Integer Ratio Self-Synchronization in Pairs of Limit Cycle Oscillators

The existence of multiple stable states of higher order m:n locking in coupled limit cycle oscillators has been studied by prior authors in the context of injection-locking in systems driven by an external periodic force. The current work builds on this concept to study the higher order locking characteristics of pairs of limit cycle oscillators self-synchronizing under coupling forces. To this end we analyze three oscillator systems: Van der Pol oscillators using numerical analysis, a simplified model for MEMS oscillators using numerical analysis as well as perturbation theory, and a full model of thermo-optically driven MEMS oscillators using numerical analysis. For the Van der Pol system, higher order locking is observed for the strongly nonlinear case corresponding to relaxation oscillations and the transition from weak to strong nonlinearity is studied using a parameter sweep. Additionally, coupling of a different nature such as quadratic coupling is also capable of inducing higher order coupling in Van der Pol oscillators. For the MEMS systems with linear coupling, higher order locking is observed when a strong cubic stiffness nonlinearity exists. Devil’s staircase-like structures are obtained for the coupling strength-frequency ratio parameter space which suggest overlapping Arnold locking regions for m:n locks corresponding to different integers.

Resilience Viewed through the Lens of Climate Change and Water Management

Resilience is not a new idea but there has been an upsurge in efforts to operationalize the concept within water management. This review begins with a synopsis of related themes around persistent and emerging pressures on freshwaters; environmental thresholds (or tipping points); ‘safe’ operating conditions; multiple stable states; regime shifts. A case is made for viewing and managing the resilience of water systems at nested scales. Indicators are needed to track evolving climate risks as well as to measure socio-ecological responses. Catchment properties can identify those river systems that are more or less likely to return to a pre-disturbance state; resilience further depends on institutional and social landscapes. Ideally, allied notions of resistance and reliability are applied alongside resilience to broaden the portfolio of adaptation measures. Water managers would also benefit from more consistent use of resilience terminology; incentives to build back better after catastrophes; strategic monitoring of incipient threats and tipping points; availability of long-term adaptation indicators; coordinated efforts to reduce non-climatic pressures on freshwaters (especially in headwaters); evidence-based, practical guidance on adaptation measures that build resilience.

Biocrust as one of multiple stable states in global drylands

Biocrusts cover ~30% of global drylands with a prominent role in the biogeochemical cycles. Theoretically, biocrusts, vascular plants, and bare soil can represent multiple stable states in drylands. However, no empirical evidence for the existence of a biocrust stable state has been reported. Here, using a global drylands dataset, we found that biocrusts form an alternative stable state (biocrust cover, ~80%; vascular cover, ≤10%) besides bare soil (both biocrust and vascular cover, ≤10%) and vascular plants (vascular cover, >50%; biocrust cover, ~5%). The pattern of multiple stable states associated with biocrusts differs from the classic fold bifurcation, and values of the aridity index in the range of 0 to 0.6 define a bistable region where multiple stable states coexist. This study empirically demonstrates the existence and thresholds of multiple stable states associated with biocrusts along climatic gradients and thus may greatly contribute to conservation and restoration of global drylands.

Combustor-inlet interactions in a low-order dynamic model of ramjet engines

ABSTRACTThe coexistence of multiple stable states is indicative of self-organising processes occurring in the course of the combustor-inlet interactions in a ramjet engine and give rise to the appearance of various nonlinear phenomena. This paper provides a dynamic model that can describe the multiple stable states and the corresponding nonlinear effects to further investigate the dynamic interactions between combustor and inlet in a ramjet engine. Our study shows the whole engine can display distinct dynamic behaviours ranging from irreversibility to hysteresis and to various mode transitions, depending on different physical parameters. With the model, we also illustrate the role of the instability of the normal shock wave in impacting the whole engine’s nonlinear dynamics. Additionally, we extend the previous studies of the classification of combustor-inlet interactions from a static framework to a dynamic framework, which helps to clarify the transient processes of the nonlinear interactions. This work offers a quantitative illustration of the combustor-inlet interactions in a ramjet engine by revealing its nonlinear dynamics and associated characteristics, therefore advancing our understanding of the nonlinear phenomena that exhibit in ramjet engines.