Dampeddriven systems are ubiquitous in nature from Physics to
Sociology. Heuristically, the concept of dampeddriven systems is
quite simple. If we consider the undamped unforced ordinary
differential equation (ODE) x'' + ax = 0; a > 0, we know that
the solutions can only be sinusoidal. If we now add a damping
term, bx', the solutions become more complex with exponential terms
multiplying the sinusoidal terms. Finally, if we drive the ODE
with some nonautonomous forcing f(t), the system exhibits even more
complex behavior and can even become chaotic if it no longer satisfies
the hypothesis of the Poincaré–Bendixson theorem.
It
has been known for decades that, given proper conditions, a fluid drop
can be made to bounce on a vibrating fluid bath for long times
scales. In recent years, bouncing droplets have been observed to
bifurcate from the bouncing state (no horizontal motion) to the
walking state (horizontal motion). Experiments with walking
droplets (called walkers) exhibit analogs of waveparticle
duality and more specifically quantumlike phenomena. Studying
walkers can enhance our understanding of quantum mechanics and suggest viable alternatives to the Copenhagen interpretation.
My research has focused on modeling walking droplet dynamics to analyze the longtime statistics and on developing experiments producing quantumlike behavior. Recently, this has manifested
in modeling walkers on annulus and elliptical corrals with impurities,
and analyzing those models via rigorous dynamical systems theory.
From the analysis we observe and prove that the droplets experience
similar bifurcation behavior that is canonical to a wide class of
dampeddriven systems. This theoretical will allow us to study the
dynamical possibilities of such a system in parameter regimes that
would not be possible on the macroscale, thereby aiding us in asking
questions about what happens at quantumscales. Further, in
collaboration with Nathan Kutz and the AI Institute in Dynamic Systems, with equipment which Steve Brunton
graciously allowed us to use, my students and I are conducting
experiments on walking droplets to produce quantumlike behavior. We
then use datadriven techniques such as statistical and machine learning
to track the droplets and model the data in the form of differential
equations.
Relevant Publications:
Logical
circuits are an integral part of modern life that are traditionally
designed with minimal uncertainty. While this is straightforward
to achieve with electronic logic, other logic families such as fluidic,
chemical, and biological circuits naturally exhibit uncertainties due to
the slower timescales of Boolean operations. In addition, chaotic
logical circuits have the potential to be employed in random number
generation, encryption, and fault tolerance. However, in order to
exploit the properties of various nonlinear circuits they need to be
studied further. Since experiments with large systems become
difficult, tractable mathematical models that are amenable to analysis
via dynamical systems theory are of particular value.
We developed a modeling framework for the chaotic Set/Reset flipflop
circuit and two types of chaotic NOR gates. Through this
framework we derive discrete dynamical models for Set/Reset  type
circuits with any number of NOR gates. Since the models are
recurrence relations, the computational expense is quite low.
Further, we conducted experiments to test these models, which shows both
qualitatively and quantitatively close behavior between the theory and
experiments.
Relevant Publications:
Similar phenomena often appear across many different branches of
Science. Cancer dynamics, infectious disease, and information
cascades, often involve interactions of various agents. Similar to
Statistical Mechanics, these interactions appear random at the scale of
a few agents, but average out at the scale of an entire population.
According
to the National Cancer Institute, almost 40% of people in the
United States end up developing cancer in their lifetime, and total
national expenditure on cancer is $125 Billion. While cancer
deaths have fallen, the treatment of cancer is still predominant a trial
and error process. This may result in delays to administer the
correct treatment, the use of more invasive procedures, or an increase
in toxicity due to superfluous treatments. Although these
procedures may end up saving the patient, the treatment may also have an
adverse effect on their quality of life. To remedy this cruel
paradox my collaborators and I have been developing a framework of
complete scientific investigations for optimal treatment through the use
of robust mathematical models and biological experiments rooted in the
rigors of physics. Moreover, the President's recent moonshot has
opened a singular opportunity for cancer modeling studies.
Dynamical models for cancer often describe either transport (of
cells, drugs, nutrients, etc.) or averaged population dynamics, but
seldom both. By coupling the two processes we are able to model a
complete treatment strategy. First, I developed a simplest case
model for drug injection into a tumor coupled with tumor cell
death. Here (shown in the insert on the left) drug transport is
modeled as radial diffusion with a leaky boundary obeying Newton's law
of cooling, and the injection is modeled as an initial profile.
The population is modeled as a binary process where a cell that is
exposed to higher amounts of a drug will have a greater chance of death,
which on aggregate becomes deterministic: the regions of the
tumor with a higher than threshold concentration
will ablate and the regions with lower concentration will remain.
This simplicity allows for fast computation, which is often necessary
in real world scenarios.
More recently I have been collaborating with Katherine Owens and
Ivana Bozic on the spatiotemporal dynamics of Chimeric antigen receptor
T cell (CAR Tcell) therapies. Treatments supplemented by CAR
Tcells have shown clinical success in varieties of Leukemias and
Lymphomas, and there have been promising studies on solid tumors as
well. Since the use of CAR Tcell therapies is still in its
infancy, our work presents a singular opportunity for mathematical
