# Denis Blackmore

### Contact Info

Title: | Professor |

Email: | denis.l.blackmore@njit.edu |

Office: | CULM 517 |

Hours: | Mon.: 11:30 - 12:30; Thurs.: 4:30 - 5:30 pm; Fri.: 1 - 2 pm |

Phone: | 973-596-3495 |

Dept: | Mathematical Sciences |

## About

### About Me

I am a Professor of Mathematical Sciences at the NJIT, a founding member of the Center for Applied Mathematics and Statistics, and a member of the Particle Technology Center at NJIT. My research focus is on dynamical systems theory and applications. Dynamical systems (nonlinear dynamics) theory is a rich amalgam of techniques from algebra, analysis, chaos theory, differential equations, differential geometry, algebraic and differential topology, fractal geometry, integrability theory, geometry and singularity theory, and has important applications in every branch of science and engineering, which may account for my eclectic interests and the time I have devoted to collaborative research in various fields of engineering and science. In addition to several fundamental theoretical contributions, I have worked on many applied problems in such areas as acoustics, automated assembly, biological populations, computational topology, computer aided geometric design, fluid mechanics, granular flows, plant growth (phyllotaxis), relativistic and quantum physics, rough surface analysis and vortex dynamics. Some of my research has been supported by grants from the National Science Foundation and Office of Naval Research.

In addition to my research, for which I received the Harlan Perlis Research Award from NJIT in 1993, I am devoted to instruction at both the undergraduate and graduate level and have won several teaching awards, including a Mathematical Association of America Award for Distinguished College or University Teaching of Mathematics in 2015. I have developed perhaps a dozen graduate and undergraduate courses and was a principal contributor to the development of the Ph.D. Program in Mathematical Sciences at NJIT.

I am an associate editor of Mechanics Research Communications and serve on the editorial boards of several mathematics and mathematical physics journals and a book series in mathematical physics.

### Education

- PhD, Mathematics, Polytechnic University (now NYU –Tandon School of Engineering), 1971
- MS, Mathematics, Polytechnic University (now NYU –Tandon School of Engineering), 1966
- BS, Aerospace Engineering, Polytechnic University (now NYU-Tandon School of Engineering), 1965

## Teaching

### Classes Taught

- Math 656 (Complex Variables) - Spring17
- Math 335 (Vector Analysis) - Spring17
- Math 491.725 (Independent Study in Algebraic Topology) - Spring17
- Math 332 (Complex Variables) – Fall16
- Math 745 (Analysis II) – Spring16
- Math 331 (Introduction to Partial Differential Equations) – Spring16
- Math 645 (Analysis I) – Fall15
- Math 111H (Honors Calculus I) – Fall15
- Math 111 (Calculus I) – Spring15
- Math 725 (Independent Study in Functional Analysis) – Spring15
- Math 491 (Independent Study in Chaos/with A. Rahman) – Spring15
- Math 545 (Introduction to Mathematical Analysis) – Fall14
- Math 480 (Introduction to Mathematical Analysis) – Fall14
- Math 491 (Independent Study in Topology) – Fall14
- Math 211 (Calculus IIIa) – Fall14

## Research

### Research Interests

- Dynamical systems theory and applications
- Integrable and near-integrable Hamiltonian dynamical systems
- Dynamics of granular and other material flows
- Fluid - especially vortex - dynamics
- Analysis of equations of mathematical physics
- Differential and algebraic topology and applications including computational differential topology
- Computer aided geometric design
- Applications of fractal geometry – especially in metrology
- Analysis and applications of differentiable varieties

### Current Research

- Chaotic strange attractors: identification, analysis, construction, applications, and related SRB measures
- New exotic dynamical bifurcations associated with dynamical systems models of walking droplets and other phenomena
- Development of novel methods for determining and exploiting the integrability or near-integrabilty of infinite-dimensional Hamiltonian dynamical systems arising in classical and modern physics
- Dynamical system modeling and analysis of magnetic particle flows
- Modeling and analysis of capillarity driven fluid flows
- Dissipative solitons and their applications
- Higher dimensional vortex dynamics associated with Bose—Einstein condensates and other fields

