Systematic experimental exploration of bifurcations with noninvasive control

Emergent structure and dynamics of tropical forest-grassland landscapes

Previous work indicates that tropical forest can exist as an alternative stable state to savanna. Therefore, perturbation by climate change or human impact may lead to crossing of a tipping point beyond which there is rapid forest dieback that is not easily reversed. A hypothesized mechanism for such bistability is feedback between fire and vegetation, where fire spreads as a contagion process on grass patches. Theoretical models have largely implemented this mechanism implicitly, by assuming a threshold dependence of fire spread on flammable vegetation. Here, we show how the nonlinear dynamics and bistability emerge spontaneously, without assuming equations or thresholds for fire spread. We find that the forest geometry causes the nonlinearity that induces bistability. We demonstrate this in three steps. First, we model forest and fire as interacting contagion processes on grass patches, showing that spatial structure emerges due to two counteracting effects on the forest perimeter: forest expansion by dispersal and forest erosion by fires originating in adjacent grassland. Then, we derive a landscape-scale balance equation in which these two effects link forest geometry and dynamics: Forest expands proportionally to its perimeter, while it shrinks proportionally to its perimeter weighted by adjacent grassland area. Finally, we show that these perimeter quantities introduce nonlinearity in our balance equation and lead to bistability. Relying on the link between structure and dynamics, we propose a forest resilience indicator that could be used for targeted conservation or restoration.

Mean-field models of dynamics on networks via moment closure: An automated procedure

In the study of dynamics on networks, moment closure is a commonly used method to obtain low-dimensional evolution equations amenable to analysis. The variables in the evolution equations are mean counts of subgraph states and are referred to as moments. Due to interaction between neighbors, each moment equation is a function of higher-order moments, such that an infinite hierarchy of equations arises. Hence, the derivation requires truncation at a given order and an approximation of the highest-order moments in terms of lower-order ones, known as a closure formula. Recent systematic approximations have either restricted focus to closed moment equations for SIR epidemic spreading or to unclosed moment equations for arbitrary dynamics. In this paper, we develop a general procedure that automates both derivation and closure of arbitrary order moment equations for dynamics with nearest-neighbor interactions on undirected networks. Automation of the closure step was made possible by our generalized closure scheme, which systematically decomposes the largest subgraphs into their smaller components. We show that this decomposition is exact if these components form a tree, there is independence at distances beyond their graph diameter, and there is spatial homogeneity. Testing our method for SIS epidemic spreading on lattices and random networks confirms that biases are larger for networks with many short cycles in regimes with long-range dependence. A Mathematica package that automates the moment closure is available for download.

Control-Based Continuation: A New Approach to Prototype Synthetic Gene Networks

Control-Based Continuation (CBC) is a general and systematic method to carry out the bifurcation analysis of physical experiments. CBC does not rely on a mathematical model and thus overcomes the uncertainty introduced when identifying bifurcation curves indirectly through modeling and parameter estimation. We demonstrate, in silico, CBC applicability to biochemical processes by tracking the equilibrium curve of a toggle switch, which includes additive process noise and exhibits bistability. We compare the results obtained when CBC uses a model-free and model-based control strategy and show that both can track stable and unstable solutions, revealing bistability. We then demonstrate CBC in conditions more representative of an in vivo experiment using an agent-based simulator describing cell growth and division, cell-to-cell variability, spatial distribution, and diffusion of chemicals. We further show how the identified curves can be used for parameter estimation and discuss how CBC can significantly accelerate the prototyping of synthetic gene regulatory networks.

Dynamics of weakly coupled parametrically forced oscillators

The dynamics of two weakly coupled parametric oscillators are studied in the neighborhood of the primary subharmonic instability. The nature of both primary and secondary instabilities depends in a critical way on the permutation symmetries, if any, that remain after coupling is considered, and this depends on the relative phases of the parametric forcing terms. Detailed bifurcation sets, revealing a complex series of transitions organized in part by Bogdanov-Takens points, are calculated for representative sets of parameters. In the particular case of out-of-phase forcing the predictions of the coupled oscillator model are compared with direct numerical simulations and with recent experiments on modulated cross waves. Both the initial Hopf bifurcation and the subsequent saddle-node heteroclinic bifurcation are confirmed.

On the experimental identification of unstable static equilibria

This paper shows how the presence of unstable equilibrium configurations of elastic continua is reflected in the behaviour of transients induced by large perturbations. A beam that is axially loaded beyond its critical state typically exhibits two buckled stable equilibrium configurations, separated by one or more unstable equilibria. If the beam is then loaded laterally (effectively like a shallow arch) it may snap-through between these states, including the case in which the loading is applied dynamically and of short duration, i.e. an impact. Such impacts, if applied at random locations and of random strength, will generate an ensemble of transient trajectories that explore the phase space. Given sufficient variety, some of these trajectories will possess initial energy that is close to (just less than or just greater than) the energy required to cause snap-through and will have a tendency to slowdown as they pass close to an unstable configuration: a saddle point in a potential energy surface, for example. Although this close-encounter is relatively straightforward in a system characterized by a single degree of freedom, it is more challenging to identify in a higher order or continuous system, especially in a (necessarily) noisy experimental system. This paper will show how the identification of unstable equilibrium configurations can be achieved using transient dynamics.