Nonsmooth folds as tipping points

Abstract

A nonsmooth fold is where an equilibrium or limit cycle of a nonsmooth dynamical system hits a switching manifold and collides and annihilates with another solution of the same type. We show that beyond the bifurcation the leading-order truncation to the system in general has no bounded invariant set. This is proved for boundary equilibrium bifurcations of Filippov systems, hybrid systems, and continuous piecewise-smooth ODEs, and grazing-type events for which the truncated form is a continuous piecewise-linear map. The omitted higher-order terms are expected to be incapable of altering the local dynamics qualitatively, implying the system has no local invariant set on one side of a nonsmooth fold, and we demonstrate this with an example. Thus if the equilibrium or limit cycle is attracting the bifurcation causes the local attractor of the system to tip to a new state. The results also help explain global aspects of the bifurcation structures of the truncated systems.

Figures (10)

• Figure 1: Schematic phase portraits of two-dimensional nonsmooth systems with one switching manifold (vertical line). In (a) the ODEs are continuous but non-differentiable on the switching manifold. In (b) the ODEs are discontinuous on the switching manifold and Filippov's convention is used to specify sliding motion on the switching manifold. In (c) the system involves ODEs and a map (reset law) that instantaneously transports the system state from the top half of the switching manifold to the bottom half of the switching manifold. Such hybrid systems are commonly used to model mechanical systems with hard impacts BlCz99Br99Ib09.
• Figure 2: A bifurcation diagram and sample phase portraits of a piecewise-smooth continuous ODE system that experiences a persistence-type BEB as a parameter $\mu$ is varied. The blue curves indicate the location of the admissible equilibrium in relation to the switching manifold.
• Figure 3: A bifurcation diagram and sample phase portraits of a piecewise-smooth continuous ODE system that experiences a nonsmooth fold BEB as a parameter $\mu$ is varied. To the left of the bifurcation the system has two admissible equilibria; to the right of the bifurcation it has no admissible equilibria. Throughout this paper stable solutions are coloured blue and unstable solutions are coloured red.
• Figure 4: A bifurcation diagram of Stommel's ocean circulation model with $\alpha = 5$ and $\beta = 0.2$. The blue and red curves are branches of stable and unstable equilibria. The black solution uses $\dot{\mu} = -0.01$ and experiences a tipping point by passing the nonsmooth fold at $\mu = 1$.
• Figure 5: A bifurcation diagram and sample phase portraits of a piecewise-smooth continuous ODE system that experiences a nonsmooth fold BEB as a parameter $\mu$ is varied. The bifurcation creates a large amplitude limit cycle due to global features of the dynamics.

Theorems & Definitions (15)

• Definition 1
• Theorem 1
• proof : Proof of Theorem \ref{['th:m']}
• Theorem 2
• proof
• Definition 2
• Definition 3
• Theorem 3
• proof
• Definition 4