Denis Blackmore

• PhD
• Professor at New Jersey Institute of Technology

We report on the development of a recurrent neural network (RNN) that models the density relaxation process in initially disordered assemblies of monodisperse spheres within a tapped, three-dimensional container. The RNN model is trained on coordinate data sets generated from granular dynamics simulations to examine microstructure development. In p...

There is studied the integrability of a generalized Gurevich-Zybin dynamical system based on the differential-algebraic and geometrically motivated gradient-holonomic approaches. There is constructed the corresponding Lax type represenation, compatible Poisson structures as well as the integrability of the related Hunter-Saxton reduction. In partic...

We describe a class of self-dual dark nonlinear dynamical systems \textit{a priori} allowing their quasi-linearization, whose integrability can be effectively studied by means of a geometrically motivated gradient-holonomic approach. Using this integrability scheme, we study a new self-dual, dark nonlinear dynamical system on a smooth functional ma...

We review some analytic, measure-theoretic and topological techniques for studying ergodicity and entropy of discrete dynamical systems, with a focus on Boole-type transformations and their generalizations. In particular, we present a new proof of the ergodicity of the 1-dimensional Boole map and prove that a certain 2-dimensional generalization is...

We show that there is a mildly nonlinear three-dimensional 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, whi...

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 wa...

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...

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the...

We review a modern differential geometric description of the fluid isotropic motion and featuring it the diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. There is analyzed the adiabatic liquid dynamics, within which, following the general approach, th...

The aim of this paper is to experimentally and analytically study the solutocapillary flow induced in a waterbody due to the presence of a solute source on its surface and the mixing induced by this flow of the solutes and gases dissolved at and near the surface into the waterbody. According to the analytic solution, the induced flow is analogous t...

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 (pilot-wave) phenomena - are introduced and illust...

We present a study of the dynamics of a system of masses connected by springs and repelling by a 1∕r potential in 1D. The present study focuses on the dynamics in the regime where the repulsive force dominates the dynamics of the system. We conjecture that such a system may be approximately modeled by an alignment of repelling rigid bar magnets tha...

There are studied algebraic properties of quadratic Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Their relations both with derivations of symmetric tensor algebras and Yang–Baxter structures on the adjacent Lie algebras are demonstrated. Special attention is paid to quadratic Poisson...

The differential-geometric and topological structures related to the Delsarte transmutation operators and the Gelfand–Levitan–Marchenko equations that describe these operators are studied by using suitable differential de Rham–Hodge–Skrypnik complexes. The correspondence between the spectral theory and special Berezansky-type congruence properties...

A rigorous dynamical systems-based hierarchy is established for the definitions of entropy of Shannon (information), Kolmogorov–Sinai (metric) and Adler, Konheim & McAndrew (topological). In particular, metric entropy, with the imposition of some additional properties, is proven to be a special case of topological entropy and Shannon entropy is sho...

We present a brief survey of the original results obtained by the authors in the theory of Delsarte–Lions transmutations of multidimensional spectral differential operators based on the classical works by Yu. M. Berezansky, V. A. Marchenko, B. M. Levitan, and R. G. Newton, on the well-known L. D. Faddeev’s survey, the book by L. P. Nyzhnyk, and the...

We show that there is a mildly nonlinear three-dimensional 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, whi...

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebrai...

Our review is devoted to Lie-algebraic structures and integrability properties of an interesting class of nonlinear dynamical systems called the dispersionless heavenly equations, which were initiated by Plebanski and later analyzed in a series of articles. The AKS-algebraic and related $\mathcal{R}$-structure schemes are used to study the orbits o...

We have constructed a new fractional pseudo-differential metrized operator Lie algebra on the axis, enabling within the general Adler–Kostant–Symes approach the generation of infinite hierarchies of integrable nonlinear differential-fractional Hamiltonian systems of Korteweg–de Vries, Schrödinger and Kadomtsev–Petviashvili types. Using the natural...

