary methodology for understanding the interactive behaviour within typical industrial ultrasonic cutting components. Ultrasonic energy is generated within a piezoelectric exciter, which is inherently nonlinear, and which is coupled to a bar-horn or block-horn to one, or more, material cutting blades, for example. The horn/blade configurations are also nonlinear, and within the whole system there are response features which are strongly reminiscent of positive and negative cubic stiffness effects. The two degree-of-freedom model is analysed and it is shown that a practically useful mitigating effect on the overall nonlinear response of the system can be created under certain conditions when one of the cubic stiffnesses is varied. It has also bfeen shown experimentally that coupling of ultrasonic components with different nonlinear characteristics can strongly influence the performance of the system and that the general behaviour of the hypothetical theoretical model is indeed borne out in practice

Mammalian Brain As a Network of Networks

Acknowledgements AZ, SG and AL acknowledge support from the Russian Science Foundation (16-12-00077). Authors thank T. Kuznetsova for Fig. 6.Peer reviewedPublisher PD

Non-Smooth Spatio-Temporal Coordinates in Nonlinear Dynamics

This paper presents an overview of physical ideas and mathematical methods for implementing non-smooth and discontinuous substitutions in dynamical systems. General purpose of such substitutions is to bring the differential equations of motion to the form, which is convenient for further use of analytical and numerical methods of analyses. Three different types of nonsmooth transformations are discussed as follows: positional coordinate transformation, state variables transformation, and temporal transformations. Illustrating examples are provided.Comment: 15 figure

Basin boundary, edge of chaos, and edge state in a two-dimensional model

In shear flows like pipe flow and plane Couette flow there is an extended range of parameters where linearly stable laminar flow coexists with a transient turbulent dynamics. When increasing the amplitude of a perturbation on top of the laminar flow, one notes a a qualitative change in its lifetime, from smoothly varying and short one on the laminar side to sensitively dependent on initial conditions and long on the turbulent side. The point of transition defines a point on the edge of chaos. Since it is defined via the lifetimes, the edge of chaos can also be used in situations when the turbulence is not persistent. It then generalises the concept of basin boundaries, which separate two coexisting attractors, to cases where the dynamics on one side shows transient chaos and almost all trajectories eventually end up on the other side. In this paper we analyse a two-dimensional map which captures many of the features identified in laboratory experiments and direct numerical simulations of hydrodynamic flows. The analysis of the map shows that different dynamical situations in the edge of chaos can be combined with different dynamical situations in the turbulent region. Consequently, the model can be used to develop and test further characterisations that are also applicable to realistic flows.Comment: 24 pages, 9 color figure

Nonlinear dynamics and pattern formation in turbulent wake transition

Results are reported on direct numerical simulations of transition from two-dimensional to three-dimensional states due to secondary instability in the wake of a circular cylinder. These calculations quantify the nonlinear response of the system to three-dimensional perturbations near threshold for the two separate linear instabilities of the wake: mode A and mode B. The objectives are to classify the nonlinear form of the bifurcation to mode A and mode B and to identify the conditions under which the wake evolves to periodic, quasi-periodic, or chaotic states with respect to changes in spanwise dimension and Reynolds number. The onset of mode A is shown to occur through a subcritical bifurcation that causes a reduction in the primary oscillation frequency of the wake at saturation. In contrast, the onset of mode B occurs through a supercritical bifurcation with no frequency shift near threshold. Simulations of the three-dimensional wake for fixed Reynolds number and increasing spanwise dimension show that large systems evolve to a state of spatiotemporal chaos, and suggest that three-dimensionality in the wake leads to irregular states and fast transition to turbulence at Reynolds numbers just beyond the onset of the secondary instability. A key feature of these ‘turbulent’ states is the competition between self-excited, three-dimensional instability modes (global modes) in the mode A wavenumber band. These instability modes produce irregular spatiotemporal patterns and large-scale ‘spot-like’ disturbances in the wake during the breakdown of the regular mode A pattern. Simulations at higher Reynolds number show that long-wavelength interactions modulate fluctuating forces and cause variations in phase along the span of the cylinder that reduce the fluctuating amplitude of lift and drag. Results of both two-dimensional and three-dimensional simulations are presented for a range of Reynolds number from about 10 up to 1000

Chaotic Oscillations in CMOS Integrated Circuits

Chaos is a purely mathematical term, describing a signal that is aperiodic and sensitive to initial conditions, but deterministic. Yet, engineers usually see it as an undesirable effect to be avoided in electronics. The first part of the dissertation deals with chaotic oscillation in complementary metal-oxide-semiconductor integrated circuits (CMOS ICs) as an effect behavior due to high power microwave or directed electromagnetic energy source. When the circuit is exposed to external electromagnetic sources, it has long been conjectured that spurious oscillation is generated in the circuits. In the first part of this work, we experimentally and numerically demonstrate that these spurious oscillations, or out-of-band oscillations are in fact chaotic oscillations. In the second part of the thesis, we exploit a CMOS chaotic oscillator in building a cryptographic source, a random number generator. We first demonstrate the presence of chaotic oscillation in standard CMOS circuits. At radio frequencies, ordinary digital circuits can show unexpected nonlinear responses. We evaluate a CMOS inverter coupled with electrostatic discharging (ESD) protection circuits, designed with 0.5 μm CMOS technology, for their chaotic oscillations. As the circuit is driven by a direct radio frequency injection, it exhibits a chaotic dynamics, when the input frequency is higher than the typical maximum operating frequency of the CMOS inverter. We observe an aperiodic signal, a broadband spectrum, and various bifurcations in the experimental results. We analytically discuss the nonlinear physical effects in the given circuit : ESD diode rectification, DC bias shift due to a non-quasi static regime operation of the ESD PN-junction diode, and a nonlinear resonant feedback current path. In order to predict these chaotic dynamics, we use a transistor-based model, and compare the model's performance with the experimental results. In order to verify the presence of chaotic oscillations mathematically, we build on an ordinary differential equation model with the circuit-related nonlinearities. We then calculate the largest Lyapunov exponents to verify the chaotic dynamics. The importance of this work lies in investigating chaotic dynamics of standard CMOS ICs that has long been conjectured. In doing so, we experimentally and numerically give evidences for the presence of chaotic oscillations. We then report on a random number generator design, in which randomness derives from a Boolean chaotic oscillator, designed and fabricated as an integrated circuit. The underlying physics of the chaotic dynamics in the Boolean chaotic oscillator is given by the Boolean delay equation. According to numerical analysis of the Boolean delay equation, a single node network generates chaotic oscillations when two delay inputs are incommensurate numbers and the transition time is fast. To test this hypothesis physically, a discrete Boolean chaotic oscillator is implemented. Using a CMOS 0.5 μm process, we design and fabricate a CMOS Boolean chaotic oscillator which consists of a core chaotic oscillator and a source follower buffer. Chaotic dynamics are verified using time and frequency domain analysis, and the largest Lyapunov exponents are calculated. The measured bit sequences do make a suitable randomness source, as determined via National Institute of Standards and Technology (NIST) standard statistical tests version 2.1