More Like this

1. Rotating spirals in oscillatory media with nonlocal interactions and their normal form

   https://doi.org/10.3934/dcdss.2022085  

   Jaramillo, Gabriela (January 2022, Discrete and Continuous Dynamical Systems - S)

   Biological and physical systems that can be classified as oscillatory media give rise to interesting phenomena like target patterns and spiral waves. The existence of these structures has been proven in the case of systems with local diffusive interactions. In this paper the more general case of oscillatory media with nonlocal coupling is considered. We model these systems using evolution equations where the nonlocal interactions are expressed via a diffusive convolution kernel, and prove the existence of rotating wave solutions for these systems. Since the nonlocal nature of the equations precludes the use of standard techniques from spatial dynamics, the method we use relies instead on a combination of a multiple-scales analysis and a construction similar to Lyapunov-Schmidt. This approach then allows us to derive a normal form, or reduced equation, that captures the leading order behavior of these solutions. 

   more » « less
2. Nonspreading Solutions and Patch Formation in An Integro-Difference Model With a Strong Allee Effect and Overcompensation

   https://doi.org/10.21203/rs.3.rs-953137/v1  

   Otto, Garrett; Fagan, William F.; Li, Bingtuan (October 2021, Research Square)

   Abstract Previous work involving integro-difference equations of a single species in a homogenous environment has emphasized spreading behaviour in unbounded habitats. We show that under suitable conditions, a simple scalar integro-difference equation incorporating a strong Allee effect and overcompensation can produce solutions where the population persists in an essentially bounded domain without spread despite the homogeneity of the environment. These solutions are robust in that they occupy a region of full measure in the parameter space. We develop bifurcation diagrams showing various patterns of nonspreading solutions from stable equilibria, period two, to chaos. We show that from a relatively uniform initial density with small stochastic perturbations a population consisting of multiple isolated patches can emerge. In ecological terms this work suggests a novel endogenous mechanism for the creation of patch boundaries.AMS subject classification. 92D40, 92D25 

   more » « less
3. Oscillations and coupling in interconnections of two-dimensional brain networks

   E. Nozari, J. Cortés (July 2019, Proceedings of the American Control Conference)

   Oscillations in the brain are one of the most ubiquitous and robust patterns of activity and correlate with various cognitive phenomena. In this work, we study the existence and properties of oscillations in simple mean-field models of brain activity with bounded linear-threshold rate dynamics. First, we obtain exact conditions for the existence of limit cycles in two-dimensional excitatory-inhibitory networks (E-I pairs) and provide generalizations for networks with one inhibitory and multiple excitatory nodes. Building on these results, we study networks of multiple E-I pairs, provide exact conditions for the lack of stable equilibria, and numerically show that this is a tight proxy for the existence of oscillatory behavior. Finally, we study cross-frequency coupling between pairs of oscillators each consisting of an E-I pair. We find that while both phase-phase coupling (synchronization) and phase-amplitude coupling (PAC) monotonically increase with inter-oscillator connection strength, there exists a tradeoff in increasing frequency mismatch between the oscillators as it de-synchronizes them while enhancing their PAC. 

   more » « less
4. Homoclinic snaking of contact defects in reaction-diffusion equations

   https://doi.org/10.1088/1361-6544/ae1376  

   Roberts, Timothy_V; Sandstede, Björn (November 2025, Nonlinearity)

   Abstract We apply spatial dynamical-systems techniques to prove that certain spatiotemporal patterns in reversible reaction-diffusion equations undergo snaking bifurcations. That is, in a narrow region of parameter space, countably many branches of patterned states coexist that connect at towers of saddle-node bifurcations. Our patterns of interest are contact defects, which are one-dimensional time-periodic patterns with a spatially oscillating core region that at large distances from the origin in space resemble pure temporally oscillatory states and arise as natural analogues of spiral and target waves in one spatial dimension. We show that these solutions lie on snaking branches that have a more complex structure than has been seen in other contexts. In particular, we predict the existence of families of asymmetric traveling defect solutions with arbitrary background phase offsets, in addition to symmetric standing target and spiral patterns. We prove the presence of these additional patterns by reconciling results in classic ordinary differential equation studies with results from the spatial-dynamics study of patterns in partial differential equations and using geometrical information contained in the stable and unstable manifolds of the background wave trains and their natural equivariance structure. 

   more » « less
5. On non-unique solutions in mean field games

   https://doi.org/10.1109/CDC40024.2019.9029906  

   Hajek, Bruce; Livesay, Michael (December 2019, 2019 58th Conference on Decision and Control)

   The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric $$\epsilon$$-Nash Markov equilibria for $$N$$ players with $$\epsilon$$ converging to zero as $$N$$ goes to infinity. In contrast to the mean field game, there is a unique Nash equilibrium for finite $N.$ It is shown that fluid limits arising from the Nash equilibria for finite $$N$$ as $$N$$ goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping. 

   more » « less