Rate-induced tipping: thresholds, edge states and connecting orbits

Dangerous bifurcations have a discontinuity in the parametrised family (or branch) of attractors at the bifurcation point and include, for example, saddle-node and subcritical Hopf bifurcations [125].

Often referred to as an 'alternative stable state'.

We give non-degeneracy conditions for connecting orbits in remark 7.3.

Note that if t is in units second and r is in units inverse second then τ is dimensionless.

This is the flow $x(\tau) = \varphi(\tau,\tau_0,x_0)$ written as a process [64] with the r dependence explicitly shown. Given a solution $x^{[r]}(\tau,x_0,\tau_0)$ to system (3), one can easily obtain the corresponding solution to system (2) by setting $t = \tau/r$ and $t_0 = \tau_0/r$. However, it is important to note that, for different r > 0, a fixed initial state $(x_0,\tau_0)$ in system (3) corresponds to a fixed value of the external input $\Lambda(rt_0)$, but different initial states $(x_0,t_0) = (x_0,\tau_0/r)$ in system (2).

The smallest closed subset of $\mathbb{R}^d$ containing S.

See appendix A.4 for the definition of an attractor.

Notions of convergence to invariant sets η are discussed in appendix A.1.

Note that $\eta^+$ is contained in its stable manifold, that is $\eta^+\subseteq W^s(\eta^+)$.

We recall some notions used in discussion of differentiable manifolds in appendix A.2.

Note that the stable invariant manifold of η contains η.

See appendix A.4 for the definition of an attractor.

Not to be confused with the 'static' notion of 'basin stability' introduced in [81] as a measure related to the volume of the basin of attraction.

Equivalently, a moving sink $e(\Lambda(\tau))$ is 'forward basin stable' if, at each point in time, the basin of attraction of $e(\Lambda(\tau))$ contains all the previous positions of $e(\Lambda(\tau))$.

The signed distance $d_s(x,S)$ is discussed in appendix A.3.

Note that $\eta^+$ is not necessarily a regular edge state from definition 4.4(c); it may be a saddle with more than one unstable direction, or even a repeller of codimension-two or higher, and/or not necessarily hyperbolic.

In the one dimensional case, recall that the moving regular threshold and edge state are one and the same.

We say a moving sink $e(\Lambda(\tau))$ on I is forward threshold stable if there are no $\theta(\Lambda(\tau))$ and finite $\tau_a\lt\tau_b\in I$ that can satisfy condition (15).

Recall the notation introduced in section 2.4.

Here, we define

$$\begin{align*} \lim_{r\to r_c^+} \mbox{trj}^{[r]}(x_0,\tau_0) = \bigcap_{r\gt r_c}\; \overline{\bigcup_{r_c \lt s \lt r} \mbox{trj}^{[s]}(x_0,\tau_0)} \;\;\mbox{and}\;\; \lim_{r\to r_c^-} \mbox{trj}^{[r]}(x_0,\tau_0) = \bigcap_{r\lt r_c}\; \overline{\bigcup_{r \lt s \lt r_c} \mbox{trj}^{[s]}(x_0,\tau_0)}. \end{align*}$$

See appendix A.4 for the definition of an attractor.

By abuse of notation, we use τ to denote both the independent variable and the additional dependent variable.

$\Lambda(\tau)$ is denoted $\Gamma(t)$ in [132].

For example, when $\Lambda(\tau) \sim C\, e^{\mp\tilde{\rho}\tau}\,\tau^{n\le0}$ as $\tau\to\pm\infty$.

The notion of Hausdorff distance d_H is discussed in appendix A.1.

Note that a fixed $(x_0,t_0)$ in nonautonomous system (2) gives a rate-dependent $(x_0,s_0^{[r]})$ in the compactified system (26).

Note that $\Lambda(\tau)$ passes through λ_a before λ_b , though it may pass through either or both of these values several times.

Note that the subscript in τ_α is not related to the compactification parameter α.

The notion of Hausdorff distance d_H is discussed in appendix A.1.

The notion of signed distance d_s is discussed in appendix A.3.