Bernd Krauskopf

Wild chaos is a higher-dimensional form of chaotic dynamics characterized by the robust presence of homoclinic tangencies. It can only arise in vector fields of dimension at least four or in diffeomorphisms of dimension at least three. We study a four-dimensional Lorenz-like system known to exhibit a wild pseudohyperbolic attractor. Its global bifurcation structure in a two-parameter plane has previously been analyzed numerically by computing kneading diagrams and Lyapunov spectra, revealing dense accumulations of Shilnikov-type homoclinic bifurcations and parameter regions satisfying necessary conditions for wild chaos. Here, we build on this analysis by finding and continuing the codimension-one bifurcation curves associated with two fundamental symmetric periodic orbits and their asymmetric counterparts. We determine the criticality of symmetry-breaking, period-doubling, fold, and torus bifurcations, and construct the global two-parameter bifurcation diagram in a compactified parameter plane. These bifurcation curves are shown to form a nested structure that organizes both periodic windows and chaotic regions. In particular, they delimit transitions between periodic dynamics, chaotic attractors generated by period-doubling cascades, and chaotic attractors that satisfy a necessary condition for being wild pseudohyperbolic. While none of the computed bifurcation curves forms a sharp boundary of the regions where this necessary condition is satisfied, torus and fold bifurcations of symmetric periodic orbits are found to play a significant role in organizing the nature of attractors in the system; this finding is supported by the numerical computation of Lyapunov spectra for said attractors.

Wild chaos is a higher-dimensional form of chaotic dynamics that can only arise in vector fields of dimension at least four (or in diffeomorphisms of dimension at least three). It is characterized by the persistent presence of tangencies between stable and unstable manifolds of an invariant set. We study a four-dimensional extension of the classic Lorenz system, which was recently shown (by Gonchenko et al 2021 Nonlinearity 34 ) to exhibit a so-called wild pseudohyperbolic attractor of spiral type for a specific choice of parameters. Pseudohyperbolicity guarantees that every trajectory in the attractor has a positive maximal Lyapunov exponent, and this property persists under small perturbations of the system. We employ a combination of different advanced numerical techniques to investigate how wild chaotic attractors arise geometrically by analysing the overall bifurcation structure in the ( ϱ , μ ) -parameter plane. Here, ϱ is the standard (Rayleigh) parameter in the classic Lorenz system, and μ is a new parameter that introduces spiralling dynamics near the origin. We begin by identifying the bifurcation structure inherited from the Lorenz system and continuing its homoclinic bifurcations as curves. The computation of kneading diagrams, in combination with Lin’s method, allows us to uncover and continue additional global bifurcations intrinsic to the four-dimensional system. To identify different types of attractors, we also compute the Lyapunov spectrum of the attractors on which the one-dimensional unstable manifold of the origin accumulates. Our detailed numerical investigations provide insight into the parameter regions where wild pseudohyperbolic attractors may exist.

The Atlantic Meridional Overturning Circulation (AMOC) is a climate-relevant ocean current system responsible for the meridional heat transport in the Atlantic. The AMOC strength is affected by a meridional density difference, where the density in the northern North Atlantic is controlled by an advective and a convective feedback. Here, we introduce and study a conceptual mathematical model of the variability of the AMOC strength by representing these feedbacks as delayed terms in a scalar delay differential equation (DDE) for the salinity in the northern North Atlantic. After scaling and without external input, this DDE has the associated delay times τ and σ as its only parameters. We perform a numerical bifurcation analysis of this deceptively simple-looking DDE AMOC model with the continuation software package DDE-BifTool. We find and characterize intricate dynamical regimes, including those exhibiting complicated oscillations associated with homoclinic connections. These results are presented as bifurcation diagrams in the ( τ , σ ) -plane, where we identify a codimension-two Belyakov transition as an organizing center for nearby complicated dynamics. Moreover, we present a detailed analysis of different attractor regions in the ( τ , σ ) -plane, which we identify by computing the (strong) unstable manifold of a physically relevant equilibrium. As a general picture, we find that attractor regions repeat as the values of either σ or τ increase, including in physically relevant regions of these two (scaled) delay times. In this way, we clarify where different types of dynamics — such as periodic orbits of different periods, invariant tori, and chaotic dynamics — can be observed, and how they emerge or disappear.

