TY  JOUR
TI  Timedependent saddlenode bifurcation: Breaking time and the point of no return in a nonautonomous model of critical transitions
AU  Li, Jeremiah H.
AU  Ye, Felix X.F.
AU  Qian, Hong
AU  Huang, Sui
T2  Physica D. Nonlinear Phenomena
AB  There is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddlenode bifurcations. In particular, the term "tipping", or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddlenode bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a nonautonomous system with a timedependent parameter p(τ) and its corresponding "dynamic" (timedependent) saddlenode bifurcation by the modern theory of nonautonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the breaking time
τ
^
at which the corresponding autonomous system with a timeindependent parameter
p
a
=
p
(
τ
^
)
undergoes a bifurcation. A dimensionless parameter
α
=
λ
p
0
3
V

2
is introduced, in which λ is the curvature of the autonomous saddlenode bifurcation according to parameter p(τ), which has an initial value of p0 and a constant rate of change V. We find that the breaking time
τ
^
is always less than the actual point of no return τ∗ after which the critical transition is irreversible; specifically, the relation
τ
*

τ
^
≃
2.338
(
λ
V
)

1
3
is analytically obtained. For a system with a small λV, there exists a significant window of opportunity (
τ
^
, τ∗) during which rapid reversal of the environment can save the system from catastrophe.
DA  2019///
PY  2019
DO  10.1016/j.physd.2019.02.005
DP  PubMed
VL  395
SP  7
EP  14
J2  Physica D
LA  eng
SN  01672789
ST  Timedependent saddlenode bifurcation
ER 