# TREND Fair 2003

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**August 15, 2003**

### On this page... TREND 2003 Presentations

- Theoretical and Computational Studies of Chaos and Nonlinear Dynamics, Alexander Glasser and Marshal Miller
- Generalized Synchronization of Spatio-Temporal Chaos, Rita Kalra
- The Effect of Baffle Configuration on the Magnetic Field of a Dynamo, Daniel Blum
- Lorentz Forces and Power Dissipation in Turbulent Flows, Barbara Brawn
- Ripple-Wall Mode Converters for High Power Microwave Applications, Michelle Esteban
- Optical Studies of the Structure and Dynamics of Actin Networks, Stephen Paul Freese
- Magnetic Reconnection and the Dynamics of Energetic Particles, James McIlhargey
- Pattern Formation in a Vertically Vibrated 2-D System of Bidisperse Magnetic Particles, Zachary Smith
- Synchronization and Communication Using Mackey-Glass Electronic Circuit Systems, David Sodaitis

### TREND 2003 Results

#### First Place Winners:

- Theoretical and Computational Studies of Chaos and Nonlinear Dynamics, Alexander Glasser and Marshal Miller
- Generalized Synchronization of Spatio-Temporal Chaos, Rita Kalra

### Theoretical and Computational Studies of Chaos and Nonlinear Dynamics

#### Alexander Glasser, Harvard University and

Marshal Miller, University of Maryland College Park

Advisors: Professors Thomas Antonsen and Edward Ott

In a broad sense, our project helps us gain an understanding of chaotic dynamical systems through our studies of past discoveries in the world of chaos theory, combined with an exploration of the physical application of such theory in a specific electrical circuit. References we are using to acquaint ourselves with the concepts of chaos theory include: "Chaos in Dynamical Systems" (Ott)," Nonlinear Dynamics and Chaos" (Strogatz), and "Chaos Theory Tamed" (Williams). We intend to explore the behavior of an electrical circuit comprised of a generator connected by transmission lines to a diode (that can be represented by a capacitor and resistor). We suspect that the voltage across this diode will exhibit chaotic behavior when we allow the parameters of the capacitor and resistor to vary nonlinearly with respect to the voltage. We hope to utilize well-established numerical methods, as well as develop some of our own, to construct a mathematical model of our circuit. Theorists are often left unconvinced of a system's tendency for chaotic behavior until a theoretical model of the system demonstrates such behavior; it is indeed this model that we are attempting to construct. We have begun our analysis with a model that makes use of order-two and order-four Runge-Kutta methods and have compared this model with an expected phasor form steady state representation of the circuit. Soon we will begin to use more complex methods of analysis to construct more and more complex models of our circuit. In summary, we hope to create a mathematical model of our circuit that demonstrates its chaotic behavior. We will subsequently attempt to achieve such results in the laboratory thereby confirming our expectations.

### Generalized Synchronization of Spatiotemporal Chaos

#### Rita Kalra, Stony Brook University

Advisor: Professor Rajarshi Roy

We demonstrate generalized synchronization (GS) of spatiotemporal chaos. It is a known phenomenon that if two chaotic systems are coupled to each other, they may synchronize. In the case of one system driving another, GS occurs when there is a functional relationship between the drive signal and the resulting response signal. The auxiliary method of coupling two or more identical response systems to the same drive system is used to determine the existence of this relationship. If the response signals starting from different initial conditions are identical after transients have disappeared, we can conclude that there exists a functional relationship and therefore GS between the drive and response systems. In this experiment, spatiotemporal chaotic signals were generated in an optoelectronic feedback loop comprised of a laser, a CCD camera, a computer, and a liquid crystal spatial light modulator (SLM). A drive signal was recorded as a movie by the computer and fed back into the SLM repeatedly in order to create response movies, which were given different initial conditions. Pixel-by-pixel synchronization error analysis between the temporal response patterns driven by the same chaotic pattern revealed GS after some transient time, which was found to vary inversely with bias voltage. Synchronization error measurements once the systems were synchronized were approximately the same for all bias voltages, which implies that generalized synchronization occurs regardless of detuning parameters. Synchronization occurs in a wide variety of natural phenomena and can have engineering applications to the noninvasive testing and monitoring of structures and materials.