## Publications

### Selected Publications

- Joshi, Y., Blackmore, D. and Rahman, A., Generalized attracting horseshoes and chaotic strange attractors, (arxiv) (submitted 2017)
- Artemovych, O., Blackmore, D. and Prykarpatski, A., Poisson brackets, Novikov—Leibniz structures and integrable Riemann hydrodynamic systems, J. Nonlin. Math. Phys. 24 (2017), 41-72.
- Rahman, A. and Blackmore, D., Neimark—Sacker bifurcations and evidence of chaos in a discrete dynamical system model of walkers, Chaos, Solitons & Fractals 91 (2016), 339-349.
- Bogolubov (Jr.), N., Prykarpatski, A. and Blackmore, D., Maxwell--Lorentz electrodynamics models revisited via the Lagrangian formalism and the Feynman proper time paradigm, Mathematics 3 (2015), 190 – 257; doi:10.3390/math 3020190.
- Joshi, Y. and Blackmore, D., Strange attractors for asymptotically zero maps, Chaos, Solitons & Fractals 68 (2014), 123-138.
- Blackmore, D. and Prykarpatsky, A., Dark equations and their light integrability, J. Nonlin. Math. Phys. 21 (2014), 407-428.
- Blackmore, D., Rosato, A., Tricoche, X., Urban, K. and Zuo, L., Analysis, simulation and visualization of 1D tapping dynamics via reduced dynamical models, Physica D 273-274 (2014), 14-27.
- Prykarpatsky, A. and Blackmore, D., New vortex invariants in magneto-hydrodynamics and a related helicity theorem, Chaotic Modeling and Simulation 2 (2013), 239-245.
- Blackmore, D., Rosato, A., Tricoche, X., Urban, K. and Ratnaswamy, V., Tapping dynamics for a column of particles and beyond, J. Mech. Materials & Structures 6 (2011), 71-86.
- Blackmore, D., Prykarpatsky, A., Samoylenko, V., Nonlinear Dynamical Systems of Mathematical Physics: Spectral and Symplectic Integrability Analysis, World Scientific, Singapore, 2011.
- Blackmore, D., Urban, K. and Rosato, A., Integrability analysis of regular and fractional Blackmore-Samulyak-Rosato fields, Condensed Matter Phys. 13 (2010), 43403: 1-7.
- Joshi, Y. and Blackmore, D., Bifurcation and chaos in higher dimensional pioneer-climax systems, Int’l. Electronic J. Pure and Appl. Math. 1 (3) (2010), 303-337.
- Prykarpatsky, A. and Blackmore, D., A solution set analysis of a nonlinear operator equation using a Leray--Schauder type fixed point approach, Topology 48 (2009), 182-185.
- Rohn, E. and Blackmore, D., A unified localizable emergency events scale, Int. J. Information Systems for Crisis Response & Management (IJISCRAM) 1 (2009), 1-14.
- Wang, C., Blackmore, D. and Wang, X., Upper and lower solutions method for a superlinear Duffing equation, Commun. Appl. Nonlin. Anal. 16 (2009), 19-29.
- Blackmore, D., Rahman, A. and Shah, J., Discrete dynamical modeling and analysis of the R-S flip-flop circuit, Chaos, Solitons and Fractals 42 (2009), 951-963.
- Gafiychuk, V., Datsko, B., Meleshko, V. and Blackmore, D., Analysis of the solutions of coupled nonlinear fractional reaction-diffusion equations, Chaos, Solitons and Fractals 41 (2009), 1095-1104.
- Blackmore, D., Brøns, M. and Goullet, A., A coaxial vortex ring model for vortex breakdown, Physica D 237 (2008), 2817-2844.
- Pillapakkam, S., Singh, P., Blackmore, D., and Aubry, N., Transient and steady state motion of rising bubble in a viscoelastic fluid, J. Fluid Mech. 589 (2007), 215-252.
- Blackmore, D., Ting, L. and Knio, O., Studies of perturbed three vortex dynamics, J. Math. Phys. 48 (2007), 065402 (30 pages).
- Blackmore, D. and Mileyko, Y., Computational differential topology, Appl. Gen. Topology 8, No. 1 (2007), 35-92.
- Champanerkar, J. and Blackmore, D., Pitchfork bifurcations of invariant manifolds, Topology and Its Applications 154 (2007), 1650-1663.
