Network dynamics and synchronization

2025

1. Math. Biosci.

   A conceptual framework for modeling a latching mechanism for cell cycle regulation

   Punit Gandhi and Yangyang Wang

   Mathematical Biosciences, Mar 2025

   Abs DOI arXiv Bib

   Two identical van der Pol oscillators with mutual inhibition are considered as a conceptual framework for modeling a latching mechanism for cell cycle regulation. In particular, the oscillators are biased to a latched state in which there is a globally attracting steady-state equilibrium without coupling. The inhibitory coupling induces stable alternating large-amplitude oscillations that model the normal cell cycle. A homoclinic bifurcation within the model is found to be responsible for the transition from normal cell cycling to endocycles in which only one of the two oscillators undergoes large-amplitude oscillations.

   @article{gandhi2025conceptual,
     title = {A conceptual framework for modeling a latching mechanism for cell cycle regulation},
     journal = {Mathematical Biosciences},
     volume = {382},
     pages = {109396},
     year = {2025},
     issn = {0025-5564},
     doi = {https://doi.org/10.1016/j.mbs.2025.109396},
     url = {https://www.sciencedirect.com/science/article/pii/S0025556425000227},
     author = {Gandhi, Punit and Wang, Yangyang},
     keywords = {Coupled oscillators, Cell cycle regulation, Homoclinic bifurcation, Symmetry breaking, Bifurcation}
   }

2022

1. Lett. Biomath.

   Effects of contact tracing and self-reporting in a network disease model

   Punit Gandhi, Michael A. Robert, John P. Palacios, and David Chan

   Letters in Biomathematics, Mar 2022

   Abs DOI Bib

   Contact tracing can be an effective measure to control emerging infectious diseases, but the efficacy of contact tracing measures can depend upon the willingness of individuals to get be tested even when they are symptomatic. In this paper, we examine the effects of symptomatic individuals getting tested and the use of contact tracing in a network model of disease transmission. We utilize a network model to resolve the influence of contact patterns between individuals as apposed to assuming mass action where all individuals are connected to each other.  We find that the effects of self-reporting and contact tracing vary depending on the structure of the network. We also compare the results from the network model with an analogous ODE model that assumes mass action and demonstrate how the results can be dramatically different.

   @article{gandhi2022contact,
     title = {Effects of contact tracing and self-reporting in a network disease model},
     author = {Gandhi, Punit and Robert, Michael A. and Palacios, John P. and Chan, David},
     journal = {Letters in Biomathematics},
     volume = {9},
     number = {1},
     pages = {23--39},
     year = {2022},
     url = {https://lettersinbiomath.org/manuscript/index.php/lib/article/view/143},
     doi = {10.30707/LiB9.1.1681913305.219107}
   }

2020

1. SIADS

   Bifurcations on fully inhomogeneous networks

   Punit Gandhi, Martin Golubitsky, Claire Postlethwaite, Ian Stewart, and Yangyang Wang

   SIAM Journal on Applied Dynamical Systems, Mar 2020

   Abs DOI Bib

   Center manifold reduction is a standard technique in bifurcation theory, reducing the essential features of local bifurcations to equations in a small number of variables corresponding to critical eigenvalues. This method can be applied to admissible differential equations for a network, but it bears no obvious relation to the network structure. A fully inhomogeneous network is one in which all nodes and couplings can be different. For this class of networks, there are general circumstances in which the center manifold reduced equations inherit a network structure of their own. This structure arises by decomposing the network into path components, which connect to each other in a feedforward manner. Critical eigenvalues can then be associated with specific components, and the network structure on the center manifold depends on how these critical components connect within the network. This observation is used to analyze codimension-1 and codimension-2 local bifurcations. For codimension-1, only one critical component is involved, and generic local bifurcations are saddle-node and standard Hopf. For codimension-2, we focus on the case when one component is downstream from the other in the feedforward structure. This gives rise to four cases: steady or Hopf upstream combined with steady or Hopf downstream. Here the generic bifurcations, within the realm of network-admissible equations, differ significantly from generic codimension-2 bifurcations in a general dynamical system. In each case, we derive singularity-theoretic normal forms and unfoldings, present bifurcation diagrams, and tabulate the bifurcating states and their stabilities.

   @article{gandhi2020bifurcations,
     title = {Bifurcations on fully inhomogeneous networks},
     author = {Gandhi, Punit and Golubitsky, Martin and Postlethwaite, Claire and Stewart, Ian and Wang, Yangyang},
     journal = {SIAM Journal on Applied Dynamical Systems},
     volume = {19},
     number = {1},
     pages = {366--411},
     year = {2020},
     publisher = {SIAM},
     doi = {10.1137/18M1230736},
     url = {https://doi.org/10.1137/18M1230736},
     eprint = { https://doi.org/10.1137/18M1230736 }
   }

2015

1. PRE

   Dynamics of phase slips in systems with time-periodic modulation

   Punit Gandhi, Edgar Knobloch, and Cédric Beaume

   Physical Review E, Mar 2015

   Abs DOI arXiv Bib

   The Adler equation with time-periodic frequency modulation is studied. A series of resonances between the period of the frequency modulation and the time scale for the generation of a phase slip is identified. The resulting parameter space structure is determined using a combination of numerical continuation, time simulations, and asymptotic methods. Regions with an integer number of phase slips per period are separated by regions with noninteger numbers of phase slips and include canard trajectories that drift along unstable equilibria. Both high- and low-frequency modulation is considered. An adiabatic description of the low-frequency modulation regime is found to be accurate over a large range of modulation periods.

   @article{gandhi2015periodic,
     title = {Dynamics of phase slips in systems with time-periodic modulation},
     author = {Gandhi, Punit and Knobloch, Edgar and Beaume, C\'edric},
     journal = {Physical Review E},
     volume = {92},
     number = {6},
     pages = {062914},
     year = {2015},
     doi = {10.1103/PhysRevE.92.062914},
     url = {https://link.aps.org/doi/10.1103/PhysRevE.92.062914}
   }