Critical transitions and perturbation growth directions

Critical transitions occur in a variety of dynamical systems. Here we employ quantifiers of chaos to identify changes in the dynamical structure of complex systems preceding critical transitions. As suitable indicator variables for critical transitions, we consider changes in growth rates and directions of covariant Lyapunov vectors. Studying critical transitions in several models of fast-slow systems, i.e., a network of coupled FitzHugh-Nagumo oscillators, models for Josephson junctions, and the Hindmarsh-Rose model, we find that tangencies between covariant Lyapunov vectors are a common and maybe generic feature during critical transitions. We further demonstrate that this deviation from hyperbolic dynamics is linked to the occurrence of critical transitions by using it as an indicator variable and evaluating the prediction success through receiver operating characteristic curves. In the presence of noise, we find the alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents to be more successful in announcing critical transitions than common indicator variables as, e.g., finite-time estimates of the variance. Additionally, we propose a new method for estimating approximations of covariant Lyapunov vectors without knowledge of the future trajectory of the system. We find that these approximated covariant Lyapunov vectors can also be applied to predict critical transitions.

Figure 1

Estimating approximations of covariant Lyapunov vectors with the repetitive iteration method yields results which mimic the dynamics of covariant Lyapunov vectors computed through Ginelli et al.'s method. The absolute value of the cosine of the angle between the first and the second covariant Lyapunov vectors computed with Ginelli et al.'s method, solid black line, are compared with the results from the repetitive iteration method, dashed blue line, for a FitzHugh-Nagumo model with $a=0.4,b=0.3,e=0.01$, and $D=0$.

Figure 2

Alignment of covariant Lyapunov vectors during noise-induced transitions in a FitzHugh-Nagumo oscillator with $a=1$, $b=0.3$, $\epsilon =0.01$, and $D=0.2$. (a) Phase-space portrait of the oscillator with black lines indicating the nullclines and the discontinuous lines representing trajectories from different initial points. (b) The dotted red line indicates a typical trajectory in the phase space. Green and orange vectors show the first and the second covariant Lyapunov vector, respectively, both vectors align during transitions. (c) The cosine of the angle between the first and the second vector is shown while the system is drifting on the left and transitioning to the right branch (black dashed line) and drifting on the right and transitioning to the left branch (red line). (d) Time series of the fast variable, i.e., the observable of the system. (e) Time series of the cosine of the angle between the first and the second vector. (f) Time series of the first finite-time Lyapunov exponent and (g) time series of the second finite-time Lyapunov exponent.

Figure 3

The finite-time growth rate of the first and second covariant Lyapunov vector converge during the transition in a FitzHugh-Nagumo oscillator. The first (dotted green line) and the second (dotted red line) finite-time Lyapunov exponent are compared to the finite-time growth rate of the second covariant Lyapunov vector (solid black line). Note that the growth rate of the first covariant Lypunov vector is by definition the same as the first finite-time Lyapunov exponent. While the trajectory is slowly drifting on the nullcline the growth rate of the second covariant vector is almost the same as the second finite-time Lyapunov exponent. Merging of the first and the second covariant Lyapunov vector during the transition manifests it's self in converging of the growth rate of the second covariant vector to the first finite-time exponent. (a) The time series of the fast variable. (b) The growth rates computed with Ginelli et al.'s method. (c) The growth rates computed with repetitive iterations method.

Figure 4

Alignment of covariant Lyapunov vectors in a single FitzHugh-Nagumo oscillator with $a=0.4$, $b=0.3$, $\epsilon =0.01$, and $D=0.2$. (a) Phase-space portrait of the oscillator with black lines indicating the nullclines and the discontinuous lines representing trajectories from different initial points. (b) The dotted red line indicates a typical trajectory in the phase space. The green and the orange vectors show the first and the second covariant Lyapunov vectors, respectively, both vectors align during transitions. (c) The cosine of the angle between the first and the second vectors is shown while the system is drifting on the left and transitioning to the right branch (black dashed line) and drifting on the right and transitioning to the left branch (red line). (d) Time series of the fast variable, i.e., the observable of the system. (e) Time series of the cosine of the angle between the first and the second vector. (f) Time series of the first finite-time Lyapunov exponent and (g) time series of the second finite-time Lyapunov exponent.

