To understand the occurrence of tipping points in periodically forced systems, we take a very simple dynamical system as a useful conceptual model example, where the evolution of state x over time t in this system is determined by

$$\frac{{\bf{d}}{\bf{x}}}{{\bf{d}}{\bf{t}}}=-{{\bf{x}}}^{3}+{\bf{x}}-{\bf{D}}({\bf{t}})$$ (1)

The black lines in Fig. 1a represent this system’s equilibria (dx/dt = 0) for constant D, which we refer to as the equilibrium system. D is an external (prescribed) driver representing environmental conditions, which makes the system non-autonomous (meaning that it explicitly depends on time). In case the driver D changes slowly enough, the system follows a stable equilibrium branch which ends at a bifurcation point. After the driver passes this point the system switches to the remaining stable equilibrium – the case of a classical tipping point.

We now investigate the behavior of the system in response to a driver that oscillates around a mean D m with amplitude D a and period T.

$${\bf{D}}({\bf{t}})={{\bf{D}}}_{{\bf{m}}}+{{\bf{D}}}_{{\bf{a}}}\,\cos (\frac{2{\boldsymbol{\pi }}{\bf{t}}}{{\bf{T}}})$$ (2)

Our conceptual model is hence determined by Eqs 1 and 2, and is well-known as the overdamped limit of the Duffing oscillator. We integrate the system for different sets of the three parameters, D m , D a and T, until it reaches a stable attractor in form of a periodic oscillation. Like the driver, the system’s solution is periodic in time with period T (also involving harmonics that can arise close to a bifurcation point)17,18. Figure 2 illustrates how the periodic solutions of the system can change when one of the three parameters is varied. Varying D m (Fig. 2a) is similar to the classical scenario of a tipping point, only that the stable solutions oscillate in time: if D m is increased across the bifurcation, the upper branch of stable solutions disappears and the system has to shift to the lower branch. Interestingly, such tipping-point behavior can also be obtained by increasing the driver’s period T or its amplitude D a , without any change in D m (Fig. 2b–d). For example, when the driver is fast (small T) the system regularly passes the bifurcation point of the equilibrium system but returns in time to keep the system close to the upper branch. Therefore, there can be two alternative periodic solutions. However, when the driver becomes slower, the system destabilizes and propels itself toward the only remaining solution around the lower branch. Figure 2e illustrates this forced transition as a blue trajectory in the phase space spanned by driver D and state x.

Figure 2 Periodic solutions of the conceptual model (Eqs 1 and 2) in dependence on the driver’s mean D m (a), its period T (b), and its amplitude D a (c,d). The blue arrows show the start and end points of linear parameter changes discussed in the text and shown as blue trajectories in (e,f). The arrows in e-f indicate the overall direction of the change. The two red orbits in each bottom panel are the asymptotic solutions for the parameters at the beginning and end of each transition. See Supplementary Information for details on the parameter choices. Full size image

A similar result is obtained when D a is increased (Fig. 2c,f) because the relaxation toward the lower branch becomes too strong. A systematic sampling of the parameter space spanned by D m , D a and T shows that D a and T essentially have a very similar effect in the model (Figs S1,S2) as they both reduce the D m -range in which multiple solutions occur. When both become large, the dependence on initial conditions is destroyed (Fig. S3). This is due to another type of abrupt transition, which occurs at D a = 0.8 in Fig. 2d. At this point, the system reaches a regime where it makes a full cycle involving both branches of the equilibrium system, flipping up and down between the branches (Fig. 2g). As the system’s response becomes very slow at the bifurcation point of the equilibrium system, the transition to the alternative state occurs after the bifurcation point is reached19. Nonetheless, if the driver’s amplitude is large enough, two abrupt shifts occur within each full cycle of the system under constant parameter conditions. When this regime becomes active, the amplitude of the solution increases rapidly with D a . In contrast to the bifurcations mentioned previously, such sudden amplitude change is reversible: for intermediate D a in Fig. 2d, there are no multiple solutions because all movement occurs along one continuous stable branch far from the second one above. When D a increases, the driver suddenly encloses both bifurcation points. During a cycle, the system now switches to the upper branch, but falls down again later on the other side. It should be noted that the amplitude change is only a sharp transition if T is sufficiently large (compare Fig. 2c with 2d). Otherwise, the time is too short for the system to get close to the alternative branch to which it is suddenly attracted in a short period during each cycle.

This reminds us that Fig. 2a–d displays only asymptotic solutions where the system ends up after an infinite time. In similarity to tipping points in an equilibrium system, the time it takes do make a transition from one to the other solution (blue trajectories in Fig. 2e–g), depends on the response time of the system. This response time can be idealized as a single relaxation time τ. For example, in the equilibrium system given by Eq. 1 and constant D, the relaxation time can be shown to be

$${\boldsymbol{\tau }}={\bf{1}}/({\bf{3}}{{\bf{x}}}_{{\bf{eq}}}^{{\bf{2}}}-{\bf{1}}),$$ (3)

where x eq is a stable equilibrium and lies on either of the continuous black lines in Fig. 1a.

The ratio between τ and the driver’s period T determines the abruptness of every one of the transitions discussed above. Three regimes can be distinguished: when T ≫ τ, the system closely follows the stable state of the equilibrium system. This limit case is usually addressed with the classical tipping point concept. When T ≪ τ, the system cannot follow the rapid forcing at all and remains static. It can then be described with an equivalent time-independent system. In the intermediate regime, where the time scales of the system and driver are similar, irreversible amplitude or timescale-induced shifts can occur.