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1. Constructing continuous multi-behavioral planar systems through motivation dynamics and bifurcations

   https://doi.org/10.1109/CDC45484.2021.9683759  

   Baxevani, Kleio; Tanner, Herbert G. (January 2021, 60th IEEE Conference on Decision and Control)

   This paper offers new analytical conditions on the system parameters of a particular class of planar dynamical systems which would allow them to undergo a Hopf bifurcation. These systems are constructed as a means of generating multiple behaviors from the same single continuous dynamical system model, without resorting to switching between distinct component continuous dynamics associated to each behavioral mode. This work builds on recent advances which introduced motivation dynamics as an efficient way to design multi-behavioral systems. The contribution of this paper is that it expands the scope of the motivation dynamics approach, and offers explicit analytic conditions on the system parameters to guarantee the existence of bifurcations, which can then be utilized to better engineer the structure and location of the resulting equilibria. Numerical simulations confirm the theoretical predictions for the onset of the Hopf bifurcations. 

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2. Multi-modal Swarm Coordination via Hopf Bifurcations

   https://doi.org/10.1007/s10846-023-01966-4  

   Baxevani, Kleio; Tanner, Herbert G (October 2023, Journal of Intelligent & Robotic Systems)

   This paper outlines a methodology for the construction of vector fields that can enable a multi-robot system moving on the plane to generate multiple dynamical behaviors by adjusting a single scalar parameter. This parameter essentially triggers a Hopf bifurcation in an underlying time-varying dynamical system that steers a robotic swarm. This way, the swarm can exhibit a variety of behaviors that arise from the same set of continuous differential equations. Other approaches to bifurcation-based swarm coordination rely on agent interaction which cannot be realized if the swarm members cannot sense or communicate with one another. The contribution of this paper is to offer an alternative method for steering minimally instrumented multi-robot collectives with a control strategy that can realize a multitude of dynamical behaviors without switching their constituent equations. Through this approach, analytical solutions for the bifurcation parameter are provided, even for more complex cases that are described in the literature, along with the process to apply this theory in a multi-agent setup. The theoretical predictions are confirmed via simulation and experimental results with the latter also demonstrating real-world applicability. 

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3. Bifurcations and global dynamics of a predator–prey mite model of Leslie type

   https://doi.org/10.1111/sapm.12675  

   Yang, Yue; Xu, Yancong; Rong, Libin; Ruan, Shigui (May 2024, Studies in Applied Mathematics)

   Abstract In this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature. 

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4. Widespread neuronal chaos induced by slow oscillating currents

   https://doi.org/10.1063/5.0248001  

   Scully, James; Hinsley, Carter; Bloom, David; Meijer, Hil_G E; Shilnikov, Andrey L (March 2025, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   This paper investigates the origin and onset of chaos in a mathematical model of an individual neuron, arising from the intricate interaction between 3D fast and 2D slow dynamics governing its intrinsic currents. Central to the chaotic dynamics are multiple homoclinic connections and bifurcations of saddle equilibria and periodic orbits. This neural model reveals a rich array of codimension-2 bifurcations, including Shilnikov–Hopf, Belyakov, Bautin, and Bogdanov–Takens points, which play a pivotal role in organizing the complex bifurcation structure of the parameter space. We explore various routes to chaos occurring at the intersections of quiescent, tonic spiking, and bursting activity regimes within this space and provide a thorough bifurcation analysis. Despite the high dimensionality of the model, its fast–slow dynamics allow a reduction to a one-dimensional return map, accurately capturing and explaining the complex dynamics of the neural model. Our approach integrates parameter continuation analysis, newly developed symbolic techniques, and Lyapunov exponents, collectively unveiling the intricate dynamical and bifurcation structures present in the system. 

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5. Beyond the Bristol book: Advances and perspectives in non-smooth dynamics and applications

   https://doi.org/10.1063/5.0138169  

   Belykh, Igor; Kuske, Rachel; Porfiri, Maurizio; Simpson, David J. W. (January 2023, Chaos: An Interdisciplinary Journal of Nonlinear Science)

   Non-smooth dynamics induced by switches, impacts, sliding, and other abrupt changes are pervasive in physics, biology, and engineering. Yet, systems with non-smooth dynamics have historically received far less attention compared to their smooth counterparts. The classic “Bristol book” [di Bernardo et al., Piecewise-smooth Dynamical Systems. Theory and Applications (Springer-Verlag, 2008)] contains a 2008 state-of-the-art review of major results and challenges in the study of non-smooth dynamical systems. In this paper, we provide a detailed review of progress made since 2008. We cover hidden dynamics, generalizations of sliding motion, the effects of noise and randomness, multi-scale approaches, systems with time-dependent switching, and a variety of local and global bifurcations. Also, we survey new areas of application, including neuroscience, biology, ecology, climate sciences, and engineering, to which the theory has been applied. 

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