- Abdel-Malek, K., Yang, J., Blackmore, D. and Joy, K., Swept volumes: foundations, perspectives and applications, Int. J. Shape Modeling 12 (2006), 87-127.
- Blackmore, D., New models for chaotic dynamics, Regular & Chaotic Dynamics (Special Poincaré 150th Anniversary Issue) 10 (2005), 307-321.
- Aboobaker, N., Blackmore, D. and Meegoda, J., Mathematical modeling of the movement of suspended particles subjected to acoustic and flow fields, Applied Math. Modeling 29 (2005), 515-532.
- Blackmore, D. and Wang, C., Morse index for autonomous linear Hamiltonian systems, Int. J. Diff. Eqs. and Appl. 7 (2003), 295-309.
- Rosato, A., Blackmore, D., Zhang, N. and Lan, Y., A perspective on vibration-induced size segregation of granular materials, J. Chem. Eng. Science. 57 (2002), 265-275.
- Chen, J. and Blackmore, D., On the exponentially self-regulating population model, Chaos, Solitons and Fractals 14 (2002), 1433-1450.
- Blackmore, D., Chen, J., Perez, J. and Savescu, M., Dynamical properties of discrete Lotka-Volterra equations, Chaos, Solitons and Fractals 12 (2001), 2553-2568.
- Blackmore, D. and Knio, O., Hamiltonian structure for vortex filament flows, ZAMM 81S (2001), 145 - 148.
- Abdel-Malek, K., Yang, J. and Blackmore, D., On swept volume formulations: implicit surfaces, Computer-Aided Design 33 (2001), 113-121.
- Blackmore, D. and Knio, O., KAM theory analysis of the dynamics of three coaxial vortex rings, Physica D 140 (2000), 321-348.
- Wang, W., Hsu, C.T. and Blackmore, D., Geometrical formulation for strip yielding model with variable cohesion and its analytical solution, Int. J. Solids Struc. 37 (2000), 7533-7546.
- Blackmore, D., Samulyak, R. and Leu, M.C., Trimming swept volumes, Computer-Aided Design 31 (1999), 215-223.
- Blackmore, D., Samulyak, R. and Rosato, A., New mathematical models for particle flow dynamics, J. Nonlin. Math. Phys. 6 (1999), 198-221.
- Blackmore, D. and Zhou, G., A new fractal model for anisotropic surfaces, Int. J. Mach. Tools Manufact. 38 (1998), 551-557.
- Blackmore, D., Leu, M.C., Wang, L.P. and Jiang, H., Swept volumes: a retrospective and prospective view, Neural, Parallel & Scientific Computations 5 (1997), 81-102.
- Blackmore, D., Leu, M.C. and Wang, L.P., The sweep-envelope differential equation algorithm and its application to NC machining verification, Computer-Aided Design 29 (1997), 629-637.
- Blackmore, D. and Kappraff, J., Integrable discrete dynamics and Fibonacci sequences, ZAMM 76 S (1996), 49-52.
- Blackmore, D. and Zhou, G., A general fractal distribution function for rough surface profiles, SIAM J. Appl. Math. 56 (1996), 1694-171.
- Blackmore, D., Leu, M.C. and Shih, F., Analysis and modelling of deformed swept volume, Computer-Aided Design 26 (1994), 315-326.
- Zhou, G., Leu, M.C. and Blackmore, D., Fractal geometry model for wear prediction, Wear 170 (1993), 91-101.
- Blackmore, D. and Leu, M.C., Analysis of swept volume via Lie groups and differential equations, Int. J. Robot. Res. 11 (1992), 516-537.
- Blackmore, D. and Ting, L., Surface integral of its mean curvature, SIAM Rev. 27 (1985), 569-572.
- Blackmore, D., The describing function for bounded nonlinearities, IEEE Trans. Circuits Syst. 28 (1981), 442-447.
- Blackmore, D., An example of a local flow on a manifold, Proc. Amer. Math. Soc. 42 (1974), 208-213.
- Blackmore, D., On the local normalization of a vector field at a degenerate critical point, J. Diff. Eqs. 14 (1973), 338-359.
- Blackmore, D., Flows about a critical point with a single zero characteristic root, J. Diff. Eqs. 13 (1973), 403-431.