models to have an immense tangible impact on treatments that are
currently being developed and those that will be developed in the near
future. For our seminal study, we model the interaction between
CAR Tcells and a tumor as a coupled reactiondiffusion model as
illustrated on the right.
I
develop similar physicsinspired models for infectious disease.
The surprisingly mercurial Covid19 pandemic highlights the need for
spatiotemporal models in order to predict the next ``hot zone'' and
properly allocate resources to those regions. My collaborators, Angie Peace, Souparno Ghosh,
Ramesh Kesawan, and I modeled highly asymptomatic disease transmission
using countywise data from the current pandemic, which importantly
captures captures the evolution of hot zones, which was not previously
possible with standard population dynamics. Some preliminary
results on the model simulation versus the actual data is shown on the
left.
My MS student, Trevor, and I also developed a recurrence relation
model for information cascades in social networks inspired by physical
problems. Using Twitter as an example, through this model we are
able to forecast the virality of a tweet. While most posts are
innocuous, forecasting the virality of tweets that spread misinformation
could help mitigate such a spread through targeted interventions.
For more information please refer to my research statement.
Relevant Publications:
Upon discovering a new global bifurcation in discrete dynamical
models of walking droplets, we developed a generic dynamical system in
order to generalize the theory of such bifurcations. From a
graphical point of view, the invariant circle from an initial
NeimarkSacker bifurcation "blinks" on and off. This is due to
the transition from tangential to transverse intersections of the
unstable and stable manifolds of the saddle fixed point.
Animation of colliding invariant circles
Relevant Publications:
We had previously shown that attracting horseshoes may be generalized to be contained within a quadrilateral trapping region. Through the NJIT Provost high school internship and the
Provost Phase 1 undergraduate research grant, I mentored Karthik murthy
(Bridgewater  Raritan High School) and Parth Sojitra (NJITECE) in finding numerical evidence of
generalized attracting horseshoes (GAH) in Poincare maps of the Rossler
attractor. I developed the algorithms to go from a first return
map to a Poincare map and finally to find the quadrilateral trapping
region for the supposed GAH. I then guided our students in writing
the MATLAB codes to carry out the algorithms. While finding the
trapping region numerically is not a proof, it does give us confidence
that there exists a GAH in the Poincare map of the Rossler attractor.
Relevant Publications:
Chaotic scattering has been studied from the early 70s and
80s in solitary wave collisions from the PhiFour equation (called KinkAntikink
collisions). These were mainly numerical studies that
gave insight into the phenomena. However, since the
equation is so difficult to work with there has been very
little analysis done. In more recent years reduction
techniques have been used to approximate the PhiFour PDE
with a system of ODEs and also as an iterated map.
We have gone further and developed a mechanical analog (a
ball rolling on a special surface) of chaotic scattering in
KinkAntikink collisions. This was done in order to
conduct experiments. In addition to experiments we
have analyzed the system thoroughly, including the
dissipation that comes from friction. The experimental setup is
shown bellow.
Relevant Publications:
Peixoto's theorem is one of the most important theorems in Dynamical Systems. It was proved by Dr. Mauricio Matos Peixoto in 1962. This proof is extremely involved  far too involved for most undergraduate students to follow. We develop an alternate  pedagogical proof of the simpler 1D case, with the goal of allowing senior undergraduate students to follow and understand the proof and consequently some of the ideas involved in the much bigger proof of the 2D case.
Relevant Publications:
We numerically simulated the beam dynamics of the Energy Recovery Linear Particle Accelerator (ERL) design for Argonne National Lab's (ANL) Advanced Photon Source (APS). The code BI, created by Ivan Bazarov, is benchmarked against our own code for simpler Accelerators. Then the full ERL was simulated and the results were analyzed. We concluded that the ERL, would be stable, and therefore a reasonable design for the specifications.
Relevant Publications:
Authors:  J. Nathan Kutz, A. Rahman, M. Ebers, J. Koch, J. Bramburger. 
Sources:  (under review) arxiv (preliminary manuscript) 
Abstract:  Dampeddriven systems are ubiquitous in engineering and science. Despite the diversity of physical processes observed in a broad range of applications, the underlying instabilities observed in practice have a universal characterization which is determined by the overall gain and loss curves of a given system. The universal behavior of dampeddriven systems can be understood from a geometrical description of the energy balance with a minimal number of assumptions. The assumptions on the energy dynamics are as follows: the energy increases monotonically as a function of increasing gain, and the losses become increasingly larger with increasing energy, i.e. there are many routes for dissipation in the system for large input energy. The intersection of the gain and loss curves define an energy balanced solution. By constructing an iterative map between the loss and gain curves, the dynamics can be shown to be homeomorphic to the logistic map, which exhibits a period doubling cascade to chaos. Indeed, the loss and gain curves allow for a geometrical description of the dynamics through a simple Verhulst diagram (cobweb plot). Thus irrespective of the physics and its complexities, this simple geometrical description dictates the universal set of logistic map instabilities that arise in complex dampeddriven systems. More broadly, dampeddriven systems are a class of nonequilibrium pattern forming systems which have a canonical set of instabilities that are manifest in practice. 
Notes:  
Supplementary Materials  

