Death and revival of chaos

Characterizing chaos in systems subjected to parameter drift

To characterize chaos in systems subjected to parameter drift, where a number of traditional methods do not apply, we propose viable alternative approaches, both in the qualitative and quantitative sense. Qualitatively, following stable and unstable foliations is shown to be efficient, which are easy to approximate numerically, without relying on the need for the existence of an analog of hyperbolic periodic orbits. Chaos originates from a Smale horseshoe-like pattern of the foliations, the transverse intersections of which indicate a chaotic set changing in time. In dissipative cases, the unstable foliation is found to be part of the so-called snapshot attractor, but the chaotic set is not dense on it if regular time-dependent attractors also exist. In Hamiltonian cases stable and unstable foliations turn out to be not equivalent due to the lack of time-reversal symmetry. It is the unstable foliation, which is found to correlate with the so-called snapshot chaotic sea. The chaotic set appears to be locally dense in this sea, while tori with originally quasiperiodic character might break up, their motion becoming chaotic as time goes on. A quantity called ensemble-averaged pairwise distance evaluated in relation to unstable foliations is shown to be an appropriate tool to provide the instantaneous strength of time-dependent chaos.

A resilience concept based on system functioning: A dynamical systems perspective

We introduce a new framework for resilience, which is traditionally understood as the ability of a system to absorb disturbances and maintain its state, by proposing a shift from a state-based to a system functioning-based approach to resilience, which takes into account that several different coexisting stable states could fulfill the same functioning. As a consequence, not every regime shift, i.e., transition from one stable state to another, is associated with a lack or loss of resilience. We emphasize the importance of flexibility-the ability of a system to shift between different stable states while still maintaining system functioning. Furthermore, we provide a classification of system responses based on the phenomenological properties of possible disturbances, including the role of their timescales. Therefore, we discern fluctuations, shocks, press disturbances, and trends as possible disturbances. We distinguish between two types of mechanisms of resilience: (i) tolerance and flexibility, which are properties of the system, and (ii) adaptation and transformation, which are processes that alter the system's tolerance and flexibility. Furthermore, we discuss quantitative methods to investigate resilience in model systems based on approaches developed in dynamical systems theory.

Chaos in conservative discrete-time systems subjected to parameter drift

Based on the example of a paradigmatic area preserving low-dimensional mapping subjected to different scenarios of parameter drifts, we illustrate that the dynamics can best be understood by following ensembles of initial conditions corresponding to the tori of the initial system. When such ensembles are followed, snapshot tori are obtained, which change their location and shape. Within a time-dependent snapshot chaotic sea, we demonstrate the existence of snapshot stable and unstable foliations. Two easily visualizable conditions for torus breakup are found: one in relation to a discontinuity of the map and the other to a specific snapshot stable manifold, indicating that points of the torus are going to become subjected to strong stretching. In a more general setup, the latter can be formulated in terms of the so-called stable pseudo-foliation, which is shown to be able to extend beyond the instantaneous chaotic sea. The average distance of nearby point pairs initiated on an original torus crosses over into an exponential growth when the snapshot torus breaks up according to the second condition. As a consequence of the strongly non-monotonous change of phase portraits in maps, the exponential regime is found to split up into shorter periods characterized by different finite-time Lyapunov exponents. In scenarios with plateau ending, the divided phase space of the plateau might lead to the Lyapunov exponent averaged over the ensemble of a torus being much smaller than that of the stationary map of the plateau.

How can contemporary climate research help understand epidemic dynamics? Ensemble approach and snapshot attractors

Standard epidemic models based on compartmental differential equations are investigated under continuous parameter change as external forcing. We show that seasonal modulation of the contact parameter superimposed upon a monotonic decay needs a different description from that of the standard chaotic dynamics. The concept of snapshot attractors and their natural distribution has been adopted from the field of the latest climate change research. This shows the importance of the finite-time chaotic effect and ensemble interpretation while investigating the spread of a disease. By defining statistical measures over the ensemble, we can interpret the internal variability of the epidemic as the onset of complex dynamics-even for those values of contact parameters where originally regular behaviour is expected. We argue that anomalous outbreaks of the infectious class cannot die out until transient chaos is presented in the system. Nevertheless, this fact becomes apparent by using an ensemble approach rather than a single trajectory representation. These findings are applicable generally in explicitly time-dependent epidemic systems regardless of parameter values and time scales.

Weak tracking in nonautonomous chaotic systems

Previous studies have shown that rate-induced transitions can occur in pullback attractors of systems subject to "parameter shifts" between two asymptotically steady values of a system parameter. For cases where the attractors limit to equilibrium or periodic orbit in past and future limits of such an nonautonomous systems, these can occur as the parameter change passes through a critical rate. Such rate-induced transitions for attractors that limit to chaotic attractors in past or future limits has been less examined. In this paper, we identify a new phenomenon is associated with more complex attractors in the future limit: weak tracking, where a pullback attractor of the system limits to a proper subset of an attractor of the future limit system. We demonstrate weak tracking in a nonautonomous Rössler system, and argue there are infinitely many critical rates at each of which the pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor. We also state some necessary conditions that are needed for weak tracking.

