- Materials Science
- Neuroscience
- Climate modeling
- Power electronics
- Anti-lock braking systems
- Robotic walking
- Lorentz gas dynamics

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| Materials Science |

- Building a simple Lattice Spring Model with massless nodes and changing topology of connections
- Converting the LSM into a Moreau Sweeping
Process with changing geometry of moving constraint
*C*(*t*) (differential equation*x*'=0 constrained by*C*(*t*)) - Studying stability and convergence of the dynamics

- Building a simple Lattice Spring Model with masses in nodes
- Converting the LSM into a Perturbed
Sweeping Process (differential equation
*x*'=*f*(*x*) with a moving constraint) - Studying routes to different types of dynamic behavior

- [1] I. Gudoshnikov, O. Makarenkov. Stabilization of quasistatic evolution of elastoplastic systems subject to periodic loading. arXiv preprint.
- [2] M. Kamenskii, O. Makarenkov, L. Niwanthi Wadippuli, P. Raynaud de Fitte. Global stability of almost periodic solutions to monotone sweeping processes and their response to non-monotone perturbations. Nonlinear Anal. Hybrid Syst. 30 (2018) 213224.

- [1] J. J. Moreau, On unilateral constraints, friction and plasticity. New variational techniques in mathematical physics (Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo, Bressanone, 1973), pp. 171322. Edizioni Cremonese, Rome, 1974.
- [2] J. Bastien, C.-H. Lamarque, Persozs gephyroidal model described by a maximal monotone differential inclusion, Arch. Appl. Mech. 78 (2008) 393-407.
- [3] M. Ostoja-Starzewski, Lattice models in micromechanics, Appl. Mech. Rev. 55 (2002) 35-60.
- [4] G. A. Buxton, C. M. Care, D. J. Cleaver, A lattice spring model of heterogeneous materials with plasticity, Modell. Simul. Mater. Sci. Eng. 9 (2001) 485497.
- [5] H. Chen, E. Lin, Y. Jiao, Y. Liu, A generalized 2D non-local lattice spring model for fracture simulation, Comput. Mech. 54 (2014) 15411558.
- [6] M. Dembo , D. C. Torney, K. Saxman, D. Hammer, The reaction-limited kinetics of membrane-to-surface adhesion and detachment, Proc R Soc Lond B Biol Sci. 22 (1988) 55-83.
- [7] A. Geitmann, J.K.E. Ortega, Mechanics and modeling of plant cell growth, Trends Plant Sci. 14 (2009) 467-478.

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| Neuroscience |

- Studying the theory of Hopf bifurcation of sub-threshold oscillations
- Studying the theory of grazing bifurcation of spiking oscillations
- Computing the ratio of the area of attraction of a) and b) and compare this ration against the length of clusters of spikes coming from simulations with noise

- Studying the theory of grazing bifurcations of limit cycles in mechanical oscillators
- Extending a) to the case of multiple impacts per period
- Applying the result to a normal form linear system with neuron-type resets

- [1] Makarenkov, O,; Phung, A. (2018) Dwell time for local stability of switched affine systems with application to non-spiking neuron models. Appl. Math. Lett., accepted 6/19/2018.
- [2] Makarenkov, O. (2017) A new test for stick-slip limit cycles in dry-friction oscillators with small nonlinear friction characteristics, Meccanica, 52 (11-12), 2631-2640.
- [3] Makarenkov, O. & Lamb, J. S. W. (2012) Dynamics and bifurcations of nonsmooth systems: a survey. Phys. D, 241 (22), 1826-1844.

- [1] Coombes, S.; Thul, R.; Wedgwood, K. C. A. (2012) Nonsmooth dynamics in spiking neuron models. Phys. D 241 (22), 2042-2057.
- [2] Izhikevich, E. M. (2007) Dynamical systems in neuroscience: the geometry of excitability and bursting. Computational Neuroscience. MIT Press, Cambridge, MA, xvi+441 pp.
- [3] Khajeh Alijani, A. (2009) Mode locking in a periodically forced resonate-and-fire neuron model. Physical Review E, 80 (5), 051922, 12 pp.
- [4] Nicola, W. & Campbell, S. A. (2013) Bifurcations of large networks of two-dimensional integrate and fire neurons. J. Comput. Neurosci. 35 (1) 87-108.

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| Climate modeling |

- Studying the piecewise-smooth Budyko-Widiasih model of glacial cycles
- Finding stable switched equilibria of fold-fold type
- Identifying the class of fold-fold singularities that lead to the occurrence of limit cycles under varying parameters

- Studying the piecewise-smooth Budyko-Widiasih model of glacial cycles
- Extending to 3D the available result about limit cycles of piecewise-smooth systems with hysteresis switching

- [1] O. Makarenkov, E. Widiasih, Bifurcation of limit cycles from a fold-fold singularity in a glacial cycles model, work in progress.
- [2] O. Makarenkov, Bifurcation of limit cycles from a fold-fold singularity in planar switched systems, SIAM J. Appl. Dyn. Syst. 16 (2017) 1340-1371.

- [1] J. Walsh, E. Widiasih, J. Hahn, R. McGehee, Periodic orbits for a discontinuous vector field arising from a conceptual model of glacial cycles, Nonlinearity 29 (2016) 1843-1864.
- [2] D. Paillard and F. Parrenin, The Antarctic ice sheet and the triggering of deglaciations, Earth and Planetary Science Letters 227 (2004), 263-271.