Codes:  Coming soon 
Animations: 
Authors:  G. Ferrandez Quinto^{ug}, A. Rahman. 
Sources:  (under review) arxiv (preliminary manuscript) 
Abstract:  Recently discrete dynamical models of walking droplet dynamics have allowed for fast numerical simulations of their horizontal chaotic motion and consequently the longtime statistical distribution of the droplet position. We develop a discrete model for walkers on an elliptical corral with two dominant eigenmodes. These eigenmodes are excited periodically with each having random, but correlated, weights. At each iteration the model computes the horizontal propulsion of the droplet due to impacts with the fluid bath. The propulsion is calculated based on the interaction of the droplet with the wavefield produced by the eigenmode excitations. We record the longtime statistics, and incredibly for such a simple model, they are qualitatively similar to that of experiments. 
Notes:  
Supplementary Materials  

Codes:  Coming soon 
Animations: 
Authors:  G. Zhang^{ug}, I. C. Christov, A. Rahman. 
Sources:  (under review) arxiv (preliminary manuscript) 
Abstract:  Discrete dynamical models of walking droplets (walkers) have allowed swift numerical experiments revealing heretofore unobserved quantum statistics and related behaviors in a classical hydrodynamic system. One such model shows evidence of the empirically elusive nbounce resonance and chaotic scattering in solitarywave interactions investigated using covariant nonlinear field theory with polynomial selfinteraction. We present experimentally testable predictions of nbounce resonances between walkers. An exhaustive numerical investigation of the model reveals the usual fractal structure of resonances in the velocity invelocity out diagram for colliding walkers. We suggest some avenues for further theoretical analysis of walker collisions, which may connect this discrete model back to the fieldtheoretic setting. 
Notes:  
Supplementary Materials  

Codes:  Coming soon 
Animations: 
Authors:  A. Rahman, A. Peace., R. Kesawan.^{g}, S. Ghosh.. 
Sources:  (under review) arxiv (preliminary manuscript) 
Abstract:  The surprisingly mercurial Covid19 pandemic has highlighted the need to not only accelerate research on infectious disease, but to also study them using novel techniques and perspectives. A major contributor to the difficulty of containing the current pandemic is due to the highly asymptomatic nature of the disease. In this investigation, we develop a modeling framework to study the spatiotemporal evolution of diseases with high rates of asymptomatic transmission, and we apply this framework to a hypothetical country with mathematically tractable geography; namely, square counties uniformly organized into a rectangle. We first derive a model for the temporal dynamics of susceptible, infected, and recovered populations, which is applied at the county level. Next we use likelihoodbased parameter estimation to derive temporally varying disease transmission parameters on the statewide level. While these two methods give us some spatial structure and show the effects of behavioral and policy changes, they miss the evolution of hot zones that have caused significant difficulties in resource allocation during the current pandemic. It is evident that the distribution of cases will not be stagnantly based on the population density, as with many other diseases, but will continuously evolve. We model this as a diffusive process where the diffusivity is spatially varying based on the population distribution, and temporally varying based on the current number of simulated asymptomatic cases. With this final addition coupled to the SIR model with temporally varying transmission parameters, we capture the evolution of "hot zones" in our hypothetical setup. 
Notes:  
Supplementary Materials  

Codes:  Coming soon 
Animations: 
Authors:  A. Rahman 
Sources:  Physical Review arxiv (preliminary manuscript) 
Abstract:  In recent decades computeraided technologies have become prevalent in medicine, however, cancer drugs are often only tested on in vitro cell lines from biopsies. We derive a full threedimensional model of inhomogeneousanisotropic diffusion in a tumor region coupled to a binary population model, which simulates in vivo scenarios faster than traditional cellline tests. The diffusion tensors are acquired using diffusion tensor magnetic resonance imaging from a patient diagnosed with glioblastoma multiform. Then we numerically simulate the full model with finite element methods and produce drug concentration heat maps, apoptosis hotpots, and doseresponse curves. Finally, predictions are made about optimal injection locations and volumes, which are presented in a form that can be employed by doctors and oncologists. 
Notes:  
Supplementary Materials  

Codes:  Codes [zip] 
Animations:  N/A 
Authors:  E. Kara, G. Zhang^{ug}, J. J. Williams^{g}, G. FerrandezQuinto^{ug}, L. J. Rhoden^{ug}, M. Kim^{ug}, J. Nathan Kutz, A. Rahman. 
Sources:  Springer arxiv (preliminary manuscript) 
Abstract:  We present a deeplearning based tracking objects of interest in walking droplet and granular intruder experiments. In a typical walking droplet experiment, a liquid droplet, known as walker, propels itself laterally on the free surface of a vibrating bath of the same liquid. This lateral motion stems from the interaction between the droplet and the wave it generates upon successive bounces off the vibrating liquid surface. A walker can exhibit a highly irregular trajectory over the course of its motion, including rapid acceleration and complex interactions with the other walkers present in the bath. In analogy with the hydrodynamic experiments, the granular matter experiments consist of a vibrating bath of very small solid particles and a larger solid called intruder. Like the fluid droplets, the intruder interacts with and travels the domain due to the waves of the bath but tends to move much slower and much less smoothly than the droplets. When multiple intruders are introduced, they also exhibit complex interactions with each other. We leverage the stateoftheart object detection model YOLO (You Only Look Once) and the Hungarian Algorithm to accurately extract the trajectory of a walker or intruder in realtime. Our proposed methodology is capable of tracking individual walker(s) or intruder(s) in digital images acquired from a broad spectrum of experimental settings and does not suffer from any identityswitch issues. Thus, the deep learning approach developed in this work could be used to automatize the efficient, fast and accurate extraction of observables of interests in walking droplet, granular intruder experiments and similar particle tracking experiments. Such extraction capabilities are critically enabling for downstream tasks such as building datadriven dynamical models for the coarsegrained dynamics and interactions of the objects of interest. 
Notes:  To feature in SIAM News 
Supplementary Materials  