The dynamics of micro-electromechanical systems (MEMS) have been thesubject of extensive research for several decades. Owing to their inherentlynonlinear behaviour, MEMS have attracted significant interest fromresearchers in the fields of physics, applied mechanics, mathematics, as wellas experimentalists and technology developers, who regard them as a richplatform for studying nonlinear dynamics, bifurcations, and chaos. Whilethe nonlinear dynamics of passively operated oscillators are well understoodand widely exploited in technological applications, actively operated MEMSremain comparatively underexplored. In addition to sharing the nonlinearcharacteristics of passive systems, active MEMS exhibit a significantly morecomplex dynamical landscape due to the presence of additional bifurcationphenomena, such as Hopf and Neimark–Sacker bifurcations, which are typicallyabsent in passive devices. While the understanding of active MEMSremains limited, their dynamic performance significantly exceed that of passivecounterparts, as evidenced by natural biosensors. In this work, we introducean actively operated MEMS cantilever with integrated self-sensing andself-actuation capabilities that enable adaptive dynamic behaviour throughactive self-coupling. A theoretical framework is developed to show that Hopfbifurcations require both system coupling and an energy-pumping mechanism,which are implemented by means of thermo-mechanical interactionsand a digital feedback mechanism. A reduced-order model incorporating mechanical, thermal, and filtering dynamics is analysed to identify conditions for Hopf-induced self-oscillations. Numerical continuation reveals richnonlinear dynamics, including compressive amplification, multi-stability, andcomplex frequency response behaviour, tunable via coupling parameters.Experimental control-based continuation validates theoretical findings andreconstructs the system’s bifurcation structure, including unstable solutionbranches. The results highlight the potential of active self-coupling to induceadaptive damping and enhance sensitivity, enabling tunable and highperformanceMEMS devices without the need for structural redesign. Thisapproach opens new avenues for nonlinear sensing applications, particularlyin high-damping environments as those of biosensors.

Micro-and nano-electromechanical systems (M/NEMS) are in high demand for cutting-edge future technological solutions. Their strongly nonlinear nature is regarded as beneficial for optimising performance metrics for a wide range of engineering applications. As model-free experimentation remains limited to forward time observations, important dynamic features can be left uncharted, resulting in an incomplete dynamical landscape. Here we address this issue by employing experimental continuationa model-free technique for constructing bifurcation diagrams by tracking steady-states and/or periodic responses directly in a physical experiment. This approach is unexplored for investigating micro-scale systems with fast timescales. Our state-of-the-art experiment investigates an active MEMS cantilever operated as a self-oscillator with a natural frequency of 100 kHz; the fastest timescales of a mechanical system probed with experimental continuation. We explore the cantilever's nonlinear responses to external, periodic excitation via a sequence of one-parameter response curves. By experimentally mapping out the MEMS cantilever's stable and unstable periodic orbits, we expose the dynamic landscape by rendering a multi-valued response surface across a range of forcing frequencies and amplitudes. An atypical, non-Duffing-like bifurcation structure is revealed.

Motivated by recent experimental observations concerning the polarization dynamics in an excitable microlaser with saturable absorber coupled to an external feedback mirror reported in Ruschel et al. (Opt Lett 50(8):2618, 2025), we propose here an in-depth theoretical investigation of the locked dynamics of regenerative vectorial pulse trains that this system produces. We perform a numerical bifurcation analysis of self-sustained pulse trains of a corresponding spin-flip model with delayed feedback. Its focus is on strongly resonant regimes, where the modulation of the peak intensities of polarized regenerative pulse trains locks to two, three and four times the pulse regeneration time. Specifically, we identify points of strongly resonant rotation numbers on curves of torus bifurcations in the parameter plane of the amplitude and phase anisotropy parameters, and continue emerging curves of fold and period-doubling bifurcations to identify locking regions. In this way, we clarify where pulse trains show strongly resonant polarization dynamics and highlight that even weak polarization coupling in delay-coupled microlasers creates considerable dynamical richness. These results may have impact on the design of future coupled microlaser systems for neuroinspired on-chip computing, where the polarization of an excitable pulse can be used to encode or process information.

Motivated by recent experimental observations concerning the polarization dynamics in an excitable microlaser with saturable absorber coupled to an external feedback mirror reported in [Ruschel et al. (2025) Opt. Lett. 50 (8) 2618], we propose here an in-depth theoretical investigation of the locked dynamics of regenerative vectorial pulse trains that this system produces. We perform a numerical bifurcation analysis of self-sustained pulse trains of a corresponding spin-flip model with delayed feedback. Its focus is on strongly resonant regimes, where the modulation of the peak intensities of polarized regenerative pulse trains locks to 2, 3 and 4 times the pulse regeneration time. Specifically, we identify points of strongly resonant rotation numbers on curves of torus bifurcations in the parameter plane of the amplitude and phase anisotropy parameters, and continue emerging curves of fold and period-doubling bifurcations to identify locking regions. In this way, we clarify where pulse trains show strongly resonant polarization dynamics and highlight that even weak polarization coupling in delay-coupled microlasers creates considerable dynamical richness. These results may have impact on the design of future coupled microlasers systems for neuroinspired on-chip computing, where the polarization of an excitable pulse can be used to encode or process information.

We revisit the conceptual Welander model for the evolution of temperature and salinity in two vertically stacked boxes for surface and deep water, which interact through diffusion and/or convective adjustment. When the switching between them is assumed to be instantaneous, the model exhibits oscillations with phases of weak diffusive mixing (deep-decoupling) interspersed with strong convective mixing (deep-coupling). We present a comprehensive study of oscillations in the Welander model with a smooth, non-instantaneous switching function, where we distinguish four types in terms of their phases of diffusive vs convective mixing and show where they exist in parameter space. The characteristic Welander deep-(de)coupling oscillations still exist, but they require switching that is considerably faster than needed for sustaining oscillatory behavior. We also demonstrate how a gradual freshwater influx can lead to transitions between different types of oscillations.