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Bernd Krauskopf

Publications and source records attributed to Bernd Krauskopf.

At least 19 recordsLinked to original sources
arXiv2026

A distributed-delay model for the El Niño Southern Oscillation with a minimum delay: a case study of the shifted linear chain trick

When modelling a system of interest with a delay differential equation, a distributed delay may be the appropriate modelling choice when the delayed response occurs over a significant range of times rather than with a single constant delay. In systems with a minimum physical transit or processing time, however, the delay kernel should respect a fixed and positive minimum delay. A shifted Erlang kernel is a convenient choice, because the shifted linear chain trick allows one to replace the convolution associated with this type of kernel by a finite auxiliary-variable representation with one constant delay. As we demonstrate with a case study of the Ghil-Zaliapin-Thompson (GZT) model of the El Niño Southern Oscillation with seasonal forcing and distributed delayed oceanic feedback, this reduction makes it possible to perform a bifurcation analysis with numerical continuation techniques for any width of the Erlang distribution. Specifically, we present the bifurcation and resonance structure of the distributed-delay GZT model in the plane of forcing strength and delay for different widths of the distribution. To ensure a like-for-like comparison and determine the influence of delay distribution, we present these results in terms of the effective delay and, moreover, rescale the feedback strength to account for its attenuation with increasing width of the distribution.

math.DS↗
arXiv2026

Resonance structure of a periodically forced delay differential equation model for the El Niño--Southern Oscillation

We study resonance phenomena in the periodically forced Suarez--Schopf delay differential equation, which is a conceptual climate model for the El Niño--Southern Oscillation (ENSO). The system serves as a prototypical forced delayed-action oscillator whose self-sustained oscillations, when subjected to periodic forcing, give rise to attracting invariant tori. We provide a comprehensive bifurcation analysis of both the unforced and the forced model; for the latter, we propose a method to compute the rotation number of normally hyperbolic attracting invariant tori. With it we show that resonance tongues in parameter space are organized by critical points of the graph of the rotation number, both along torus bifurcation curves and within the region of invariant tori. We also show that the resonance structure repeats for large delays, which constitutes a reappearance mechanism not previously reported in the literature. Furthermore, depending on the feedback strength, we find bistability between period-one orbits and invariant tori. This regime involves non-classical bifurcation sequences, including `saddle-node' and `gluing' bifurcations of tori.

math.DS↗
arXiv2026

Devil's terraces: determining the organization of resonance tongues in a periodically forced dynamical system

In periodically forced dynamical systems, resonance tongues are open regions of a parameter plane in which the dynamics on an invariant torus locks to a stable periodic orbit. While individual resonance tongues are well understood, the principles governing their global arrangement remain largely unexplored. We develop a topological framework, grounded in applied topology and Morse theory, whose central object is the two-dimensional resonance surface, defined as the graph of the rotation number $ρ$ over a parameter plane. Within this framework, resonance tongues appear as terraces of the resonance surface at rational values of $ρ$, and their global arrangement is determined by the singularities of this surface. Resolving the resonance surface requires the accurate computation of $ρ$, and we present an algorithm that does so efficiently and at high resolution. As a specific example, we examine a periodically forced model of vertical mixing in the North Atlantic, a process relevant to the Atlantic Meridional Overturning Circulation, and study how its resonance surface changes under variation of a third parameter. We identify six distinct resonance-tongue arrangements and show that the resonance transitions between them are due to changes in the number and type of singularities on the boundary of the resonance surface.

math.DS↗
arXiv2026

Regular and chaotic Welander oscillations in a four-dimensional conceptual model for the Atlantic Meridional Overturning Circulation