Codes:  Available with coauthor E. Kara, github 
Animations:  N/A 
Authors:  A. Rahman, J. Nathan Kutz. 
Sources:  SIAM J. Appl. Dyn. Syst. arxiv (preliminary manuscript) 
Abstract:  We consider the dynamics of a droplet on a vibrating fluid bath. This hydrodynamic quantum analog system is shown to elicit the canonical behavior of dampeddriven systems, including a period doubling route to chaos. By approximating the system as a compositional map between the gain and loss dynamics, the underlying nonlinear dynamics can be shown to be driven by energy balances in the systems. The gainloss iterative mapping is similar to a normal form encoding for the pattern forming instabilities generated in such spatiallyextended system. Similar to modelocked lasers and rotating detonation engines, the underlying bifurcations persist for general forms of the loss and gain, both of which admit explicit representations in our approximation. Moreover, the resulting geometrical description of the particlewave interaction completely characterizes the instabilities observed in experiments. 
Notes: 

Supplementary Materials  

Codes:  Codes [zip] 
Animations: 

Authors:  A. Rahman, D. Blackmore. 
Sources:  SIAM Review arxiv (preliminary manuscript) 
Summary:  Peixoto's structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs. In this module, we formulate and prove the onedimensional analogs of Peixoto's theorems in an intuitive and fairly simple way using only concepts and results that for the most part should be familiar to upper level undergraduate students in the mathematical sciences or related fields. The intention is to provide students who may be interested in further study in dynamical systems with an accessible onedimensional treatment of structural stability theory that should help make Peixoto's theorems and their more recent generalizations easier to appreciate and understand. Further, we believe it is important and interesting for students to know the historical context of these discoveries since the mathematics was not done in isolation. The historical context is perhaps even more appropriate as it is the 100^{th} anniversary of Marília Chaves Peixoto and Mauricio Matos Peixoto's births, February 24^{th} and April 15^{th} 1921, respectively. 
Notes:  Cover article 
Supplementary Materials  

Codes:  N/A Analysis 
Animations:  N/A Analysis 
Authors:  K. Murthy^{hs}, I. Jordan^{ug}, P. Sojitra^{ug}, A. Rahman, D. Blackmore. 
Sources:  Symmetry arxiv (preliminary manuscript) 
Abstract:  We show that there is a mildly nonlinear threedimensional system of ordinary differential equations—realizable by a rather simple electronic circuit—capable of producing a generalized attracting horseshoe map. A system specifically designed to have a Poincaré section yielding the desired map is described, but not pursued due to its complexity, which makes the construction of a circuit realization exceedingly difficult. Instead, the generalized attracting horseshoe and its trapping region is obtained by using a carefully chosen Poincaré map of the Rössler attractor. Novel numerical techniques are employed to iterate the map of the trapping region to approximate the chaotic strange attractor contained in the generalized attracting horseshoe, and an electronic circuit is constructed to produce the map. Several potential applications of the idea of a generalized attracting horseshoe and a physical electronic circuit realization are proposed. 
Notes:  Invited special issue on Symmetry in Modeling and Analysis of Dynamic Systems. 
Supplementary Materials  

Codes: 

Animations:  N/A 
Authors:  A. Rahman, D. Blackmore. 
Sources:  World Scientific arxiv (preliminary manuscript) 
Abstract:  Over the past decade the study of fluidic droplets bouncing and skipping (or "walking") on a vibrating fluid bath has gone from an interesting experiment to a vibrant research field. The field exhibits challenging fluids problems, potential connections with quantum mechanics, and complex nonlinear dynamics. We detail advancements in the field of walking droplets through the lens of Dynamical Systems Theory, and outline questions that can be answered using dynamical systems analysis. The article begins by discussing the history of the fluidic experiments and their resemblance to quantum experiments. With this physics backdrop, we paint a portrait of the complex nonlinear dynamics present in physical models of various walking droplet systems. Naturally, these investigations lead to even more questions, and some unsolved problems that are bound to benefit from rigorous Dynamical Systems Analysis are outlined. 
Notes:  Invited World Scientific review article 
Supplementary Materials  

