On Bifurcations in Nonlinear Consensus Networks

The theory of consensus dynamics is widely employed to study various linear behaviors in networked control systems. Moreover, nonlinear phenomena have been observed in animal groups, power networks and in other networked systems. These observations inspire the development in this paper of three novel approaches to define distributed nonlinear dynamical interactions. The resulting dynamical systems are akin to higher-order nonlinear consensus systems. Over connected undirected graphs, the resulting dynamical systems exhibit various interesting behaviors that we rigorously characterize.     

Analysis of a Class of Symmetric Equilibrium Configurations for a Territorial Model

<正>Motivated by an animal territoriality model,we consider a centroidal Voronoi tessellation algorithm from a dynamical systems perspective.In doing so,we discuss the stability of an aligned equilibrium configuration for a rectangular domain that exhibits interesting symmetry properties.We also demonstrate the procedure for performing a center manifold reduction on the system to extract a set of coordinates which capture the long term dynamics when the system is close to a bifurcation.Bifurcations of the system restricted to the center manifold are then classified and compared to numerical results.Although we analyze a specific set-up,these methods can in principle be applied to any bifurcation point of any equilibrium for any domain.     

Canards in a Surface Oxidation Reaction

Summary. Canards are periodic orbits for which the trajectory follows both the attracting and repelling parts of a slow manifold. They are associated with a dramatic change in the amplitude and period of a periodic orbit within a very narrow interval of a control parameter. It is shown numerically that canards occur in an appropriate parameter range in two- and three-dimensional models of the platinum-catalyzed oxidation of carbon monoxide. By smoothly connecting associated stable and unstable manifolds in an asymptotic limit, we predict parameter values at which such canards exist. The relationship between the canards and saddle-loop bifurcations for these models is also demonstrated. Excellent agreement is found between the numerical and analytical results.     

Weakly coupled parametrically forced oscillator networks: existence,stability, and symmetry of solutions

In this paper, we discuss existence, stability, and symmetry of solutions for networks of parametrically forced oscillators. We consider a nonlinear oscillator model with strong 2:1 resonance via parametric excitation. For uncoupled systems, the 2:1 resonance property results in sets of solutions that we classify using a combinatorial approach. The symmetry properties for solution sets are presented as are the group operators that generate the isotropy subgroups. We then impose weak coupling and prove that solutions from the uncoupled case persist for small coupling by using an appropriate Poincaré map and the Implicit Function Theorem. Solution bifurcations are investigated as a function of coupling strength and forcing frequency using numerical continuation techniques. We find that the characteristics of the single oscillator system are transferred to the network under weak coupling. We explore interesting dynamics that emerge with larger coupling strength, including anti-synchronized chaos and unsynchronized chaos. A classification for the symmetry-breaking that occurs due to weak coupling is presented for a simple example network.     

Optimal switching between collective motion states for two agents

We show that steering control can be chosen to give bistability between parallel and anti-parallel collective motion states for a continuous-time kinetic model of two agents moving in the plane with unit speed. Variational methods are used to determine the optimal input to the steering control of one of the agents which leads to switching between these collective states. For any given time interval of switching, such an optimal input is shown to exist and to be unique. The properties of optimal inputs are interpreted by considering the phase space geometry of the Euler–Lagrange equations associated with the optimization.     

Heteroclinic dynamics in a model of Faraday waves in a square container

We study periodic orbits associated with heteroclinic bifurcations in a model of the Faraday system for containers with square cross-section and single-frequency forcing. These periodic orbits correspond to quasiperiodic surface waves in the physical system. The heteroclinic bifurcations are related to a continuum of heteroclinic connections in the integrable Hamiltonian limit, some of which persist in the presence of small damping. The dynamics in the neighborhood of one of the heteroclinic bifurcations are examined in detail using approximate Poincaré maps, with predictions that agree with numerical computations. The results suggest a great richness of possible dynamics of Faraday waves even in simple geometries and with single-frequency forcing.     

Feedback control of canards

We present a control mechanism for tuning a fast-slow dynamical system undergoing a supercritical Hopf bifurcation to be in the canard regime, the tiny parameter window between small and large periodic behavior. Our control strategy uses continuous feedback control via a slow control variable to cause the system to drift on average toward canard orbits. We apply this to tune the FitzHugh-Nagumo model to produce maximal canard orbits. When the controller is improperly configured, periodic or chaotic mixed-mode oscillations are found. We also investigate the effects of noise on this control mechanism. Finally, we demonstrate that a sensor tuned in this way to operate near the canard regime can detect tiny changes in system parameters.