The Atlantic Meridional Overturning Circulation (AMOC) is a key component of the Earth's climate. Evidence indicates a twentieth-century weakening, and enhanced freshwater input to the subpolar North Atlantic may further reduce overturning strength. We present and study a conceptual four-dimensional, single-hemisphere box model with three compartments: a tropical surface box, a subpolar surface box, and a large deep-water box. Advective exchange couples the surface boxes and vertical exchange with the deep ocean is represented by a smooth convective-adjustment scheme. A comprehensive bifurcation analysis reveals an equilibrium structure with up to four coexisting overturning states, together with regimes of bistability and tristability. We identify families of periodic solutions and chaotic attractors with a clear timescale separation: a millennial oscillation is modulated by faster decadal-to-centennial variability arising from episodic shutdowns of subpolar convection. As prescribed freshwater fluxes increase, shutdown events become more frequent and the background overturning weakens. Additionally, for certain values of freshwater influx, the dynamics become chaotic, producing an irregular on-off switching of convection.

math.DS↗
arXiv2025

Strongly resonant polarization dynamics of pulse trains in a spin-flip model for an excitable microlaser with delayed self-feedback

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.

physics.optics↗
arXiv2025

Nonlinear Dynamics of a Dicke Model for V-Type Atoms

We study the nonlinear, semiclassical dynamics of an open spin-1 (three-level) variant of the traditional Dicke model. In particular, we focus on V-type energy-level configurations with varying degrees of energy-level asymmetry. We also allow for unbalanced coupling -- where co-rotating and counter-rotating Hamiltonian terms are independently tunable. We characterise the system with a dynamical systems approach, where different behaviors map to definite dynamical objects, and phase transitions to bifurcations. We find the emergence of both periodic and two-frequency oscillations, as well as multistability and chaotic dynamics.

quant-ph↗
arXiv2024

Periodic solutions for a pair of delay-coupled excitable theta neurons

We consider a pair of identical theta neurons in the excitable regime, each coupled to the other via a delayed Dirac delta function with the same delay. This simple network can support different periodic solutions, and we concentrate on two important types: those for which the neurons are perfectly synchronous, and those where the neurons are exactly half a period out of phase and fire alternatingly. Owing to the specific type of pulsatile feedback, we are able to determine these solutions and their stability analytically. More specifically, (infinitely many) branches of periodic solutions of either type are created at saddle-node bifurcations, and they gain stability at symmetry-breaking bifurcations when their period as a function of delay is at its minimum. We also determine the respective branches of symmetry-broken periodic solutions and show that they are all unstable. We demonstrate by considering smoothed pulse-like coupling that the special case of the Dirac delta function can be seen as a sort of normal form: the basic structure of the different periodic solutions of the two theta neurons is preserved, but there may be additional changes of stability along the different branches.

nlin.PS↗
arXiv2024

A kneading map of chaotic switching oscillations in a Kerr cavity with two interacting light fields

Optical systems that combine nonlinearity with coupling between various subsystems offer a flexible platform for observing a diverse range of nonlinear dynamics. Furthermore, engineering tolerances are such that the subsystems can be identical to within a fraction of the wavelength of light; hence, such coupled systems inherently have a natural symmetry that can lead to either delocalization or symmetry breaking. We consider here an optical Kerr cavity that supports two interacting electric fields, generated by two symmetric input beams. Mathematically, this system is modeled by a four-dimensional $\mathbb{Z}_2$-equivariant vector field with the strength and detuning of the input light as control parameters. Previous research has shown that complex switching dynamics are observed both experimentally and numerically across a wide range of parameter values. Here, we show that particular switching patterns are created at specific global bifurcations through either delocalization or symmetry breaking of a chaotic attractor. We find that the system exhibits infinitely many of these global bifurcations, which are organized by $\mathbb{Z}_2$-equivariant codimension-two Belyakov transitions. We investigate these switching dynamics by means of the continuation of global bifurcations in combination with the computation of kneading invariants and Lyapunov exponents. In this way, we provide a comprehensive picture of the interplay between different switching patterns of periodic orbits and chaotic attractors.