Codes:  N/A Review article 
Animations:  N/A Review article 
Authors:  E. Kara^{g}, A. Rahman, E. Aulisa, S. Ghosh. 
Sources:  Physical Review arxiv (preliminary manuscript) 
Abstract:  In recent decades computeraided technologies have become prevalent in medicine, however, cancer drugs are often only tested on in vitro cell lines from biopsies. We derive a full threedimensional model of inhomogeneousanisotropic diffusion in a tumor region coupled to a binary population model, which simulates in vivo scenarios faster than traditional cellline tests. The diffusion tensors are acquired using diffusion tensor magnetic resonance imaging from a patient diagnosed with glioblastoma multiform. Then we numerically simulate the full model with finite element methods and produce drug concentration heat maps, apoptosis hotpots, and doseresponse curves. Finally, predictions are made about optimal injection locations and volumes, which are presented in a form that can be employed by doctors and oncologists. 
Notes:  
Supplementary Materials  

Codes:  Available with coauthor E. Kara 
Animations:  N/A 
Authors:  A. Rahman, D. Blackmore. 
Sources:  CNSNS arxiv (preliminary manuscript) 
Abstract:  We identify two types of (compound) dynamical bifurcations generated primarily by interactions of an invariant attracting submanifold with stable and unstable manifolds of hyperbolic fixed points. These bifurcation types  inspired by recent investigations of mathematical models for walking droplet (pilotwave) phenomena  are introduced and illustrated. Some of the oneparameter bifurcation types are analyzed in detail and extended from the plane to higherdimensional spaces. A few applications to walking droplet dynamics are analyzed. 
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Supplementary Materials  

Codes: 

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Authors:  S.R. Dhruba^{g}, A. Rahman, R. Rahman^{g}, S. Ghosh., R. Pal. 
Sources:  BMC 
Abstract:  Clinical studies often track doseresponse curves of subjects over time. One can easily model the doseresponse curve at each time point with Hill equation, but such a model fails to capture the temporal evolution of the curves. On the other hand, one can use Gompertz equation to model the temporal behaviors at each dose without capturing the evolution of time curves across dosage. 
Notes:  Invited special issue on Computational Network Biology: Modeling, Analysis, and Control. 
Supplementary Materials  

Codes:  Available with coauthor S.R. Dhruba 
Animations:  N/A 
Authors:  A. Rahman, S. Ghosh., R. Pal. 
Sources:  Physical Review arxiv (preliminary manuscript) 
Abstract:  It has been shown recently that changing the fluidic properties of a drug can improve its efficacy in ablating solid tumors. We develop a modeling framework for tumor ablation and present the simplest possible model for drug diffusion in a porous spherical tumor with leaky boundaries and assuming cell death eventually leads to ablation of that cell effectively making the two quantities numerically equivalent. The death of a cell after a given exposure time depends on both the concentration of the drug and the amount of oxygen available to the cell, which we assume is the same throughout the tumor for further simplicity. It can be assumed that a minimum concentration is required for a cell to die, effectively connecting diffusion with efficacy. The concentration threshold decreases as exposure time increases, which allows us to compute doseresponse curves. Furthermore, these curves can be plotted at much finer time intervals compared to that of experiments, which may possibly be used to produce a dosethresholdresponse surface giving an observer a complete picture of the drug's efficacy for an individual. In addition, since the diffusion, leak coefficients, and the availability of oxygen is different for different individuals and tumors, we produce artificial replication data through bootstrapping to simulate error. While the usual datadriven model with sigmoidal curves use 12 free parameters, our mechanistic model only has two free parameters, allowing it to be open to scrutiny rather than forcing agreement with data. Even so, the simplest model in our framework, derived here, shows close agreement with the bootstrapped curves and reproduces wellestablished relations, such as Haber's rule. 
Notes:  
Supplementary Materials  

Codes:  Codes [zip] 
Animations:  Animation of partial ablation
Animation of partial ablation of a tumor.Animation of full ablation of a tumor. 
Authors:  A. Rahman. 
Sources:  Chaos arxiv (preliminary manuscript) 
Abstract:  Recent experiments on walking droplets in an annular cavity showed the existence of complex dynamics including chaotically changing velocity. This article presents models, influenced by the kicked rotator/standard map, for both single and multiple droplets. The models are shown to achieve both qualitative and quantitative agreement with the experiments, and make predictions about heretofore unobserved behavior. Using dynamical systems techniques and bifurcation theory, the single droplet model is analyzed to prove dynamics suggested by the numerical simulations. 
Notes:  Invited special issue on Hydrodynamic Quantum Analogs.

Supplementary Materials  

Codes:  Codes [zip]

Animations:  Animation of a walker with constant speed
Animation of a single droplet walking around the annulus with constant speed.Animation of a single walker chaotically changing speed in an annulus.Animation of multiple droplet walking around the annulus with constant speed.Animation of multiple droplet walking around the annulus with speeds starting to destabilize.Animation of multiple droplet walking around the annulus in a chaotic fashion. 
Authors:  Y. Joshi, D. Blackmore, A. Rahman. 
Sources:  arxiv (preliminary manuscript) 
Abstract:  A generalized attracting horseshoe is introduced as a new paradigm for describing chaotic strange attractors (of arbitrary finite rank) for smooth and piecewise smooth maps f from Q to Q, where Q is a homeomorph of the unit interval in real mspace for any integer m > 1. The main theorems for generalized attracting horseshoes are shown to apply to Henon and Lozi maps, thereby leading to rather simple new chaotic strange attractor existence proofs that apply to a range of parameter values that includes those of earlier proofs. 
Notes: 

