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In these lectures we derive (mostly) the equations of viscous �uid dynamics. We then show how they may be generalized to the problem of Rayleigh-Benard convection�the problem of a �uid heated from below. Later we show how the RB problem itself may be reduced to the famous Lorenz equations. The highlights of these lectures are as follows: � Navier-Stokes equations of �uid dynamics (mass and momentum conser_vation). � Reynolds number � Phenomenology of RB convection � Rayleigh number � Equations of RB convection. Thus far we have dealt almost exclusively with the temporal behavior of a few variables. In these lectures we digress, and discuss the evolution of a continuum.

In these lectures we derive (mostly) the equations of viscous �uid dynamics. We then show how they may be generalized to the problem of Rayleigh-Benard convection�the problem of a �uid heated from below. Later we show how the RB problem itself may be reduced to the famous Lorenz equations. The highlights of these lectures are as follows: � Navier-Stokes equations of �uid dynamics (mass and momentum conser_vation). � Reynolds number � Phenomenology of RB convection � Rayleigh number � Equations of RB convection. Thus far we have dealt almost exclusively with the temporal behavior of a few variables. In these lectures we digress, and discuss the evolution of a continuum.

Linearity _ if _(t) is a solution, then so is �_(t), � real. This is incompatible with bounded oscillations (i.e., _max < _). We therefore introduce an equation with � a nonlinearity; and � an energy source that compensates viscous damping.

Linearity _ if _(t) is a solution, then so is �_(t), � real. This is incompatible with bounded oscillations (i.e., _max < _). We therefore introduce an equation with � a nonlinearity; and � an energy source that compensates viscous damping.

Several turbulent and nonturbulent solutions of the Navier-Stokes equations are obtained. The unaveraged equations are used numerically in conjunction with tools and concepts from nonlinear dynamics, including time series, phase portraits, Poincare sections, Liapunov exponents, power spectra, and strange attractors. Initially neighboring solutions for a low-Reynolds-number fully developed turbulence are compared. The turbulence is sustained by a nonrandom time-independent external force. The solutions, on the average, separate exponentially with time, having a positive Liapunov exponent. Thus, the turbulence is characterized as chaotic. In a search for solutions which contrast with the turbulent ones, the Reynolds number (or strength of the forcing) is reduced. Several qualitatively different flows are noted. These are, respectively, fully chaotic, complex periodic, weakly chaotic, simple periodic, and fixed-point. Of these, we classify only the fully chaotic flows as turbulent. Those flows have both a positive Liapunov exponent and Poincare sections without pattern. By contrast, the weakly chaotic flows, although having positive Liapunov exponents, have some pattern in their Poincare sections. The fixed-point and periodic flows are nonturbulent, since turbulence, as generally understood, is both time-dependent and aperiodic.

Chaos and nonlinear economic dynamics are addressed for a quantum coupled map lattice model of an artificial economy, with quantized supply and demand equilibrium conditions. The measure theoretic properties and the patterns that emerge in both the economic business volume dynamics' diagrams as well as in the quantum mean field averages are addressed and conclusions are drawn in regards to the application of quantum chaos theory to address signatures of chaotic dynamics in relevant discrete economic state variables.

Chaos and nonlinear economic dynamics are addressed for a quantum coupled map lattice model of an artificial economy, with quantized supply and demand equilibrium conditions. The measure theoretic properties and the patterns that emerge in both the economic business volume dynamics' diagrams as well as in the quantum mean field averages are addressed and conclusions are drawn in regards to the application of quantum chaos theory to address signatures of chaotic dynamics in relevant discrete economic state variables.

Chaos and nonlinear economic dynamics are addressed for a quantum coupled map lattice model of an artificial economy, with quantized supply and demand equilibrium conditions. The measure theoretic properties and the patterns that emerge in both the economic business volume dynamics' diagrams as well as in the quantum mean field averages are addressed and conclusions are drawn in regards to the application of quantum chaos theory to address signatures of chaotic dynamics in relevant discrete economic state variables.

We discuss the recently proposed quantum action - its interpretation, its motivation, its mathematical properties and its use in physics: quantum mechanical tunneling, quantum instantons and quantum chaos.

We discuss the recently proposed quantum action - its interpretation, its motivation, its mathematical properties and its use in physics: quantum mechanical tunneling, quantum instantons and quantum chaos.