We study algebraic properties of Poisson brackets on non-associative non-commutative algebras, compatible with their multiplicative structure. Special attention is paid to the Poisson brackets of the Lie-Poisson type, related with the special Lie-structures on the differential-topological torus and brane algebras, generalizing those studied before...

It has been observed through experiments and SPICE simulations that logical circuits based upon Chua's circuit exhibit complex dynamical behavior. This behavior can be used to design analogs of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modeled as systems of ordina...

The complete integrability of a generalized Riemann type hydrodynamic hierarchy is studied by means of a novel combination of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, a Lax representation and a related infinite hierarchy of conservation laws are constructed. The current investigation provid...

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 two-dimensional diffeomorphisms that pr...

In Fall 2011, the Newark College of Engineering (NCE) in collaboration with the College of Science and Liberal Arts (CSLA) began implementing two initiatives focused on the first year experience. Community Connections, NJIT's learning community program, was initiated through curricular-based cohorts in an effort to improve student engagement. Acade...

The decay of energy within particulate media subjected to an impulse is an issue of significant scientific interest, but also one with numerous important practical applications. In this paper, we study the dynamics of a granular system exposed to energetic impulses in the form of discrete taps from a solid surface. By considering a one-dimensional...

Chaotic Set/Reset (RS) flip-flop circuits are investigated once again in the context of discrete planar dynamical system models of the threshold voltages, but this time starting with simple bilinear (minimal) component models derived from first principles. The dynamics of the minimal model is described in detail, and shown to exhibit some of the ex...

We identify two rather novel 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 (pilot-wave) phenomena - are introduc...

The particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) techniques are used to study the flow induced on the surface of a body of saltwater when a drop impinges on its surface or when a source is present on the surface. The measurements show that the impingement of a fresh water drop causes a strong axisymmetric solutocapi...

The particle image velocimetry (PIV) technique is used to study the flow induced on the surface of a salt waterbody when a drop impinges on the surface or when a source is present on the surface. The measurements show that the impingement of a fresh water drop causes a strong axisymmetric solutocapillary flow about the vertical line passing through...

The work is devoted to recent investigations of the Lax–Sato compatible linear vector field equations, especially to the related Lie-algebraic structures and integrability properties of a very interesting class of nonlinear dynamical systems called the dispersionless heavenly type equations, which were initiated by Plebański and later analyzed in a...

Recent particulate flow research using a discrete element simulation-dynamical systems approach is described. The simulation code used is very efficient and the mathematical model is an integro-partial differential equation. Examples are presented to show the effectiveness of the approach.

A general differential-algebraic approach is devised for constructing multi-component Hamiltonian operators as differentiations on suitably constructed loop Lie algebras. The related Novikov-Leibniz algebraic structures are presented and a new non-associative “Riemann” algebra is constructed, which is closely related to the infinite multi-component...

The review is devoted to a novel Lie-algebraic approach to studying integrable heavenly type multi-dimensional dynamical systems and its relationships to old and recent investigations of the classical Buhl problem of describing compatible linear vector field equations, its general Pfeiffer and modern Lax-Sato type special solutions. Eespecially we...

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 m-space for any integer m > 1. The main theorems for generalized attracting horseshoes are shown to apply to...

We review the modern classical electrodynamics problems and present the related main fundamental principles characterizing the electrodynamical vacuum-field structure. We analyze the models of the vacuum field medium and charged point particle dynamics using the developed field theory concepts. There is also described a new approach to the classica...

Bouncing droplets on a vibrating fluid bath can exhibit wave-particle 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 integro-differential models...

We study versal deformatiions of the Pfeiffer-Lax-Sato type vector field equations, related with a centrally extended metrized Lie algebra as the direct sum of vector fields and differential forms on torus. © 2017 D. Blackmore, A. Prykarpatski, M. Vovk, P. Pukach, Ya. Prykarpatsky.

We review the modern classical electrodynamics problems and present the related main fundamental principles characterizing the electrodynamical vacuum-field structure. We analyze the models of the vacuum field medium and charged point particle dynamics using the developed field theory concepts. There is also described a new approach to the classica...

The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained...