Supplementary Materials  

Codes:  Available through coauthor Y. Joshi

Animations:  N/A 
Authors:  A. Rahman, I. Jordan^{ug}, D. Blackmore. 
Sources:  Royal Society arxiv (preliminary manuscript) 
Abstract:  It has been observed through experiments and SPICE simulations that logical circuits based upon Chua’s circuit exhibit complex dynamical behaviour. This behaviour can be used to design analogues of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modelled as systems of ordinary differential equations. However, as the number of components in newer circuits increases so does the complexity. This renders continuous dynamical systems models impractical and necessitates new modelling techniques. In recent years, some discrete dynamical models have been developed using various simplifying assumptions. To create a robust modelling framework for chaotic logical circuits, we developed both deterministic and stochastic discrete dynamical models, which exploit the natural recurrence behaviour, for two chaotic NOR gates and a chaotic set/reset flipflop. This work presents a complete applied mathematical investigation of logical circuits. Experiments on our own designs of the above circuits are modelled and the models are rigorously analysed and simulated showing surprisingly close qualitative agreement with the experiments. Furthermore, the models are designed to accommodate dynamics of similarly designed circuits. This will allow researchers to develop ever more complex chaotic logical circuits with a simple modelling framework. 
Notes: 

Supplementary Materials  

Codes:  Codes [zip] 
Animations:  N/A 
Authors:  A. Rahman, Y. Joshi, D. Blackmore. 
Sources:  Regular and Chaotic Dynamics 
Abstract:  Some interesting variants of walking droplet based discrete dynamical bifurcations arising from diffeomorphisms are analyzed in detail. A notable feature of these new bifurcations is that, like Smale horseshoes, they can be represented by simple geometric paradigms, which markedly simplify their analysis. The twodimensional diffeomorphisms that produce these bifurcations are called sigma maps or double sigma maps for reasons that are made manifest in this investigation. Several examples are presented along with their dynamical simulations. 
Notes:  Invited special issue dedicated to the memory of Vladimir Arnold (1937  2010). 
Supplementary Materials  

Codes:  Codes [zip] 
Animations:  Animation of colliding invariant circles
Animation of colliding invariant circles during a global homocliniclike bifurcation. 
Authors:  A. Rahman, D. Blackmore. 
Sources:  CSF arxiv (preliminary manuscript) 
Abstract:  Bouncing droplets on a vibrating fluid bath can exhibit waveparticle behavior, such as being propelled by interacting with its own wave field. These droplets seem to walk across the bath, and thus are dubbed walkers. Experiments have shown that walkers can exhibit exotic dynamical behavior indicative of chaos. While the integrodifferential models developed for these systems agree well with the experiments, they are difficult to analyze mathematically. In recent years, simpler discrete dynamical models have been derived and studied numerically. The numerical simulations of these models show evidence of exotic dynamics such as period doubling bifurcations, Neimark–Sacker (N–S) bifurcations, and even chaos. For example, in [1], based on simulations Gilet conjectured the existence of a supercritical NS bifurcation as the damping factor in his one dimensional path model. We prove Gilet’s conjecture and more; in fact, both supercritical and subcritical (NS) bifurcations are produced by separately varying the damping factor and waveparticle coupling for all eigenmode shapes. Then we compare our theoretical results with some previous and new numerical simulations, and find complete qualitative agreement. Furthermore, evidence of chaos is shown by numerically studying a global bifurcation. 
Notes: 

Supplementary Materials  

Codes:  Codes [zip] 
Animations: 
Animation of a walker before NS bifurcation
Animation of a walker converging to a fixed position before the NeimarkSacker bifurcations.Animation of a walker converging to a fixed path determined by an invariant circle.Animation of a walker converging to a fixed path determined by an invariant circle after bypassing a homoclinic orbit.Animation of a walker converging to a fixed path determined by an invariant circle.Animation of a walker trapped in an orbit created by intertwined invariant circle.Animation of a walker chaotically changing velocity. 
Authors:  R. Goodman, A. Rahman, M.J. Bellanich, C.N. Morrison. 
Sources:  Chaos arxiv (preliminary manuscript) 
Abstract:  We describe a simple mechanical system, a ball rolling along a speciallydesigned landscape, which mimics the wellknown twobounce resonance in solitary wave collisions, a phenomenon that has been seen in countless numerical simulations but never in the laboratory. We provide a brief history of the solitary wave problem, stressing the fundamental role collectivecoordinate models played in understanding this phenomenon. We derive the equations governing the motion of a point particle confined to such a surface and then design a surface on which to roll the ball, such that its motion will evolve under the same equations that approximately govern solitary wave collisions. We report on physical experiments, carried out in an undergraduate applied mathematics course, that seem to exhibit the twobounce resonance. 
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Supplementary Materials  

Codes:  Codes available through R. Goodman 
Animations: 