Chaos in Hamiltonian systems subjected to parameter drift

Based on the example of a paradigmatic low-dimensional Hamiltonian system subjected to different scenarios of parameter drifts of non-negligible rates, we show that the dynamics of such systems can best be understood by following ensembles of initial conditions corresponding to tori of the initial system. When such ensembles are followed, toruslike objects called snapshot tori are obtained, which change their location and shape. In their center, one finds a time-dependent, snapshot elliptic orbit. After some time, many of the tori break up and spread over large regions of the phase space; however, one may find some smaller tori, which remain as closed curves throughout the whole scenario. We also show that the cause of torus breakup is the collision with a snapshot hyperbolic orbit and the surrounding chaotic sea, which forces the ensemble to adopt chaotic properties. Within this chaotic sea, we demonstrate the existence of a snapshot horseshoe structure and a snapshot saddle. An easily visualizable condition for torus breakup is found in relation to a specific snapshot stable manifold. The average distance of nearby pairs of points initiated on an original torus at first hardly changes in time but crosses over into an exponential growth when the snapshot torus breaks up. This new phase can be characterized by a novel type of a finite-time Lyapunov exponent, which depends both on the torus and on the scenario followed. Tori not broken up are shown to be the analogs of coherent vortices in fluid flows of arbitrary time dependence, and the condition for breakup can also be demonstrated by the so-called polar rotation angle method.

The snowball Earth transition in a climate model with drifting parameters: Splitting of the snapshot attractor

Using an intermediate complexity climate model (Planet Simulator), we investigate the so-called snowball Earth transition. For certain values (including its current value) of the solar constant, the climate system allows two different stable states: one of them is the snowball Earth, covered by ice and snow, and the other one is today's climate. In our setup, we consider the case when the climate system starts from its warm attractor (the stable climate we experience today), and the solar constant is changed according to the following scenario: it is decreased continuously and abruptly, over one year, to a state, where only the Snowball Earth's attractor remains stable. This induces an inevitable transition or climate tipping from the warm climate. The reverse transition is also discussed. Increasing the solar constant back to its original value in a similar way, in individual simulations, depending on the rate of the solar constant reduction, we find that either the system stays stuck in the snowball state or returns to warm climate. However, using ensemble methods, i.e., using an ensemble of climate realizations differing only slightly in their initial conditions we show that the transition from the snowball Earth to the warm climate is also possible with a certain probability, which depends on the specific scenario used. From the point of view of dynamical systems theory, we can say that the system's snapshot attractor splits between the warm climate's and the snowball Earth's attractor.

Tipping phenomena in typical dynamical systems subjected to parameter drift

Tipping phenomena, i.e. dramatic changes in the possible long-term performance of deterministic systems subjected to parameter drift, are of current interest but have not yet been explored in cases with chaotic internal dynamics. Based on the example of a paradigmatic low-dimensional dissipative system subjected to different scenarios of parameter drifts of non-negligible rates, we show that a number of novel types of tippings can be observed due to the topological complexity underlying general systems. Tippings from and into several coexisting attractors are possible, and one can find fractality-induced tipping, the consequence of the fractality of the scenario-dependent basins of attractions, as well as tipping into a chaotic attractor. Tipping from or through an extended chaotic attractor might lead to random tipping into coexisting regular attractors, and rate-induced tippings appear not abruptly as phase transitions, rather they show up gradually when the rate of the parameter drift is increased. Since chaotic systems of arbitrary time-dependence call for ensemble methods, we argue for a probabilistic approach and propose the use of tipping probabilities as a measure of tipping. We numerically determine these quantities and their parameter dependence for all tipping forms discussed.

Leaking in history space: A way to analyze systems subjected to arbitrary driving

Our aim is to unfold phase space structures underlying systems with a drift in their parameters. Such systems are non-autonomous and belong to the class of non-periodically driven systems where the traditional theory of chaos (based e.g., on periodic orbits) does not hold. We demonstrate that even such systems possess an underlying topological horseshoe-like structure at least for a finite period of time. This result is based on a specifically developed method which allows to compute the corresponding time-dependent stable and unstable foliations. These structures can be made visible by prescribing a certain type of history for an ensemble of trajectories in phase space and by analyzing the trajectories fulfilling this constraint. The process can be considered as a leaking in history space-a generalization of traditional leaking, a method that has become widespread in traditional chaotic systems, to leaks depending on time.