This paper communicates the results of a synergistic investigation that initiates our long term research goal of developing a continuum model capable of predicting a variety of granular flows. We consider an ostensibly simple system consisting of a column of inelastic spheres subjected to discrete taps in the form of half sine wave pulses of amplit...

Logical RS flip-flop circuits are investigated once again in the context of discrete planar dynamical systems, but this time starting with simple bilinear (minimal) component models based on fundamental principles. The dynamics of the minimal model is described in detail, and shown to exhibit some of the expected properties, but not the chaotic reg...

Trying to predict whether a crisis or emergency event is going to occur is a challenge, but attempting to do so without a quantifiable scale makes the task a virtual mission impossible. A crisis scale is also needed to perform effective post-crisis analysis. The extant scales, however, are inadequate. To address these issues, we developed the unifi...

We consider the behavior of a column of spheres that are subjected to a time-dependent external load in the form of vertical taps. Of interest are various dynamical properties, such as the motion of its mass center, its response to taps of different intensities and forms, and the effect of system size and material properties. The interplay between...

We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field theory approach to the Lagrangian and Hamilto...

Trying to predict whether a crisis or emergency event is going to occur is a challenge, but attempting to do so without a quantifiable scale makes the task a virtual mission impossible. A crisis scale is also needed to perform effective post-crisis analysis. The extant scales, however, are inadequate. To address these issues, we developed the unifi...

A relatively new approach to analyzing integrability, based upon differential-algebraic and symplectic techniques, is applied to some “dark equations ”of the type introduced by Boris Kupershmidt. These dark equations have unusual properties and are not particularly well-understood. In particular, dark three-component polynomial Burgers type systems...

A low-dimensional center-of-mass dynamical model is devised as a simplified means of approximately predicting some important aspects of the motion of a vertical column comprised of a large number of particles subjected to gravity and periodic vertical tapping. This model is investigated first as a continuous dynamical system using analytical, simul...

A discrete dynamical system in Euclidean m-space generated by the iterates of an asymptotically zero map f, satisfying f(x) goes to zero as x goes to infinity, must have a compact global attracting set $A $. The question of what additional hypotheses are sufficient to guarantee that A has a minimal (invariant) subset A* that is a chaotic strange at...

The Lax type integrability of a two-component polynomial Burgers type dynamical system within a differential-algebraic approach is studied, its linear adjoint matrix Lax representation is constructed. A related recursion operator and infnite hierarchy of Lax integrable nonlinear dynamical systems of the Burgers-Korteweg-de Vries type are derived by...

A new exactly solvable spatially one-dimensional quantum superradiance model describing a charged fermionic medium interacting with an external electromagnetic field is proposed. The infinite hierarchy of quantum conservation laws and many-particle Bethe eigenstates that model quantum solitonic impulse structures are constructed. The Hamilton opera...

We present an approximate dynamical systems model for the mass center trajectory of a tapped column of N uniform, inelastic, spheres (diameter d), in which collisional energy loss is governed by the Walton-Braun linear loading-unloading soft interaction. Rigorous analysis of the model, akin to the equations for the motion of a single bouncing ball...

Invariant ergodic measures for generalized Boole type transformations are studied using an invariant quasi-measure generating function approach based on special solutions to the Frobenius--Perron operator. New two-dimensional Boole type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.

We review the modern classical electrodynamics problems and present the related main fundamental principles characterizing the electrodynamical vacuumfield structure. We analyze the models of the vacuumfield medium and charged point particle dynamics using the developed field theory concepts. There is also described a new approach to the classical...

The complete integrability of a generalized Riemann type hydrodynamic system is studied by means of symplectic and differential-algebraic tools. A compatible pair of polynomial Poissonian structures, Lax type representation and related infinite hierarchy of conservation laws are constructed.

We report our findings on the evolution of solids fraction in a tapped system of inelastic, frictional spheres as a function of the applied acceleration obtained via discrete element simulations. Animations of the simulation data reveal the propagation of a wave initiated from the base that causes local rearrangements of the particles ultimately le...