### The Effect of Baffle Configuration on the Magnetic Field of a Dynamo

#### Daniel Blum, University of North Carolina

Advisor: Professor Daniel Lathrop

This project explores induced magnetic fields that are created from a moving conducting liquid. If conditions are right, it is possible to self-generate a magnetic field from a liquid metal flow, as happens in the Earth's dynamo. Although the experiment we examine is not expected to produce a self-generating dynamo, observations are made to quantify what conditions will be needed to create self-generation and what role turbulence the liquid metal plays. In this experiment liquid sodium is driven in a stainless steel sphere 30 cm in diameter by two propellers. Baffles are welded inside the sphere to enhance poloidal flow and create turbulence. Measurements will be made of the magnetic field surrounding the sphere, the mean velocity field of the liquid sodium, and the decay rate of an external pulsed magnetic field at the apparatus. The knowledge gained in this experiment will help to design a larger experiment where a self-generating magnetic field may be possible.

### Lorentz Forces and Power Dissipation in Turbulent Flows

#### Barbara Brawn, University of Maryland College Park

Advisor: Professor Daniel Lathrop

We examine the instantaneous local power in turbulent flows, a quantity that has yet to be measured experimentally. Instantaneous local power can be expressed using the local velocity of the fluid element and the local force on that same fluid element: P=**u·F**. Our goal is to determine the power by experimentally measuring both **u** and **F** in a small volume. To obtain these measurements, we are designing a tank in which turbulent flow is produced via Lorentz force, **F _{L}**=

**J**·

**B**, where

**J**is the applied current density and

**B**is the magnetic field. The tank will then be inserted into an instrument that uses three-dimensional particle image velocimetry (PIV) to study and characterize turbulent flow in small volumes. This instrument has already been used to establish

**u**in a turbulent flow originally produced by two oscillating grids within the original experimental tank. It is our hope that, with data collection and effective measurements of

**F**, we will indeed be able to measure instantaneous local power in turbulent flows.

### Ripple-Wall Mode Converters for High-Power Microwave Applications

#### Michelle Esteban, University of San Diego

Advisors: Professor Wesley Lawson and H. Raghunathan

For many high-power microwave applications, the desired mode for the application is different from the output mode of the source, and mode converters are used for the purpose of converting the actual output mode to the required output mode. This paper presents the design, numerical analysis, and testing of several symmetric, periodic mode converters of two generic types: the uniform ripple-wall converter and the nonuniform ripple-wall converter. All designs operate at a central frequency of 17.136 GHz and convert circular electric modes (TE_{0n}) from one radial mode to the next higher mode. The design performance is seen to vary with a number of parameters, the primary one being the number of ripples (N). Simulations suggest that as the number of ripples increases, the bandwidth decreases with a nominal rate of 1/N. Furthermore, the maximum mode purity increases rapidly as the number of ripples increases at first, but stabilizes near 100% after about 4-6 ripples. Mode converters from the TE_{01} to the TE_{02} mode and from the TE_{02} to the TE_{03} mode were studied in depth. The nonuniform ripple seems to have a significant advantage over the constant design for the higher-mode converter, while the designs for the lower-mode converter are comparable. The comparison of performance characteristics of all designs is detailed in this work.

### Optical Studies of the Structure and Dynamics of Actin Networks

#### Stephen Paul Freese, University of Maryland College Park

Advisor: Assistant Professor Wolfgang Losert

Recent advances in optical manipulation have made it possible to probe the mechanical properties of actin filaments. Actin is a biological molecule that polymerizes to form a network that contributes to the structure of the cytoskeleton present in human cells. The structure and dynamics of actin networks at physiologically relevant concentrations were explored by utilizing an adaptive holographic laser tweezer array. Microrheology was performed on the actin network using 2 mm carboxylate-modified spheres to determine the polymerization temperature at a fixed salt concentration. A thermal gradient stage was modified to examine forces generated by a polymerization front. Preliminary studies led to the refinement of the experimental methodology in order to eliminate the effects of air bubbles, convection, and to ensure accurate imaging of the dispersed microspheres.