physics.optics↗
arXiv2024

Characterising blenders via covering relations and cone conditions

We present a characterisation of a blender based on the topological alignment of certain sets in phase space in combination with cone conditions. Importantly, the required conditions can be verified by checking properties of a single iterate of the diffeomorphism, which is achieved by finding finite series of sets that form suitable sequences of alignments. This characterisation is applicable in arbitrary dimension. Moreover, the approach naturally extends to establishing C1-persistent heterodimensional cycles. Our setup is flexible and allows for a rigorous, computer-assisted validation based on interval arithmetic.

math.DS↗
arXiv2024

Regenerative vectorial breathers in a delay-coupled neuromorphic microlaser with integrated saturable absorber

We report on the polarization dynamics of regenerative light pulses in a micropillar laser with integrated saturable absorber coupled to an external feedback mirror. The delayed self-coupled microlaser is operated in the excitable regime, where it regenerates incident pulses with a supra-threshold intensity -- resulting in a pulse train with inter-pulse period approximately given by the feedback delay time, in analogy with a self-coupled biological neuron. We report the experimental observation of vectorial breathers in polarization angle, manifesting themselves as a modulation of the linear polarized intensity components without significant modulation of the total intensity. Numerical analysis of a suitable model reveals that the observed polarization mode competition is a consequence of symmetry-breaking bifurcations induced by polarization anisotropy. Our model reproduces well the observed experimental results and predicts different regimes as a function of the polarization anisotropy parameters and the pump parameter. We believe that these findings are relevant for the fabrication of flexible sources of polarized pulses with controlled properties, as well as for neuroinspired on-chip computing applications, where the polarization may be used to encode or process information in novel ways.

physics.optics↗
arXiv2024

A detailed analysis of the origin of deep-decoupling oscillations

The variability of the strength of the Atlantic Meridional Overturning Circulation is influenced substantially by the formation of deep water in the North Atlantic. In many ocean models, so-called deep-decoupling oscillations have been found, whose timescale depends on the characteristics of convective vertical mixing processes. Their precise origin and sensitivity to the representation of mixing have remained unclear so far. To study this problem, we revisit a 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. The model is known to exhibit several types of deep-decoupling oscillations, with phases of weak diffusive mixing interspersed with strong convective mixing, when the switching between them is assumed to be instantaneous. We present a comprehensive study of oscillations in Welander's model with non-instantaneous switching between mixing phases, as described by a smooth switching function. A dynamical systems approach allows us to distinguish four types of oscillations, in terms of their phases of diffusive versus convective mixing, and to identify the regions in the relevant parameter plane where they exist. The characteristic deep-decoupling oscillations still exist for non-instantaneous switching, but they require switching that is considerably faster than needed for sustaining oscillatory behaviour. Furthermore, we demonstrate how a gradual freshwater influx can lead to transitions between different vertical mixing oscillations. Notably, the convective mixing phase becomes shorter and even disappears, resulting in long periods of much reduced deep water formation. The results are relevant for the interpretation of ocean-climate variability in models and (proxy) observations.

math.DS↗
arXiv2023

Bifurcations of Periodic Orbits in the Generalised Nonlinear Schrödinger Equation

We focus on the existence and persistence of families of saddle periodic orbits in a four-dimensional Hamiltonian reversible ordinary differential equation derived using a travelling wave ansatz from a generalised nonlinear Schr{ö}dinger equation (GNLSE) with quartic dispersion. In this way, we are able to characterise different saddle periodic orbits with different signatures that serve as organising centres of homoclinic orbits in the ODE and solitons in the GNLSE. To achieve our objectives, we employ numerical continuation techniques to compute these saddle periodic orbits, and study how they organise themselves as surfaces in phase space that undergo changes as a single parameter is varied. Notably, different surfaces of saddle periodic orbits can interact with each other through bifurcations that can drastically change their overall geometry or even create new surfaces of periodic orbits. Particularly we identify three different bifurcations: symmetry-breaking, period-$k$ multiplying, and saddle-node bifurcations. Each bifurcation exhibits a degenerate case, which subsequently gives rise to two bifurcations of the same type that occurs at particular energy levels that vary as a parameter is gradually increased. Additionally, we demonstrate how these degenerate bifurcations induce structural changes in the periodic orbits that can support homoclinic orbits by computing sequences of period-$k$ multiplying bifurcations.