It is shown how functional-analytic gradient-holonomic structures can be used for an isospectral integrability analysis of nonlinear dynamical systems on discrete manifolds. The approach developed is applied to obtain detailed proofs of the integrability of the discrete nonlinear Schrödinger, Ragnisco–Tu and Riemann–Burgers dynamical systems.

A general distribution function for the heights of anisotropic engineering surfaces is obtained by extending earlier work on surface profiles. The derivation starts from a functional description of surface heights that involves fractal quantities and is comprehensive enough to include almost all of the mathematical models for surface topography tha...

An approach based on the spectral and Lie - algebraic techniques for constructing vertex operator representation for solutions to a Riemann type Gurevicz-Zybin hydrodynamical hierarchy is devised. A functional representation generating an infinite hirerachy of dispersive Lax type integrable flows is obtaned.

The existence, convergence, realizability and stability of solutions of differential operator equations obtained via a novel projection-algebraic scheme are analyzed in detail. This analysis is based upon classical discrete approximation techniques coupled with a recent generalization of the Leray-Schauder fixed point theorem. An example is include...

The dynamics of a vertical stack of particles subject to gravity and a sequence of small, periodically applied taps is considered. First, the motion of the particles, assumed to be identical, is modeled as a system of ordinary differential equations, which is analyzed with an eye to observing connections with finite-dimensional Hamiltonian systems....

A novel approach based upon vertex operator representation is devised to study the AKNS hierarchy. It is shown that this method reveals the remarkable properties of the AKNS hierarchy in relatively simple, rather natural and particularly effective ways. In addition, the connection of this vertex operator based approach with Lie-algebraic integrabil...

This distinctive volume presents a clear, rigorous grounding in modern nonlinear integrable dynamics theory and applications in mathematical physics, and an introduction to timely leading-edge developments in the field—including some innovations by the authors themselves —that have not appeared in any other book. The exposition begins with an intro...

Managers of emergencies face challenges of complexity, uncertainty, and unpredictably. Triadic constraints imply requisite parsimony in describing the essence of the emergency, its magnitude and direction of development. Linguistic separation increases as the crisis management organization is more complex and made up of diverse constituents. Theref...

A regular approach to studying the Lax type integrability of the AKNS hierarchy of nonlinear Lax type integrable dynamical systems in the vertex operator representation is devised. The relationship with the Lie-algebraic integrability scheme is analyzed, the connection with the tau-function representation is briefly discussed.

A competing market model with a polyvariant profit function that assumes "zeitnot" stock behavior of clients is formulated within the banking portfolio medium and then analyzed from the perspective of devising optimal strategies. An associated Markov process method for finding an optimal choice strategy for monovariant and bivariant profit function...

This paper aims at an equation based simulation, CAD modeling and manufacturing of a fractal surface embedded on a musical cymbal. This study is a proof-of-concept of a new method of complex-surface characterization, design and manufacturing using an equation-based approach. A cymbal shape was chosen to carry the fractal profile because generally m...

The dynamics of a vertical stack of particles subject to gravity and a sequence of small, periodically ap-plied taps is considered. First, the motion of the particles, assumed to be identical, is modeled as a system of ordinary differential equations, which is analyzed with an eye to observing connections with finite-dimensional Hamiltonian sys-tem...

Blackmore-Samulyak-Rosato (BSR) fields, originally developed as a means of obtaining reliable continuum approximations for granular flow dynamics in terms of relatively simple integro-differential equations, can be used to model a wide range of physical phenomena. Owing to results obtained for one-dimensional granular flow configurations, it has be...

A simple discrete planar dynamical model for the ideal (logical) R–S flip-flop 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 flip-flop circuit, such as an intrinsic instabilit...

Managers of emergencies face challenges of complexity, uncertainty, and unpredictably. Triadic constraints imply requisite parsimony in describing the essence of the emergency, its magnitude and direction of development. Linguistic separation increases as the crisis management organization is more complex and made up of diverse constituents. Theref...

