Article
Mathematics, Applied
Noah Cheesman, Kristian Uldall Kristiansen, S. J. Hogan
Summary: This work focuses on PWS systems with isolated codimension-2 discontinuity sets, using regularization and blowup methods to study the dynamics. A general framework is presented, along with specific results for the local dynamics in a class of problems, generalizing Filippov sliding, crossing, and sliding vector fields.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2021)
Article
Automation & Control Systems
Otavio Henrique Perez, Gabriel Rondon, Paulo Ricardo da Silva
Summary: We study planar piecewise smooth differential systems with a regular value of 0. Linear regularizations and nonlinear regularizations are considered, with transition functions that may or may not be monotonic. The paper focuses on the typical singularities of slow-fast systems that arise from regularizations, such as fold, transcritical, and pitchfork singularities. The dependence of the slow-fast system on the graphical properties of the transition function is also investigated.
JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS
(2023)
Article
Mathematics, Applied
Ryan Goh, Tasso J. Kaper, Arnd Scheel, Theodore Vos
Summary: This work studies front formation in the Allen-Cahn equation with a parameter heterogeneity that slowly varies in space. The existence, stability, and interface location of positive, monotone fronts are rigorously established for slowly varying ramps. The front location depends on the transition between convective and absolute instability of the base state, causing a delay before the system jumps to a nontrivial state. For stationary ramps, the front is governed by the slow passage through the instantaneous pitchfork bifurcation.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Engineering, Mechanical
Tapan Saha, Pallav Jyoti Pal, Malay Banerjee
Summary: This paper investigates a modified Leslie-type prey-generalist predator system with piecewise-smooth Holling type I functional response. By employing geometric singular perturbation theory and blow-up technique, a wide range of interesting and complicated dynamical phenomena are revealed. Numerical simulations are performed to validate the analytical results.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics, Applied
Kristian Uldall Kristiansen, Morten Gram Pedersen
Summary: In this paper, we use geometric singular perturbation theory and blowup to study the mixed-mode oscillations (MMOs) occurring in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. We demonstrate that the MMOs in this model are not due to folded singularities, but rather due to singularities at a cusp of the critical manifold. Using blowup, we determine the number of small-amplitude oscillations (SAOs) analytically, showing that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a saddle-node bifurcation in the desingularized reduced problem, which occurs on a cusp, and not a fold.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Automation & Control Systems
Denis de Carvalho Braga, Alexander Fernandes da Fonseca, Luiz Fernando Goncalves, Luis Fernando Mello
Summary: This article investigates the existence and stability of limit cycles in a family of piecewise smooth vector fields, as well as the relationship between these crossing limit cycles and the limit cycles of the family of smooth vector fields obtained by the regularization method.
JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS
(2021)
Article
Mathematics, Interdisciplinary Applications
Jian Song, Shenquan Liu, Qixiang Wen
Summary: This paper investigates the electrical excitability of pituitary cells and the triggering relationship between their secretory function and electrical activity using mathematical modeling and analysis. The simplified model and the multi-geometric perspective reveal the intrinsic transients and long-term evolution of the system, as well as the dynamic mechanism and mixed-mode oscillations of cell firing. This study provides a new perspective on the origin of cellular spontaneous firing activity.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Applied
P. De Maesschalck, G. Kiss, A. Kovacs
Summary: The study uses geometric singular perturbation theory to investigate a two-gene system with an autoregulatory feedback loop, identifying relaxation oscillations, singular Hopf bifurcations, homoclinic loops, and presenting a new method to compute the criticality of the singular Hopf bifurcations.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
D. J. W. Simpson
Summary: This paper investigates continuous, piecewise-linear, and slow-fast systems of ODEs. The piecewise-linear setting presents three unique phenomena that hinder the extension of Fenichel theory, including the possibility of boundary equilibrium instability, coincidence with another attractor, or chaotic attraction of orbits. The paper also introduces a slow-fast version of the observer canonical form for piecewise-linear ODEs, which simplifies the analysis.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Zhihao Fang, Xingwu Chen
Summary: In this paper, we investigate the global dynamics of a piecewise smooth system with a fold-cusp using a two-parametric unfolding of a normal form. We analyze the global structure of the switching manifold and determine the precise domain of the Poincare map. The bifurcation diagram in the parameter space and the corresponding global phase portraits in the Poincare disc are obtained. Degenerate sliding homoclinic loops and fold-folds are observed at certain nonlocal parameters, leading to the emergence of sliding limit cycles and pseudo-equilibria.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Samuel Jelbart, Nathan Pages, Vivien Kirk, James Sneyd, Martin Wechselberger
Summary: This article discusses ordinary differential equations (ODEs) used to model phenomena in chemistry, biology, and neuroscience, and presents a heuristic procedure for identifying small parameters in these ODE models. The procedure is applied to a model of intracellular calcium dynamics characterized by switching and multiple time-scale dynamics. Using geometric singular perturbation theory, the existence and uniqueness of stable relaxation oscillations with three distinct time scales are proven, and an estimate for the period of the oscillations is provided.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2022)
Article
Acoustics
Misael Avendano-Camacho, Alejandra Torres-Manotas, Jose Antonio Vallejo
Summary: By combining techniques of singular geometric reduction with the more classical averaging method of Moser, we have proven the existence of closed stable orbits in a strongly coupled Wilberforce pendulum at a 1:2 resonance.