Authors:  D. Blackmore, A. Rahman, J. Shah. 
Sources:  CSF arxiv (preliminary manuscript) 
Abstract:  A simple discrete planar dynamical model for the ideal (logical) R–S flipflop circuit is developed with an eye toward mimicking the dynamical behavior observed for actual physical realizations of this circuit. It is shown that the model exhibits most of the qualitative features ascribed to the R–S flipflop circuit, such as an intrinsic instability associated with unit set and reset inputs, manifested in a chaotic sequence of output states that tend to oscillate among all possible output states, and the existence of periodic orbits of arbitrarily high period that depend on the various intrinsic system parameters. The investigation involves a combination of analytical methods from the modern theory of discrete dynamical systems, and numerical simulations that illustrate the dazzling array of dynamics that can be generated by the model. Validation of the discrete model is accomplished by comparison with certain Poincaré map like representations of the dynamics corresponding to threedimensional differential equation models of electrical circuits that produce R–S flipflop behavior. 
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Supplementary Materials  

Codes: 

Animations: 

Authors:  A. Rahman 
Sources:  NJIT Library 
Abstract:  Logical circuits and waveparticle duality have been studied for most of the 20th century. During the current century scientists have been thinking differently about these wellstudied systems. Specifically, there has been great interest in chaotic logical circuits and hydrodynamic quantum analogs. Traditional logical circuits are designed with minimal uncertainty. While this is straightforward to achieve with electronic logic, other logic families such as fluidic, chemical, and biological, naturally exhibit uncertainties due to their inherent nonlinearity. In recent years, engineers have been designing electronic logical systems via chaotic circuits. While traditional boolean circuits have easily determined outputs, which renders dynamical models unnecessary, chaotic logical circuits employ components that behave erratically for certain inputs. There has been an equally dramatic paradigm shift for studying waveparticle systems. In recent years, experiments with bouncing droplets (called walkers) on a vibrating fluid bath have shown that quantum analogs can be studied at the macro scale. These analogs help us ask questions about quantum mechanics that otherwise would have been inaccessible. They may eventually reveal some unforeseen properties of quantum mechanics that would close the gap between philosophical interpretations and scientific results. Both chaotic logical circuits and walking droplets have been modeled as differential equations. While many of these models are very good in reproducing the behavior observed in experiments, the equations are often too complex to analyze in detail and sometimes even too complex for tractable numerical solution. These problems can be simplified if the models are reduced to discrete dynamical systems. Fortunately, both systems are very naturally timediscrete. For the circuits, the states change very rapidly and therefore the information during the process of change is not of importance. And for the walkers, the position when a wave is produced is important, but the dynamics of the droplets in the air are not. This dissertation is an amalgam of results on chaotic logical circuits and walking droplets in the form of experimental investigations, mathematical modeling, and dynamical systems analysis. Furthermore, this thesis makes connections between the two topics and the various scientific disciplines involved in their studies. 
Authors:  A. Rahman 
Sources:  SIAM DSweb arxiv (preliminary manuscript) 
Abstract:  Peixoto’s structural stability and density theorems represent milestones in the modern theory of dynamical systems and their applications. Despite the importance of these theorems, they are often treated rather superficially, if at all, in upper level undergraduate courses on dynamical systems or differential equations. This is mainly because of the depth and length of the proofs. 
Notes:  Won runnerup in the 2013 DSweb pedagogy contest. 
Authors:  A. Rahman, N. Sereno, H. Shang. 
Sources:  Department of Defense Technical Note 
I have included lecture notes in the menus below.
Week  Notes  Practice problems 

1  Principles of Applied Math  
2  Basics of Modeling and Dynamical Systems  Problem set 1 
3  Phase Planes  TBD 
4  Phase Planes and Perturbation  Problem set 2 , Solutions 1 
5  Perturbations and Chaos  
6  Maps  
7  Fractals  
8  Asymptotic Integrals  
9  Heat Conduction  
10  Heat Conduction and Dimensional Analysis  
Week  Notes  Homework 

1  Sections 8.1  8.6  Homework 1 
2  Sections 8.6, 8.8, & 10.1  Homework 2 
3  Section 10.1  10.2  Homework 3 
4  Section 10.2, 11.1  11.3  No Homework 
5  Section 11.1  11.3  No Homework 
6  Section 12.1  Homework 4 
7  Section 12.2  Study for Exam I 
8  Section 12.3  No Homework 
9  Section 12.5, 13.2  Homework 5 
10  Section 13.3  Homework 6 
11  Section 13.4  Homework 7 
12  Section 13.5  Homework 8 
13  Section 13.6  Finish Homework 8 
Lecture  Notes  Homework 

1  Linear Equations and Notation  
2  Properties of Matrix Operations and Gaussian Elimination  Sec. 1.2: 26, 28, 32, 34, 38; Sec. 2.2: 16, 18, 22, 25, 28 
3  The Inverse of a Matrix  
4  Elementary Matrices  Pg 71: 2, 4, 8, 10, 16, 18, 19; Pg. 82: 9, 11, 44, and 46 
5  The Determinant  
6  Vector Spaces and Subspaces  
7  Spanning Sets, Linear Independence, Basis, and Dimension  Pg 116: 4, 6, 20, 22; Pg. 131: 1, 17, 22, 23, 28 
8 and 9  Rank and Matrix Subspaces  
10 and 13  Dot Products and Inner Products  
14 and 15  GramSchmidt  
16  Least Squares 
Chapter  Notes  Homework 

