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# Distribution of tipping events from Monte Carlo simulations and most probable paths (red and black curves) from x + to x − , with (a) attracting sliding (η = −2; 6,487 tips with N = 1.056×10 7 simulations) and (b) repelling sliding (η = 1; 1,960 tips with N = 1.676 × 10 7 simulations). Parameters used are a = p = −2, b = q = −7, c = r = 1, and σ = 0.3. All other curves are as in Figure 2.

Source publication

We develop a path integral framework for determining most probable paths in a class of systems of stochastic differential equations with piecewise-smooth drift and additive noise. This approach extends the Freidlin-Wentzell theory of large deviations to cases where the system is piecewise-smooth and may be non-autonomous. In particular, we consider...

## Contexts in source publication

**Context 1**

... performed Monte Carlo simulations for two values of η: η = 1, where there is repelling sliding along Σ, and η = −2, where there is attracting sliding. See Figure 4 for histograms of the solutions to System (2),(23) that 'tip' from x + to a neighborhood of x − . For (a) η = −2, transition paths tip over the basin boundary Σ by tracking the attracting sliding region until it ends. ...

**Context 2**

... that in Figure 4(b), we indicate both the most probable path followed by the Monte Carlo simulations (black curve) and the predicted family of nonunique most probable paths that slide (red curves). These paths coincide for x > 0 but are distinct for x ≤ 0. The observed most probable path in the left-half plane corresponds to the solutions to the EL equations with boundary conditions x(t 1 ) = (0, max y∈Σ R y) and x(t f ) = x − . ...

**Context 3**

... is, steady state solutions of Equation (26) solve the Euler-Lagrange equations. The converged solution to the IBVP in Equation (26), approximated using the Forward Time Centered Space finite difference scheme with s as the time variable and t as the space variable, is shown as the black curve in Figure 4(b). From Equation (20) we might expect the predicted most probable paths in Figure 4(b) to minimize I (t 0 ,t f ) because I (t 1 ,t f ) = 0. ...

**Context 4**

... converged solution to the IBVP in Equation (26), approximated using the Forward Time Centered Space finite difference scheme with s as the time variable and t as the space variable, is shown as the black curve in Figure 4(b). From Equation (20) we might expect the predicted most probable paths in Figure 4(b) to minimize I (t 0 ,t f ) because I (t 1 ,t f ) = 0. However, if we consider the predicted most probable path of the piecewise-smooth system in the context of the mollified system and calculate I ...

**Context 5**

... performed Monte Carlo simulations for two values of η: η = 1, where there is repelling sliding along Σ, and η = −2, where there is attracting sliding. See Figure 4 for histograms of the solutions to System (2),(23) that 'tip' from x + to a neighborhood of x − . For (a) η = −2, transition paths tip over the basin boundary Σ by tracking the attracting sliding region until it ends. ...

**Context 6**

... that in Figure 4(b), we indicate both the most probable path followed by the Monte Carlo simulations (black curve) and the predicted family of nonunique most probable paths that slide (red curves). These paths coincide for x > 0 but are distinct for x ≤ 0. The observed most probable path in the left-half plane corresponds to the solutions to the EL equations with boundary conditions x(t 1 ) = (0, max y∈Σ R y) and x(t f ) = x − . ...

**Context 7**

... is, steady state solutions of Equation (26) solve the Euler-Lagrange equations. The converged solution to the IBVP in Equation (26), approximated using the Forward Time Centered Space finite difference scheme with s as the time variable and t as the space variable, is shown as the black curve in Figure 4(b). From Equation (20) we might expect the predicted most probable paths in Figure 4(b) to minimize I (t 0 ,t f ) because I (t 1 ,t f ) = 0. ...

**Context 8**

... converged solution to the IBVP in Equation (26), approximated using the Forward Time Centered Space finite difference scheme with s as the time variable and t as the space variable, is shown as the black curve in Figure 4(b). From Equation (20) we might expect the predicted most probable paths in Figure 4(b) to minimize I (t 0 ,t f ) because I (t 1 ,t f ) = 0. However, if we consider the predicted most probable path of the piecewise-smooth system in the context of the mollified system and calculate I ...