JOURNAL OF VIBRATION AND CONTROL
(2022)
Article
Mathematics, Applied
Adam Bauer, Paul Carter
Summary: This study focuses on the one-fluid stellar wind problem for steady, radial outflow, taking into account the effects of heat conduction and viscosity. Using geometric singular perturbation techniques, stellar wind profiles are rigorously constructed in the large Reynolds number limit, identifying transonic solutions that accelerate from subsonic to supersonic speeds. These solutions are identified as folded saddle canard trajectories lying in the intersection of a subsonic saddle slow manifold and a supersonic repelling slow manifold.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2021)
Article
Mathematics
Qi Qiao
Summary: This paper investigates a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By using the geometric singular perturbation theory, the existence of a positive traveling wave connecting two constant steady states is confirmed. The monotonicity of the wave is analyzed for different parameter ranges, and spectral instability is observed in some exponentially weighted spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics, Applied
Yiorgos Patsios, Renato Huzak, Peter De Maesschalck, Nikola Popovic
Summary: MMOs are complex oscillatory patterns found in singularly perturbed systems, typically related to folded singularities and canard trajectories. We introduced a new type of MMOs based on a jump mechanism in a canonical family of slow-fast systems.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
K. Uldall Kristiansen, P. Szmolyan
Summary: In this paper, a novel type of relaxation oscillations in substrate-depletion oscillators is described using geometric singular perturbation theory and blow-up as a key technical tool. The oscillations in the planar model are shown to be produced by a complicated interplay between two stable nodes and a discontinuity set in the singular limit. This interplay introduces a new mechanism for generating relaxation-type oscillations, which is also discussed in a more general setting.
Article
Mathematics
Samuel Jelbart, Kristian Uldall Kristiansen, Martin Wechselberger
Summary: We study the transition of smooth systems to piecewise-smooth systems with a boundary-focus bifurcation as epsilon -> 0, and identify different bifurcation structures. We uncover the evolution characteristics of cycles associated with BF bifurcations in the smooth system, and prove the existence of a family of stable limit cycles.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
S. Jelbart, K. U. Kristiansen, P. Szmolyan, M. Wechselberger
Summary: This paper investigates two prototypical singularly perturbed oscillators with exponential nonlinearities, normalizing both systems to a piecewise smooth system in the limit is an element of -> 0, showing exponential convergence due to the nonlinearities studied. By extending spatial dimensions, degeneracies caused by exponentially small terms are tackled for the second model system, with (unique) limit cycles proven to exist for both systems by perturbing away from singular cycles with desirable hyperbolicity properties.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
S. Jelbart, K. U. Kristiansen, M. Wechselberger
Summary: This work completes a classification of codimension-1 singularly perturbed boundary equilibria bifurcation (BEB) in the plane, utilizing tools from PWS theory, geometric singular perturbation theory, and the method of geometric desingularization known as blow-up. Local normal forms for generating all 12 singularly perturbed BEBs are derived, and the unfolding in each case is described. Detailed quantitative results on various bifurcations involved in the unfoldings and classification are presented, including saddle-node, Andronov-Hopf, homoclinic, and codimension-2 Bogdanov-Takens bifurcations.