1  Sets  1.2 b,c; 1.3 b,c; 1.4 b,c; 1.13; 1.14; 1.25; 1.26; 1.40; 1.41b. 
2  Logic  2.13, 2.17, 2.21, 2.31c, 2.32b,c, 2.33b,c, 2.40, 2.46, 2.47, 2.53b, 2.55. 
3  Direct/contrapositive  3.2, 3.4, 3.9, 3.11, 3.13, 3.17, 3.19, 3.27, 3.29. 
5  Contradiction  5.2, 5.6, 5.13, 5.17, 5.20. 
6  Induction  6.5, 6.10, 6.22, 6.25, 6.34. 
9  Equivalence relations  9.26, 9.27, 9.37, 9.48, 9.49, 9.50. 
10  Functions  10.4c, 10.6 c and e, 10.12 b and d; 10.22, 10.42 b and c, 10.51. 
11  Functions  11.26d 
14  Functions  14.4, 14.6, 14.19, 14.21 
Week  Notes  Homework 

1  Section 1.1  
2  Section 1.2  1.3  Homework 1 
3  Section 2.1  2.4  Homework 2 
4  Section 2.5  3.1  Homework 3 
5  Section 3.2  3.3  Homework 4 
6  Section 3.4  3.5  Homework 5 
7  Section 3.5, 3.7  Exam I Review 
8  Section 4.14.2  Homework 6 
9  Section 4.2  4.3  Homework 7 
10  Section 4.3  5.1  
11  Thank you to Dr. Giorgio Bornia for covering 5.1  5.2  
12  Section 5.15.3  Homework 8 
13  Section 5.4 and 5.6  Exam II Review 
14  Section 6.1  Homework 9 
15  Section 6.26.4  Final Exam Review 
Week  Notes  Homework 

1  Sections 8.1  8.6  Homework 1 
2  Sections 8.8 & 10.1  Homework 2 
3  Section 10.2  Homework 3 
4  Section 11.1  11.3  Exam I Review 
5  Section 12.1  Exam I 
6  Sections 12.2  12.3  Homework 4 
7  Section 12.5  Homework 5 
8  Section 13.2  
9  Section 13.3  Homework 6 
10  Section 13.4  Homework 7 
11  Section 13.5  Homework 8 
12  Section 13.6  Exam II Review 
13  Exam II  Thanksgiving Holiday 
14  Section 14.1 and 14.2  Homework 9 
15  Final Exam  Final Exam Review 
Week  Notes  Hand in Homework 

1  Integration Review & Section 6.1  Sec. 6.1 # 8, 10, 16(sketch), 25(sketch), 62(a,b) 
2  Sections 6.2, 6.3, and 6.4  
3  Sections 6.5, 7.3, 8.1, 8.2, and 8.3  Sec. 6.5 # 2, 8, 19 
4  Exam I Review Solutions  Study for Exam I! 
5  Exam I Solutions  Sec. 8.4 # 1, 12, 20, 44, 57 
6  Sections 8.4, 8.5, 8.7, 8.8, and 10.1  Sec. 8.5 # 9, 18, 30, 31, 38 
7  Exam II Review Solutions  Study for Exam II! 
8  Exam II Solutions  Sec. 10.3 # 3, 6, 9, 11, 13, 19, 20, 23, 25, 27, 33, 35, 36 & MATLAB assignment 1 
9  Sections 10.2, 10.3, 10.4, and 10.5  Sec. 10.4 # 1, 4, 5, 12, 18, 19, 21, 23, 25, 28, 31, 32, 34, 36, 37, 39, 40, 41, 51, 56 & Sec. 10.5 # 5, 7, 9, 18, 19, 21, 29, 31, 35, 38, 42, 55, 56, 57, 58, 59 
10  Sections 10.6 and 10.7  Sec. 10.6 # 5, 7, 9, 10, 11, 12, 13, 15, 19, 21, 23, 27, 30, 34, 35, 37, 39, 41, 44, 47, 50, 51, 53 
11  Sections 10.8, 10.9, and 10.10  Sec. 10.7 # 22, 24, 31, 32, 37, 55 
12  Exam III Review Solutions  Study for Exam III! 
13  Exam III Solutions  MATLAB assignment 2 
My main bike is a Surly Crosscheck that I take to various
places.
Places I like to hike:
Breakneck Ridge (Bear Mountain)
Boroughs Range (Catskills: Slide Mountain, Cornell,
Whitenberg)
Veerkerderkill Falls and Ice Caves (Sam's point)
Mount Tamany (Delaware Watergap)
Palisades (Shore trail + Great Steps)
Acadia National Park
Great Smokey Mountain National Park
Green Mountain National Forest
White Mountain National Forest
Some interesting places I have biked through:
Cycling around North
Jersey
Here is a video of my bike commute:
Kearny to Newark
I used to play football in high school, and still try to
play from time to time. Now I'm more of a spectator,
though. I mostly follow the USMNT, Juventus,
Liverpool, and Sheffield Wednesday.