CAD/CAM Related Publications

1. ´´Analysis of Swept Volumes via Lie Groups and Differential Equations´´ (with M.C. Leu), International Journal of Robotics Research, 11, 1992, pp. 516-537.

2. ´´Applications of Flows and Envelopes to NC Machining´´ (with M.C. Leu and K.K. Wang), Annals of CIRP, 41, 1992, pp. 493-496.

3. ´´Fractal Geometry Model for Wear Prediction´´ (with G. Zhou and M.C. Leu), Wear, 170/1, 1993, pp. 91-101.

4. ´´The Flow Approach to CAD/CAM Modeling of Swept Volume´´(with H. Jiang and M.C. Leu), Advances in Manufacturing Systems, Elsevier, 1994, pp. 341-346.

5. ´´Analysis and Modelling of Deformed Swept Volume´´(with M.C. Leu and F. Shih), Computer Aided Design, 26, 1994, pp. 315-326.

6. ´´Improved Flow Approach for Swept Volumes´´ (with M.C. Leu and D. Qin), Proc. Japan-USA Symposium on Flexible Automation, 1994, pp. 1191-1198.

7. ´´Application of Sweep Differential Equation Approach to Nonholonomic Motion Planning´´ (with Z. Deng and M.C. Leu), Proc. Japan-USA Symposium on Flexible Automation, 1994, pp. 1025-1034.

8. ´´Implementation of the SDE Method to Represent Cutter Swept Volumes in 5-Axis NC Milling´´(with M.C. Leu, L. Wang and K. Pak), Proc. International Conference on Intelligent Manufacturing, 1995, pp. 211-220. 9. ´´Analysis of Sweep Classes: An Application of Differential Topology to Manufacturing Science´´(with M.C. Leu), SIAM J. Applied Math. (to appear)

10. ´´A General Fractal Distribution Function for Rough Surface Profiles´´(with G. Zhou), SIAM J. Applied Math. 56, 1996, pp. 1694-1719.

11. ´´Hamiltonian Structure of Axial Benney-Type Hydrodynamic and Boltzmann-Vlasov Kinetic Equations with Applications to Manufacturing Science´´(with A. Prykarpatsky and N. Bogoliubov), Nuovo Cimento. (to appear).

12. ´´The Sweep-Envelope Differential Equation Algorithm and Its Application to NC Machining Verification´´ (with M.C. Leu and L. Wang), Computer Aided Design 29, 1997, pp. 629-637.

13. ´´Swept volumes: a retrospective and prospective view´´(with M.C. Leu, L.P. Wang, and H. Jiang), Neural, Parallel & Scientific Computations 5, 1997, pp. 81-102.

14. ´´A verification program for 5-axis NC machining with general APT tools´´ (with M.C. Leu and L.P. Wang), Annals of CIRP 46, 1997, pp. 419-424.

15. ´´Simulation of NC machining with cutter deflection by modeling deformed swept volume´´(with M.C. Leu and F. Lu), Annals of CIRP 47, 1998, pp. 441-446.

16. ´´Trimming swept volumes´´(with R. Samulyak and M.C. Leu), Computer-Aided Design 31, 1999, pp. 215-223.

