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Full-Text Articles in Non-linear Dynamics
The Effects Of Finite Precision On The Simulation Of The Double Pendulum, Rebecca Wild
The Effects Of Finite Precision On The Simulation Of The Double Pendulum, Rebecca Wild
Senior Honors Projects, 2010-2019
We use mathematics to study physical problems because abstracting the information allows us to better analyze what could happen given any range and combination of parameters. The problem is that for complicated systems mathematical analysis becomes extremely cumbersome. The only effective and reasonable way to study the behavior of such systems is to simulate the event on a computer. However, the fact that the set of floating-point numbers is finite and the fact that they are unevenly distributed over the real number line raises a number of concerns when trying to simulate systems with chaotic behavior. In this research we …
A Single Particle Impact Model For Motion In Avalanches, J. J. P. Veerman, Dacian Daescu, M. J. Romero-Vallés, P. J. Torres
A Single Particle Impact Model For Motion In Avalanches, J. J. P. Veerman, Dacian Daescu, M. J. Romero-Vallés, P. J. Torres
Mathematics and Statistics Faculty Publications and Presentations
We describe the global behavior of the dynamics of a particle bouncing down an inclined staircase. For small inclinations all orbits eventually stop (independent of the initial condition). For large enough inclinations all orbits end up accelerating indefinitely (also independent of the initial conditions). There is an interval of inclinations of positive length between these two. In that interval the behavior of an orbit depends on its initial condition. In addition to stopping and accelerating orbits, there are also orbits with speeds bounded away from both zero and infinity. A second hallmark of the dynamics is that the orbits going …
A Solvable Model For Gravity Driven Granular Dynamics, J. J. P. Veerman
A Solvable Model For Gravity Driven Granular Dynamics, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
We discuss a toy model to study the dynamics of individual particles in avalanches. The model describes a particle launched from an inclined infinite staircase. The particle is not allowed to bounce when it collides with the staircase. During the collision, the particle loses some energy, and after that slides on to the end of the step it landed on. The process then repeats itself. The dynamics of this no-bounce model can essentially be completely understood. Partial versions of some results were stated and argued in previous work. Here we give a full description together with all the proofs. We …
Single-Particle Model For A Granular Ratchet, Albert J. Bae, Welles Antonio Martinez Morgado, J. J. P. Veerman, Giovani L. Vasconcelos
Single-Particle Model For A Granular Ratchet, Albert J. Bae, Welles Antonio Martinez Morgado, J. J. P. Veerman, Giovani L. Vasconcelos
Mathematics and Statistics Faculty Publications and Presentations
A simple model for a granular ratchet corresponding to a single grain bouncing off a vertically vibrating sawtooth-shaped base is studied. Depending on the model parameters, horizontal transport is observed in both the preferred and unfavoured directions. A phase diagram is presented indicating the regions in parameter space where the different regimes (no current, normal current, and current reversal) occur.
Geometrical Model For A Particle On A Rough Inclined Surface, Giovani L. Vasconcelos, J. J. P. Veerman
Geometrical Model For A Particle On A Rough Inclined Surface, Giovani L. Vasconcelos, J. J. P. Veerman
Mathematics and Statistics Faculty Publications and Presentations
A simple geometrical model is presented for the gravity-driven motion of a single particle on a rough inclined surface. Adopting a simple restitution law for the collisions between the particle and the surface, we arrive at a model in which the dynamics is described by a one-dimensional map. This map is studied in detail and it is shown to exhibit several dynamical regimes (steady state, chaotic behavior, and accelerated motion) as the model parameters vary. A phase diagram showing the corresponding domain of existence for these regimes is presented. The model is also found to be in good qualitative agreement …