We discuss the recently proposed quantum action - its interpretation, its motivation, its mathematical properties and its use in physics: quantum mechanical tunneling, quantum instantons and quantum chaos.

We discuss the recently proposed quantum action - its interpretation, its motivation, its mathematical properties and its use in physics: quantum mechanical tunneling, quantum instantons and quantum chaos.

Notes from a course on Nonlinear Dynamics and Chaos Theory taken at Northeastern University in 2015.

Notes from a course on Nonlinear Dynamics and Chaos Theory taken at Northeastern University in 2015.

This article is from Sensors (Basel, Switzerland) , volume 9 . Abstract In Atomic force microscope (AFM) examination of a vibrating microcantilever, the nonlinear tip-sample interaction would greatly influence the dynamics of the cantilever. In this paper, the nonlinear dynamics and chaos of a tip-sample dynamic system being run in the tapping mode (TM) were investigated by considering the effects of hydrodynamic loading and squeeze film damping. The microcantilever was modeled as a spring-mass-damping system and the interaction between the tip and the sample was described by the Lennard-Jones (LJ) potential. The fundamental frequency and quality factor were calculated from the transient oscillations of the microcantilever vibrating in air. Numerical simulations were carried out to study the coupled nonlinear dynamic system using the bifurcation diagram, Poincaré maps, largest Lyapunov exponent, phase portraits and time histories. Results indicated the occurrence of periodic and chaotic motions and provided a comprehensive understanding of the hydrodynamic loading of microcantilevers. It was demonstrated that the coupled dynamic system will experience complex nonlinear oscillation as the system parameters change and the effect of squeeze film damping is not negligible on the micro-scale.

Physicists experimentalists use a large number of observations of a phenomenon, where are the unknown equations that describe it, in order to play the dynamics and obtain information on their future behavior. In this article we study the possibility of reproducing the dynamics of the phenomenon using only a measurement scale. The Whitney immersion theorem ideas are presented and generalization of Sauer for fractal sets to rebuild the asymptotic behaviour of the phenomena and to investigate, chaotic behavior evidence in the reproduced dynamics. The applications are made in the financial market which are only known stock prices.

Physicists experimentalists use a large number of observations of a phenomenon, where are the unknown equations that describe it, in order to play the dynamics and obtain information on their future behavior. In this article we study the possibility of reproducing the dynamics of the phenomenon using only a measurement scale. The Whitney immersion theorem ideas are presented and generalization of Sauer for fractal sets to rebuild the asymptotic behaviour of the phenomena and to investigate, chaotic behavior evidence in the reproduced dynamics. The applications are made in the financial market which are only known stock prices.

A theoretical analysis of the subharmonic generation process in an acoustical resonator (interferometer) with plane walls is performed. It is shown that, when both the pumping wave and the generated subharmonic are detuned with respect to the resonator modes, the fields can display complex temporal behaviour such as self-pulsing and chaos. A discussion about the acoustical parameters required for the experimental observation of the phenomenon is given.

A theoretical analysis of the subharmonic generation process in an acoustical resonator (interferometer) with plane walls is performed. It is shown that, when both the pumping wave and the generated subharmonic are detuned with respect to the resonator modes, the fields can display complex temporal behaviour such as self-pulsing and chaos. A discussion about the acoustical parameters required for the experimental observation of the phenomenon is given.

This topical review focusses on the contributions of plasma physics to chaos and nonlinear dynamics bringing new methods which are or can be used in other scientific domains. It starts with the development of the theory of Hamiltonian chaos, and then deals with order or quasi order, for instance adiabatic and soliton theories. It ends with a shorter account of dissipative and high dimensional Hamiltonian dynamics, and of quantum chaos. Most of these contributions are a spin-off of the research on thermonuclear fusion by magnetic confinement, which started in the fifties. Their presentation is both exhaustive and compact. [15 April 2016]

This topical review focusses on the contributions of plasma physics to chaos and nonlinear dynamics bringing new methods which are or can be used in other scientific domains. It starts with the development of the theory of Hamiltonian chaos, and then deals with order or quasi order, for instance adiabatic and soliton theories. It ends with a shorter account of dissipative and high dimensional Hamiltonian dynamics, and of quantum chaos. Most of these contributions are a spin-off of the research on thermonuclear fusion by magnetic confinement, which started in the fifties. Their presentation is both exhaustive and compact. [15 April 2016]