17. ´´Swept volume computation for machining simulation in virtual reality applications´´(with B. Maiteh, M.C. Leu and L. Abdel-Malek), J. Materials Processing & Manufacturing Science 7, 1999, pp. 380-390.

18. ´´A singularity theory approach to swept volumes´´(with R. Samulyak and M.C. Leu), Int. J. Shape Modeling 6, 2000, pp. 105-129.

19. ´´On swept volume formulations: implicit surfaces´´(with K. Abdel-Malek and J. Yang), Compuer-Aided Design 33, 2001, pp.113-121.

20. ´´Creation of freeform solid models in virtual reality´´(with M.C. Leu and B. Maiteh), Annals of CIRP 50, 2001, pp. 73-76.

### Recent Publications

1. ´´The integrability of Lie-invariant geometric objects generated by ideals in the Grassmann algebra´´ (with Y. Prykarpatsky and R. Samulyak), J. Nonlin. Math. Phys. 5 (1998), 54-67.

2. ´´Fractal analysis of height distributions of anisotropic rough surfaces´´(with G. Zhou), Fractals 6 (1998), 43-58.

3. ´´New mathematical models for particle flow dynamics´´(with R. Samulyak and A. Rosato), J. Nonlinear Math. Physics 6, (1999), 198-221. 4. ´´KAM theory analysis of the dynamics of three coaxial vortex rings´´(with O. Knio), Physica D 140, (2000), 321-348.

5. ´´Dynamical properties of discrete Lotka-Volterra equations´´(with J. Chen, J. Perez and M. Savescu), Chaos, Solitons and Fractals 12 (2001), 2553-2568.

6. "Hamiltonian structure for vortex filament flows" (with O. Knio), ZAMM 81S (2001), 45-48.

7. "On the exponentially self-regulating population model" (with J. Chen), Chaos, Solitons and Fractals 14 (2002), 1433-1450.

8. "A perspective on vibration-induced size segregation of granular materials, J. Chem. Eng. Sci. 57 (2002), 265-275.

9." Higher order conditions for weak shocks: modified Prandtl relation" (with L. Ting), PAMM 1 (2002), 397-398.

10. "Fractionation and segregation of suspended particles using acoustic and flow fields" (with N. Aboobaker and J. Meegoda), ASCE J. Environ. Eng. 129 (2003), 427-434.

11. "Vorticity jumps across shock surfaces" (with L. Ting), Proc. 2nd MIT Conf. on Computational Fluids and Solid Mechanics, Vol. 1, K. J. Bathe, ed., Elsevier, Amsterdam, 2003, pp. 847-849.

12. "The Lax solution to a Hamilton-Jacobi equation and its generalizations: Part 2" (with Ya V. Mykytiuk and A. Prykarpatsky), Nonlin. Anal. 55 (2003), 629-640.

13. "Morse index for autonomous linear Hamiltonian systems" (with C. Wang), Int. J. Diff. Eqs. and Appl. 7 (2003), 295-309.

14. "A generalized Poincaré-Birkhoff theorem with applications to coaxial vortex ring motion" (with J. Champanerkar and C. Wang), Disc. and Contin. Dyn. Systems-B (in press).

15. "A geometrical approach to quantum holonomic computing algorithms" (with A. Samoilenko, Y. Prykarpatsky, U. Taneri and A. Prykarpatsky), Math. and Computers in Simulation (in press)

### Projects

** Manufacturing Automation**

- Analysis and Representation of Swept Volumes: We developed characterizations of swept volumes of general piecewise-smooth objects in terms of trajectories of differential equations that we call the sweep differential equation(SDE) and the sweep-envelope differential equation(SEDE). Both SDE and SEDE based methods have been applied to solve real world problems in NC machining, robot motion planning and virtual design and manufacturing.

- Swept Volume Algorithms and Software: Using SDE and SEDE theory as a framework, we developed fast, efficient and robust algorithms for computing and graphically representing swept volumes. These algorithms are being used to create computer software that can be interfaced with commercial software in order to obtain more useful software for a variety of manufacturing applications.