math.DS↗
arXiv2023

Bifurcation analysis of a conceptual model for the Atlantic Meridional Overturning Circulation

The Atlantic Meridional Overturning Circulation (AMOC) distributes heat and salt into the Northern Hemisphere via a warm surface current toward the subpolar North Atlantic, where water sinks and returns southwards as a deep cold current. There is substantial evidence that the AMOC has slowed down over the last century. We introduce a conceptual box model for the evolution of salinity and temperature on the surface of the North Atlantic Ocean, subject to the influx of meltwater from the Greenland ice sheets. Our model, which extends a model due to Welander, describes the interaction between a surface box and a deep-water box of constant temperature and salinity, which may be convective or non-convective, depending on the density difference. Its two main parameters $μ$ and $η$ describe the influx of freshwater and the threshold density between the two boxes, respectively. We use bifurcation theory to analyse two cases of the model: instantaneous switching between convective or non-convective interaction, where the system is piecewise-smooth (PWS), and the full smooth model with more gradual switching. For the PWS model we derive analytical expressions for all bifurcations. The resulting bifurcation diagram in the $(μ,η)$-plane identifies all regions of possible dynamics, which we show as phase portraits - both at typical parameter points, as well as at the different transitions between them. We also present the bifurcation diagram for the case of smooth switching and show how it arises from that of the PWS case. In this way, we determine exactly where one finds bistability and self-sustained oscillations of the AMOC in both versions of the model. In particular, our results show that oscillations between temperature and salinity on the surface North Atlantic Ocean disappear completely when the transition between the convective and non-convective regimes is too slow.

math.DS↗
arXiv2023

Complex switching dynamics of interacting light in a ring resonator

Microresonators are micron-scale optical systems that confine light using total internal reflection. These optical systems have gained interest in the last two decades due to their compact sizes, unprecedented measurement capabilities, and widespread applications. The increasingly high finesse (or $Q$ factor) of such resonators means that nonlinear effects are unavoidable even for low power, making them attractive for nonlinear applications, including optical comb generation and second harmonic generation. In addition, light in these nonlinear resonators may exhibit chaotic behavior across wide parameter regions. Hence, it is necessary to understand how, where, and what types of such chaotic dynamics occur before they can be used in practical devices. We consider a pair of coupled differential equations that describes the interactions of two optical beams in a single-mode resonator with symmetric pumping. Recently, it was shown that this system exhibits a wide range of fascinating behaviors, including bistability, symmetry breaking, chaos, and self-switching oscillations. We employ here a dynamical system approach to identify, delimit, and explain the regions where such different behaviors can be observed. Specifically, we find that different kinds of self-switching oscillations are created via the collision of a pair of asymmetric periodic orbits or chaotic attractors at Shilnikov homoclinic bifurcations, which acts as a gluing bifurcation. We present a bifurcation diagram that shows how these global bifurcations are organized by a Belyakov transition point (where the stability of the homoclinic orbit changes). In this way, we map out distinct transitions to different chaotic switching behavior that should be expected from this optical device.

physics.optics↗
arXiv2023

Bifurcation analysis of a North Atlantic Ocean box model with two deep-water formation sites