Article
Mathematics, Applied
P. Kaklamanos, N. Popovic, K. U. Kristiansen
Summary: This article studies a class of multi-parameter three-dimensional systems of ordinary differential equations that exhibit dynamics on three distinct timescales. Geometric singular perturbation theory is applied to explore the dependence of the geometry of these systems on their parameters, with a focus on mixed-mode oscillations and their bifurcations. A novel geometric mechanism is uncovered, which encodes the transition from single-epoch small-amplitude oscillations to double-epoch small-amplitude oscillations, and is demonstrated using a prototypical three-timescale system.
Article
Mathematics
R. Huzak, K. Uldall Kristiansen
Summary: In this paper, the authors extend the slow divergence-integral to smooth systems that approach piecewise smooth ones. The slow divergence-integral is based on a generalized canard cycle for a piecewise smooth bifurcation, and it is used to show that the number of limit cycles in regularized piecewise smooth polynomial systems is unbounded.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Paul Glendinning, S. John Hogan, Martin Homer, Mike R. Jeffrey, Robert Szalai
Summary: The sewed focus is a unique phenomenon in planar piecewise smooth dynamical systems, consisting of two invisible tangencies on either side of the switching manifold. Filippov showed that for analytic focus-like behavior, the approach to the singularity is in infinite time. However, we demonstrate that for non-analytic focus-like behavior, the approach to the singularity can be in finite time. Additionally, we explore the non-analytic sewed centre-focus, uncovering a multitude of different topological types of local dynamics, including cases with infinitely many stable periodic orbits and systems with periodic orbits intersecting any bounded symmetric closed set.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Article
Mathematics, Applied
Panagiotis Kaklamanos, Nikola Popovic, Kristian U. Kristiansen
Summary: We propose a novel and global reduction method for a nondimensionalized system based on geometric singular perturbation theory. By investigating the dynamics in two parameter regimes, we observe bifurcations and classify firing patterns with external current applied. Our findings reveal the underlying geometry of transitions between patterns, which has not been emphasized before despite similar patterns documented in previous studies.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Kristian Uldall Kristiansen, Morten Gram Pedersen
Summary: In this paper, we use geometric singular perturbation theory and blowup to study the mixed-mode oscillations (MMOs) occurring in two coupled FitzHugh-Nagumo units with symmetric and repulsive coupling. We demonstrate that the MMOs in this model are not due to folded singularities, but rather due to singularities at a cusp of the critical manifold. Using blowup, we determine the number of small-amplitude oscillations (SAOs) analytically, showing that they are determined by the Weber equation and the ratio of eigenvalues. We also show that the model undergoes a saddle-node bifurcation in the desingularized reduced problem, which occurs on a cusp, and not a fold.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Mathematics
K. Uldall Kristiansen
Summary: In this paper, the birth of canard limit cycles in slow-fast systems in R-3 is rigorously described through the folded saddle-node of type II and the singular Hopf bifurcation. The paper proves the existence of a family of periodic orbits born in the (singular) Hopf bifurcation and extending to (1) cycles that follow the strong canard of the folded saddle-node. The results show that unlike the explosive family of periodic orbits in R-2, the family in R-3 is not explosive and is called the dud canard.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Noah Cheesman, S. J. Hogan, Kristian Uldall Kristiansen
Summary: This paper explores the importance of Painleve's paradoxes in three dimensions, proving the existence of three critical values of the azimuthal angular velocity Psi and revealing a rich geometry in the 3D problem.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Noah Cheesman, Kristian Uldall Kristiansen, S. J. Hogan
Summary: This work focuses on PWS systems with isolated codimension-2 discontinuity sets, using regularization and blowup methods to study the dynamics. A general framework is presented, along with specific results for the local dynamics in a class of problems, generalizing Filippov sliding, crossing, and sliding vector fields.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Kristian U. Kristiansen
Summary: This paper analyzes a mass-spring-friction oscillator in a special parameter regime, showcasing new friction phenomena, resolving some open problems, and proving the existence of chaos in the fundamental oscillator system.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2021)
