Codimension-two border collision bifurcation in a two-class growth model with optimal saving and switch in behavior

We consider a two-class growth model with optimal saving and switch in behavior. The dynamics of this model is described by a two-dimensional (2D) discontinuous map. We obtain stability conditions of the border and interior fixed points (known as Solow and Pasinetti equilibria, respectively) and investigate bifurcation structures observed in the parameter space of this map, associated with its attracting cycles and chaotic attractors. In particular, we show that on the x-axis, which is invariant, the map is reduced to a 1D piecewise increasing discontinuous map, and prove the existence of a corresponding period adding bifurcation structure issuing from a codimension-two border collision bifurcation point. Then, we describe how this structure evolves when the related attracting cycles on the x-axis lose their transverse stability via a transcritical bifurcation and the corresponding interior cycles appear. In particular, we show that the observed bifurcation structure, being associated with the 2D discontinuous map, is characterized by multistability, that is impossible in the case of a standard period adding bifurcation structure.

Period-3 dominant phase synchronisation of Zelkova serrata: border-collision bifurcation observed in a plant population

The population synchrony of tree seed production has attracted widespread attention in agriculture, forestry and ecosystem management. Oaks usually show synchronisation of irregular or intermittent sequences of acorn production, which is termed ‘masting’. Tree crops such as citrus and pistachio show a clear two-year cycle (period-2) termed ‘alternate bearing’. We identified period-3 dominant phase synchronisation in a population of Zelkova serrata. As ‘period-3’ is known to provide evidence to imply chaos in nonlinear science, the observed period-3 phase synchronisation of Zelkova serrata is an attractive real-world phenomenon that warrants investigation in terms of nonlinear dynamics. Using the Hilbert transform, we proposed a procedure to determine the fractions of periods underlying the survey data and distinguished the on-year (high yield year) and the off-year (low yield year) of the masting. We quantified the effects of pollen coupling, common environmental noise and individual variability on the phase synchronisation and demonstrated how the period-3 synchronisation emerges through a border-collision bifurcation process. In this paper, we propose a model that can describe diverse behaviours of seed production observed in many different tree species by changing its parameters.

Chaotic dynamics of a pulse modulated control system for a heating unit

The dynamic modes and bifurcations in a pulse control system of a heating unit, the condition of which is described through differential equations with discontinuous right–hand sides, have been studied. It has been shown that the system under research can demonstrate a great variety of nonlinear phenomena and bifurcation transitions, such as quasiperiodicity, multistable behaviour, chaotization of oscillations through a classical period–doubling bifurcations cascade and border–collision bifurcation.