The tipping of the Atlantic Meridional Overturning Circulation (AMOC) to a 'shutdown' state due to changes in the freshwater forcing of the ocean is of particular interest and concern due to its widespread ramifications, including a dramatic climatic shift for much of Europe. A clear understanding of how such a shutdown would unfold requires analyses of models from across the complexity spectrum. For example, detailed simulations of sophisticated Earth System Models have identified scenarios in which deep-water formation first ceases in the Labrador Sea before ceasing in the Nordic Seas, en route to a complete circulation shutdown. Here, we study a simple ocean box model with two polar boxes designed to represent deep-water formation at these two distinct sites. A bifurcation analysis reveals how, depending on the differences of freshwater and thermal forcing between the two polar boxes, transitions to 'partial shutdown' states are possible. Our results shed light on the nature of the tipping of AMOC and clarify dynamical features observed in more sophisticated models.

physics.ao-ph↗
arXiv2023

Generalized and multi-oscillation solitons in the Nonlinear Schrödinger Equation with quartic dispersion

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows us to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this way, we establish the existence of connections from one periodic orbit to another, called PtoP connections. They give rise to families of homoclinic orbits to either of the two periodic orbits; in the GNLSE they correspond to generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Moreover, PtoP connections can be found in the energy level of the origin, where connections between this equilibrium and a given periodic orbit, called EtoP connections, are known to organize families of solitons. As we show here, EtoP and PtoP cycles can be assembled into different types of heteroclinic cycles that give rise to additional families of homoclinic orbits to the origin. In the GNLSE, these correspond to multi-oscillation solitons that feature several episodes of different oscillations in between their decaying tails. As for solitons organized by EtoP connections only, multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking.

nlin.PS↗
arXiv2023

Quantum Fluctuation Dynamics of Dispersive Superradiant Pulses in a Hybrid Light-Matter System

We consider theoretically a driven-dissipative quantum many-body system consisting of an atomic ensemble in a single-mode optical cavity as described by the open Tavis-Cummings model. In this hybrid light-matter system the interplay between coherent and dissipative processes leads to superradiant pulses with a build-up of strong correlations, even for systems comprising hundreds to thousands of particles. A central feature of the mean-field dynamics is a self-reversal of two spin degrees of freedom due to an underlying time-reversal symmetry, which is broken by quantum fluctuations. We demonstrate a quench protocol that can maintain highly non-Gaussian states over long time scales. This general mechanism offers interesting possibilities for the generation and control of complex fluctuation patterns, as suggested for the improvement of quantum sensing protocols for dissipative spin-amplification.

quant-ph↗
arXiv2022

Merging and disconnecting resonance tongues in a pulsing excitable microlaser with delayed optical feedback

Excitability, encountered in numerous fields from biology to neurosciences and optics, is a general phenomenon characterized by an all-or-none response of a system to an external perturbation. When subject to delayed feedback, excitable systems can sustain multistable pulsing regimes, which are either regular or irregular time sequences of pulses reappearing every delay time. Here, we investigate an excitable microlaser subject to delayed optical feedback and study the emergence of complex pulsing dynamics, including periodic, quasiperiodic and irregular pulsing regimes. This work is motivated by experimental observations showing these different types of pulsing dynamics. A suitable mathematical model, written as a system of delay differential equations, is investigated through an in-depth bifurcation analysis. We demonstrate that resonance tongues play a key role in the emergence of complex dynamics, including non-equidistant periodic pulsing solutions and chaotic pulsing. The structure of resonance tongues is shown to depend very sensitively on the pump parameter. Successive saddle transitions of bounding saddle-node bifurcations constitute a merging process that results in unexpectedly large locking regions, which subsequently disconnect from the relevant torus bifurcation curve; the existence of such unconnected regions of periodic pulsing is in excellent agreement with experimental observations. As we show, the transition to unconnected resonance regions is due to a general mechanism: the interaction of resonance tongues locally at an extremum of the rotation number on a torus bifurcation curve. We present and illustrate the two generic cases of disconnecting and of disappearing resonance tongues. Moreover, we show how a maximum and a minimum of the rotation number appears naturally when two torus bifurcation curves undergo a saddle transition (where they connect differently).

math.DS↗