### Magnetic Reconnection and the Dynamics of Energetic Particles

#### James McIlhargey, University of Maryland Baltimore County

Advisors: Professor James Drake and Dr. Marc Swisdak

Magnetic reconnection is a process in which adjacent magnetic field lines with components that are oppositely aligned annihilate and reconnect, giving their energy to nearby electrons and ions in the form of directed kinetic and thermal energies. This phenomenon has been observed in the solar corona magnetosphere and is believed to trigger substorms and solar flares. Four decades ago, scientists used magnetohydrodynamics (MHD) to explain reconnection. However, the rate at which energy is released was found to be slow, even though, observationally, the process was very fast. Since resistivity controls the rate of reconnection, scientists suggested that the rate could be increased by assuming that turbulence enhanced the resistivity above classical values. No convincing theory of this "anomaly" had been developed up to this day. In a recent paper, Dr. Drake and colleagues noted that in the presence of a magnetic guide field of certain strength, electron currents inside of the reconnection zone had started some turbulent behavior, creating electron holes that could in fact explain the anomalous resistivity invoked by scientists in the past. During reconnection, electrons accelerate more than ions due to their much smaller mass, and if the two velocity distributions of ions and electrons are separated enough, then turbulent behavior becomes very likely. But with the absence of this guide field, electrons decouple from the field lines and get a chance to heat and accelerate. This washes out any of the turbulence that might occur by broadening the electrons' velocity distribution. However, in between no guide field and a strong guide field very little is known. Using the electromagnetic particle-in-cell code, p3d, the effect of different magnitudes of this guide field on the heating of electrons near the magnetic X-line and the implications for the generation of turbulent behavior during reconnection will be explored. To do this, two-dimensional simulations "δ/δz = 0" will be run with various guide field strengths and the velocity distributions of the ions and electrons near the x-line will be studied. If time permits, a couple of two-dimensional runs in perpendicular planes will be performed to see if the turbulence can be observed.

### Pattern Formation in a Vertically Vibrated 2-D System of Bidisperse Magnetic Particles

#### Zachary Smith, Colorado School of Mines

Advisor: Assistant Professor Wolfgang Losert

We will investigate the segregation and pattern formation phenomena in a vertically vibrated 2-D system of bidisperse magnetic particles. Two sample bidisperse particle mixtures will be tested. In one group of tests, we look at the effects of differing magnetic strength at the surface of the particle; in the other, we observe the effects of a differing shape of the field. Several different particle mixtures will be shaken with various amplitudes at a selected frequency to see if segregation occurs. The pattern of particle positions and dipole orientations, during and after vibration, will be examined quantitatively as will the rates at which these phenomena take place. Patterns that have already been observed in monodisperse mixtures include two-dimensional hexagonal close-packed patterns and long polymer chains or rings. The patterns will be examined theoretically to determine stability, energetic favorability, and possible mechanisms by which they form. We finally plan to relate this experiment to real world systems such as the molecular interactions in dipolar fluids which also form molecular chains and clusters.

### Synchronization and Communication Using Mackey-Glass Electronic Circuit Systems

#### David Sodaitis, University of New Hampshire

Advisor: Professor Rajarshi Roy

Synchronization of chaotic time-delay circuits based on the Mackey-Glass system is demonstrated. The potential for communications applications is also investigated using two Mackey-Glass circuits. A message signal is encoded in the time-delay loop of a drive circuit and transmitted to a response circuit. The message is recovered by subtraction of the synchronized signal in the response system from the transmission signal. Errors in the message recovery are minimized when the parameters in each circuit are identical. Characteristics of the message recovery are investigated when the delay time is altered in both of the circuits.