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| Power electronics |

- Studying the stable convex combination method for switching between smooth systems (Bolzern-Spinelli)
- Extending the method to switching between discontinuous systems
- Implementing the method on the models of buck and boost power converters

- Computing limit cycles of piecewise affine models of power converters in closed form
- Combining a) with the theory of stability of limit cycles of switched systems (Astrom)

- [1] O. Makarenkov, A linear state feedback switching rule for global stabilization of switched nonlinear systems about a nonequilibrium point, preprint.
- [2] O. Makarenkov, Bifurcation of limit cycles from a fold-fold singularity in planar switched systems, SIAM J. Appl. Dyn. Syst. 16 (2017) 1340-1371.

- [1] K. J. Astrom, Oscillations in systems with relay feedback. Adaptive control, filtering, and signal processing (Minneapolis, MN, 1993), 125, IMA Vol. Math. Appl., 74, Springer, New York, 1995.
- [2] P. Bolzern, W. Spinelli, Quadratic stabilization of a switched affine system about a nonequilibrium point, Proceedings of the American Control Conference 5 (2004) 38903895.
- [3] P. Gupta, A. Patra, Hybrid Mode-Switched Control of DC-DC Boost Converter Circuits, IEEE Trans. Circuits and Systems 52 (2005), no. 11, 734738.
- [4] Y. M. Lu, X. F. Huang, B. Zhang, L. Y. Yin, Hybrid Feedback Switching Control in a Buck Converter, IEEE International Conference on Automation and Logistics (2008) 207210.
- [5] A. Schild, J. Lunze, J. Krupar, and W. Schwarz, Design of generalized hysteresis controllers for dc-dc switching power converters, IEEE Trans. Power Electron. 24 (2009), no. 1, 138146.
- [6] M. A. Wicks, P. Peleties, and R. A. DeCarlo. Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems, European J. Control 4 (1998) 140147.

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| Anti-lock braking systems |

Consider two vector fields (

In my paper SIAM J. Appl. Dyn. Syst. 16 (2017) the aforementioned cycles are obtained in dimension 2 through a bifurcation from an fold-fold singularity, where a suitably defined parameter μ crosses its bifurcation value μ=0. In 3-dimensional systems, the value μ=0 corresponds to the well-known Teixeira singularity (as it is termed after the pioneering work by Teixeira) or U-singularity (how it is termed in the book by Filippov). The Teixeira singularity is shown at the leftmost figure (the red point on the intersection of blue and green dashed lines) along with a sample trajectory (that sticks to the switching manifold

My interest in studying attractors of switching systems is motivated by closed-loop control problems in Anti-Lock Braking Systems (ABSs) (Drakunov et al, Bruijn et al, Tanelli et al) and switching convertors (Tse et al, Gupta et al, Sreekumar et al, Schild et al), intermittent therapy modeling in medicine (Tanaka et al), grazing management in ecology (Meza et al). My earlier results Internat. J. Control 90 (2017) on limit cycles in ABSs used perturbation theory (joint project with Tassilo Kuepper and Thomas Koppen).

My current work investigates border splitting bifurcation in Anti-Lock Braking System in 3D.

Here is an interactive diagram of a sample ABS that shows how the switching of the valves links to the motion of the phase point along a limit cycle of the respective switched systdem. The animation is created by Gudoshnikov.

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| Robotic walking |

A bipedal robot can walk down a shallow slope without any control, see this video. The simulation below reproduces the walking gait by viewing the biped as a coupled pendulum combined with a velocity jump applied when the swing leg touches the ground. The animation is created by Gudoshnikov. Garcia, Chatterjee and Ruina discovered (paper, preprint) that the walking gait observed can be mathematically established through a bifurcation of a periodic solution in the switched pendulum model when the slope of the ground crosses zero. A rigorous proof of the discovery by Garcia, Chatterjee and Ruina has been recently delivered in my course in UTD Summer School on Nonsmooth Dynamical Systems.

My current goal is to classify all the walking gaits that the bipedal robot is capable to realize and to study scenarious of gait initiation/termination (detailed research plan).

⌂ |
Dynamics of Lorentz gas |

A possible explanation of the loss of thermal equilibria in gases of hard balls (see Kinoshita et al) is over regularization of the ergodic Sinai's (i.e. dispersing) billiard. This regularization is accomplished by approximating the billiard by a Hamiltonian system with a steep potential (thus smoothing the instantaneous collisions) and the absence of thermal equilibrium in the gas corresponds to the dramatic phenomenon of the lack of ergodicity in the regularized billiard, see Turaev & Rom-Kedar for the theory and Kaplan et al for experiments.

The main assumption of paper by Turaev & Rom-Kedar is the presence of a closed orbit that hits one of the balls tangentially (grazes). My goal is to prove that the set of dispersing Sinai's billiards with periodic grazing orbits is dense in the set of all dispersing Sinai's billiards. It is well known that periodic points of the respective dynamical system are dense in the phase space (Bunimovich-Sinai-Katok-Strelcin). However, if one moves one of the discs (say, disk C) towards the closed orbit (e.g. towards the dashed red line