The properties of charged-particle motion in Hamiltonian dynamics are studied in a magnetotail-like magnetic field configuration. It is shown by numerical integration of the equation of motion that the system is generally nonintegrable and that the particle motion can be classified into three distinct types of orbits; the bounded integrable orbits, unbounded stochastic orbits and unbounded transient orbits. It is also shown that different regions of the phase space exhibit qualitatively different response to external influences. The concept of differential memory in single-particle distributions is proposed. Physical implications to the dynamical properties of the magnetotail plasmas, and the possible generation of non-Maxwellian features in the distribution function are discussed. Keywords: Chaos; Hamiltonian dynamics; Magnetotail; Quasi-neutral sheet.

The properties of charged-particle motion in Hamiltonian dynamics are studied in a magnetotail-like magnetic field configuration. It is shown by numerical integration of the equation of motion that the system is generally nonintegrable and that the particle motion can be classified into three distinct types of orbits; the bounded integrable orbits, unbounded stochastic orbits and unbounded transient orbits. It is also shown that different regions of the phase space exhibit qualitatively different response to external influences. The concept of differential memory in single-particle distributions is proposed. Physical implications to the dynamical properties of the magnetotail plasmas, and the possible generation of non-Maxwellian features in the distribution function are discussed. Keywords: Chaos; Hamiltonian dynamics; Magnetotail; Quasi-neutral sheet.

In this lecture we discuss the other two generic routes to chaos, intermittency and quasiperiodicity.

In this lecture we discuss the other two generic routes to chaos, intermittency and quasiperiodicity.

Two elastically coupled nanomechanical resonators driven independently near their resonance frequencies show intricate nonlinear dynamics. The dynamics provide a scheme for realizing a nanomechanical system with tunable frequency and nonlinear properties. For large vibration amplitudes the system develops spontaneous oscillations of amplitude modulation that also show period doubling transitions and chaos. The complex nonlinear dynamics are quantitatively predicted by a simple theoretical model.

Two elastically coupled nanomechanical resonators driven independently near their resonance frequencies show intricate nonlinear dynamics. The dynamics provide a scheme for realizing a nanomechanical system with tunable frequency and nonlinear properties. For large vibration amplitudes the system develops spontaneous oscillations of amplitude modulation that also show period doubling transitions and chaos. The complex nonlinear dynamics are quantitatively predicted by a simple theoretical model.

The final report of the ARO MURI Digital Communications Based on Chaos and Nonlinear Dynamics contains research results in the areas of chaos and nonlinear dynamics applied to wireless and optical communications. New modulation and coding approaches were developed that improved the spectral efficiency, observability and orthogonality of wireless communications devices. New numerical techniques were developed that provided for improved accuracy in the simulation and analysis of nonlinear dynamical systems. Nonlinear optical communications techniques were developed that operate at 10Gb/sec by controlling the nonlinear dynamics for modulation of the optical system. Extensive theoretical analyses were undertaken, and a firm analytical foundation was established for all of the techniques 5 developed on the program. New coding approaches were developed that reduced the probability of an error bit in a digital communications system corrupted by noise.

The final report of the ARO MURI Digital Communications Based on Chaos and Nonlinear Dynamics contains research results in the areas of chaos and nonlinear dynamics applied to wireless and optical communications. New modulation and coding approaches were developed that improved the spectral efficiency, observability and orthogonality of wireless communications devices. New numerical techniques were developed that provided for improved accuracy in the simulation and analysis of nonlinear dynamical systems. Nonlinear optical communications techniques were developed that operate at 10Gb/sec by controlling the nonlinear dynamics for modulation of the optical system. Extensive theoretical analyses were undertaken, and a firm analytical foundation was established for all of the techniques 5 developed on the program. New coding approaches were developed that reduced the probability of an error bit in a digital communications system corrupted by noise.

Summary Finally, we summarize the characteristics of dissipative systems: � Energy not conserved. � Irreversible. � Contraction of areas (volumes) in phase space.

Summary Finally, we summarize the characteristics of dissipative systems: � Energy not conserved. � Irreversible. � Contraction of areas (volumes) in phase space.