We discuss the ω-limit sets of a flow using the Conley theory, chain recurrence and Morse decompositions. Our results generalize and improve the related result of J. Schropp [Z. Angew. Math. Mech. 76, No. 6, 349–356 (1996; Zbl 0879.34049)], and we also show how they can be used as a basis for some new criteria for the existence of periodic orbits....

This paper is concerned with analysis of coupled fractional reaction–diffusion equations. As an example, the reaction–diffusion model with cubic nonlinearity and Brusselator model are considered. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the in...

a b s t r a c t Here we study the solution set of a nonlinear operator equation in a Banach subspace L n ⊂ C (X) by reducing it to a Leray–Schauder type fixed point problem. The subspace L n is of finite codimension n ∈ Z + in C (X), with X an infinite compact Hausdorff space, and is defined by conditions α * i (f) := X f (x)dµ i (x) = 0, f ∈ C (X)...

In this paper, an upper and lower solutions theory for the forced superlinear Duffing equation x" + f(t)x' + g(t, x) = 8 a.e. t ∈ [0, T] x(0)=x(T),x'(0)=x'(T) is established, and the multiplicity of periodic solutions is discussed, where / ∈ L1([O, T]), g(t, x) is a Carathéodory function, and s is a real parameter.

A simple–yet plausible–model for B-type vortex breakdown flows is postulated; one that is based on the immersion of a pair of slender coaxial vortex rings in a swirling flow of an ideal fluid rotating around the axis of symmetry of the rings. It is shown that this model exhibits in the advection of passive fluid particles (kinematics) just about al...

A simple Hamiltonian dynamical systems model for vortex breakdown of the bubble-type (B-type) is developed and analyzed. This model is constructed using the flow induced by two point vortices moving in a half-plane immersed in an ideal (= inviscid and incompressible) fluid with an ambient uniform horizontal velocity. It is shown - using a combinati...

Grobli (1877) laid the foundation for the analysis of the motion of three point vortices in a plane by deriving governing equations for triangular configurations of the vortices. Synge (1949) took this formulation one step further to that of a similar triangle of unit perimeter, via trilinear coordinates. The final reduced problem is governed by an...

This volume contains a selection of papers presented at the 2008 Conference on Frontiers of Applied and Computational Mathematics (FACM'08), held at the New Jersey Institute of Technology (NJIT), May 19-21, 2008. The papers reflect the conference themes of mathematical biology, mathematical fluid dynamics, applied statistics and biostatistics, and...

This chapter provides an overview of computational topology. The first usage of the term "computational topology" appears to have occurred in the dissertation of M. Mantyla. The focus there was upon the connective topology joining vertices, edges, and faces in geometric models, frequently informally described as the symbolic information of a solid...

The governing equations of motion of two point vortices in an ideal fluid in the plane has a Hamiltonian formulation that is completely integrable, so the dynamics are regular in the sense that one has quasiperiodic solutions confined to invariant two-dimensional tori accompanied by periodic orbits. Moreover, it is well known that the same is true...

We study an N -vortex problem having J of them forming a cluster, which means the distances between the vortices in the cluster is much smaller by O (ε) than the distances, O (ℓ), to the N – J vortices outside of the cluster. With the strengths of N vortices being of the same order, the velocity and time scales for the motion of the J vortices rela...

The motion of three point vortices in an ideal fluid in a plane comprises a Hamiltonian dynamical system – one that is completely integrable, so it exhibits numerous periodic orbits, and quasiperiodic orbits on invariant tori. Certain perturbations of three vortex dynamics, such as three vortex motion in a half-plane, are also Hamiltonian, but not...

A finite element code based on the level-set method is used to perform direct numerical simulations (DNS) of the transient and steady-state motion of bubbles rising in a viscoelastic liquid modelled by the Oldroyd-B constitutive equation. The role of the governing dimensionless parameters, the capillary number (Ca), the Deborah number (De) and the...

We develop the Cartan-Monge geometric approach to the characteristic method for nonlinear partial differential equations of the first and higher orders. The Hamiltonian structure of characteristic vector fields related with nonlinear partial differential equations of the first order is analyzed, and tensor fields of special structure are constructe...