We examine the conditions for appearance of symmetry breaking bifurcation in damped and periodically driven pendulum in the case of strong damping. We show that symmetry breaking, unlike other nonlinear phenomena, can exist at high dissipation. We prove that symmetry breaking phases exist between phases of symmetric normal and symmetric inverted oscillations. We find that symmetry broken solutions occupy a sufficiently smaller region of pendulum's parameter space in comparison to the statements made in earlier considerations [McDonald and Plischke, Phys. Rev. B 27 (1983) 201]. Our research on symmetry breaking in a strongly damped pendulum is relevant to an understanding of phenomena of dynamic symmetry breaking and rectification in a pure ac driven semiconductor superlattices.

We find the conditions for a rectification of electromagnetic wave in a lateral semiconductor superlattice with a high mobility of electrons. The rectification is assisted by a transition to a dissipative chaos at a very high mobility. We show that mechanism responsible for the rectification is a creation of warm electrons in the superlattice miniband caused by an interplay of the effects of nonlinearity and finite band width.

We consider the first order differential equation with a sinusoidal nonlinearity and periodic time dependence, that is, the periodically driven overdamped pendulum. The problem is studied in the case that the explicit time dependence has symmetries common to pure ac-driven systems. The only bifurcation that exists in the system is a degenerate pitchfork bifurcation, which describes an exchange of stability between two symmetric nonlinear modes. Using a type of Prüfer transform to a pair of linear differential equations, we derive an approximate condition of the bifurcation. This approximation is in very good agreement with our numerical data. In particular, it works well in the limit of large drive amplitudes and low external frequencies. We demonstrate the usefulness of the theory applying it to the models of pure ac-driven semiconductor superlattices and Josephson junctions. We show how the knowledge of bifurcations in the overdamped pendulum model can be utilized to describe the effects of rectification and amplification of electric fields in these microstructures.

This paper investigates the macroeconomic dynamics of consumption and real interest rates when there are constraints on debt finance, used both for insurance against income shocks and transferability of resources over time. We amend a standard continuous‐time deterministic model, with patient and impatient household sectors, introducing sector level income shocks. Shocks that push the impatient sector towards its leverage limit increase precautionary saving and result in a substantial but transient decline of the real interest rate relative to the deterministic benchmark. We discuss the resulting dynamics of consumption, leverage and interest rates, and implications for macroeconomic modelling and policy.

We are grateful for comments from Marcus Brunnermeier and from the handbook editors 1 An example of such a DSGE extension is Meh and Moran [2010] who generalise the financial accelerator to include a bank-moral hazard based on Holmstrom and Tirole [1997].