Figure 5

Merging of all covariant Lyapunov vectors forecasts spikes in the Hindmarsh-Rose model. (a) Projection of the trajectory of the Hindmarsh-Rose model and the nullclines of $x$ and $y$ on the $x\text{−}y$ plane. The trajectory (dash-dotted line) slowly moves up close to the nullclines (black solid lines) before the spike occurs. (b) The three-dimensional phase space portrait of the trajectory shows a slow drift of $z$ as the fast variables go through an excursion. The green-blue lines are the shifted projections of the trajectory on the $x\text{−}y$ plane, showing the absolute value of the cosine of the angle between the covariant Lyapunov vectors.

Figure 6

In the Hindmarsh-Rose model, all three covariant Lyapunov vectors merge during a critical transition and the finite-time Lyapunov exponents increase prior and reach their maximum during transitions. (a) Time series of the fast variable $x$. [(b)–(d)] Time series of the cosine of the angles between the covariant Lyapunov vectors. [(e)–(g)] Time series of the finite-time Lyapunov exponents.

Figure 7

In the model of Josephson junctions, the first and the second vector have tangencies during and prior to critical transitions. The angle between the first and the third vector decreases as well before transitions occur. These observations hold for both ranges of $\beta$ although the overall dynamics of covariant Lyapunov vectors is very different for different values of $\beta$. Left: $\beta =0.1$; right: $\beta$ = 2. (a) For $\beta =0.1$, the Josephson junction is effectively two dimensional. (b) For $\beta =2$, the transitions include spiraling around the nullcline. In both figures, the green, the orange, and the purple vectors represent the first, second, and third covariant Lyapunov vectors, respectively. The contour lines are shifted projections of the trajectory onto the $\varphi \text{−}\psi$ plane, showing the absolute value of the cosine of the angles between the covariant Lyapunov vectors. [(c) and (d)] Times series of the fast variable $\varphi$. [(e)–(j)] Time series of the cosine of the angles between the covariant Lyapunov vectors. [(k)–(p)] Time series of the finite-time Lyapunov exponents.

Figure 8

In a network of four coupled FitzHugh-Nagumo oscillators, covariant Lyapunov vectors also align during transitions. The finite-time Lyapunov exponents exhibit qualitatively the same behavior as observed for a single FitzHugh-Nagumo oscillator with $a=0.4$, $b=0.3$, $\epsilon =0.01$, and $D=0.2$. (a) Time series of the fast variables of four fully connected FitzHugh-Nagumo oscillators going through noise-induced transitions. (b) Time series of the angle between the first and the second covariant Lyapunov vectors. [(c) and (d)] Time series of the first and the fifth finite-time Lyapunov exponents.

Figure 9

The area under the curve (AUC) is a measure of the effectiveness of predictions. Increasing the time lag $\mathrm{\Delta }t$ decreases the success of predictions. Different ROC curves for a FitzHugh-Nagumo oscillator with noise strength $D=0.3$.

Figure 10

AUCs of the angle between the covariant Lyapunov vectors estimated without iterating to the future, ${\alpha }_{ij}$, and with Ginelli et al.'s method, ${\theta }_{ij}$, finite-time Lyapunov exponents, $\lambda$, and the variance, $\sigma$, for different noise strengths, $D$, and different lead times, $\mathrm{\Delta }t$, for a single FitzHugh-Nagumo oscillator. In the presence of noise, predictors based on covariant Lyapunov vectors can predict CTs better than conventional indicator variables, such as ${\sigma }_1^2$. The lead time, $\mathrm{\Delta }t$, indicates the time lag between the prediction and the event. The noise strengths being (a) $D=0$, (b) $D=0.3$, and (c) $D=0.6$. Yellow (gray) shaded regions represent $95\%$ confidence intervals estimated from random predictions on the same data sets.

Figure 11

AUCs of the angle between the covariant Lyapunov vectors estimated without iterating to the future, ${\alpha }_{ij}$, and with Ginelli et al.'s method, ${\theta }_{ij}$, finite-time Lyapunov exponents, $\lambda$, and the variance, $\sigma$, for different noise strengths, $D$, and different lead times, $\mathrm{\Delta }t$, for the Josephson junction. In the presence of noise, predictors based on covariant Lyapunov vectors can predict CTs better than conventional indicator variables, such as ${\sigma }_1^2$. The lead time, $\mathrm{\Delta }t$, indicates the time lag between prediction and occurrence of the CT. The noise strengths being (a) $D=0$, (b) $D=0.4$, and (c) $D=0.8$. Yellow (gray) shaded regions represent $95\%$ confidence intervals estimated from random predictions on the same data sets.