# Jan SieberUniversity of Exeter | UoE · College of Engineering, Mathematics and Physical Sciences

Jan Sieber

PhD Mathematics

## About

116

Publications

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2,304

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Introduction

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December 2016 - March 2017

## Publications

Publications (116)

This paper presents a framework to perform bifurcation analysis in laboratory experiments or simulations. We employ control-based continuation to study the dynamics of a macroscopic variable of a microscopically defined model, exploring the potential viability of the underlying feedback control techniques in an experiment. In contrast to previous e...

This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the coco software package and its explicit support for commun...

Grazing events may create coexisting attractors and cause complex dynamics in piecewise-smooth dynamical systems. This paper studies control of grazing-induced multistability in a soft impacting oscillator by using time-delayed feedback control. The control switches from one of the coexisting attractors to a desired one to suppress complex dynamics...

We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We p...

This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is sh...

This paper studies a control method for switching stable coexisting attractors of a class of non-autonomous dynamical systems. The central idea is to introduce a continuous path for the system's trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information...

This paper studies a control method for switching stable coexisting attractors of a class of non-autonomous dynamical systems. The central idea is to introduce a continuous path for the system's trajectory to transition from its original undesired stable attractor to a desired one by varying one of the system parameters according to the information...

This paper presents a rigorous framework for the continuation of solutions to nonlinear constraints and the simultaneous analysis of the sensitivities of test functions to constraint violations at each solution point using an adjoint-based approach. By the linearity of a problem Lagrangian in the associated Lagrange multipliers, the formalism is sh...

We study a scalar, first-order delay differential equation (DDE) with instantaneous and state-dependent delayed feedback, which itself may be delayed. The state dependence introduces nonlinearity into an otherwise linear system. We investigate the ensuing nonlinear dynamics with the case of instantaneous state dependence as our starting point. We p...

This paper treats comprehensively the construction of problems from nonlinear dynamics and constrained optimization amenable to parameter continuation techniques and with particular emphasis on multi-segment boundary-value problems with delay. The discussion is grounded in the context of the COCO software package and its explicit support for commun...

A new technique to derive delay models from systems of partial differential equations, based on the Mori–Zwanzig (MZ) formalism, is used to derive a delay-difference equation model for the Atlantic Multidecadal Oscillation (AMO). The MZ formalism gives a rewriting of the original system of equations, which contains a memory term. This memory term c...

This chapter presents a dynamical systems point of view of the study of systems with delays. The focus is on how advanced tools from bifurcation theory, as implemented for example in the package DDE-BIFTOOL, can be applied to the study of delay differential equations (DDEs) arising in applications, including those that feature state-dependent delay...

A new technique to derive delay models from systems of partial differential equations, based on the Mori-Zwanzig formalism, is used to derive a delay difference equation model for the Atlantic Multidecadal Oscillation. The Mori-Zwanzig formalism gives a rewriting of the original system of equations which contains a memory term. This memory term can...

Lyapunov exponents are a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise-smooth systems with time-delayed arguments one faces a lack of continuity in the variational problem. This paper studies how to build a variational equation for the efficient construction of Jacobians along trajectories of the...

The licence type in the original article was incorrect and should be CC BY and not CC BY NC.

Lyapunov exponent is a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise smooth systems with time delayed arguments one faces two difficulties: a high dimension of the discretized state space and a lack of continuity of the variational problem. This paper shows how to build a variational equation for...

The article Optimization along families of periodic and quasiperiodic orbits in dynamical systems with delay, written by Zaid Ahsan, Harry Dankowicz, and Jan Sieber, was originally published electronically on 5 November 2019 without open access.

This paper generalizes a previously conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipli...

This paper investigates the robustness against localized impacts of elastic spherical shells pre-loaded under uniform external pressure. We subjected a pre-loaded spherical shell that is clamped at its equator to axisymmetric blast-like impacts applied to its polar region. The resulting axisymmetric dynamic response is computed for increasing ampli...

Control-based continuation (CBC) is a general and systematic method to probe the dynamics of nonlinear experiments. In this paper, CBC is combined with a novel continuation algorithm that is robust to experimental noise and enables the tracking of geometric features of the response surface such as folds. The method uses Gaussian process regression...

Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, autosolitons, and spot or pulse solutions, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly stu...

Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism is introduced as a way to systematically derive delay models from systems of partial differential e...

Dynamic buckling is addressed for complete elastic spherical shells subject to a rapidly applied step in external pressure. Insights from the perspective of nonlinear dynamics reveal essential mathematical features of the buckling phenomena. To capture the strong buckling imperfection-sensitivity, initial geometric imperfections in the form of an a...

This work studies grazing-induced multistability in an impacting system and proposes to use the delayed feedback control [1] to drive the system to a desired periodic motion. The second part of this work will focus on studying the delayed feedback control via path-following methods. As there is no continuation tool which has been developed to study...

Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad-hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism is introduced as a way to systematically derive delay models from systems of partial differential e...

This paper generalizes a previously-conceived, continuation-based optimization technique for scalar objective functions on constraint manifolds to cases of periodic and quasiperiodic solutions of delay-differential equations. A Lagrange formalism is used to construct adjoint conditions that are linear and homogenous in the unknown Lagrange multipli...

Control-based continuation (CBC) is a general and systematic method to probe the dynamics of nonlinear experiments. In this paper, CBC is combined with a novel continuation algorithm that is robust to experimental noise and enables the tracking of geometric features of the response surface such as folds. The method uses Gaussian process regression...

Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solutions, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studie...

Dynamic buckling is addressed for complete elastic spherical shells subject to a rapidly applied step in external pressure. Insights from the perspective of nonlinear dynamics reveal essential mathematical features of the buckling phenomena. To capture the strong buckling imperfection-sensitivity, initial geometric imperfections in the form of an a...

We present a study of a delay differential equation (DDE) model for the Mid-Pleistocene Transition. We investigate the behavior of the model when subjected to periodic forcing. The unforced model has a bistable region consisting of a stable equilibrium along with a large amplitude stable periodic orbit. We are interested in how forcing affects solu...

The Mid-Pleistocene Transition, the shift from 41 kyr to 100 kyr glacial-interglacial cycles that occurred roughly 1 Myr ago, is often considered as a change in internal climate dynamics. Here we revisit the model of Quaternary climate dynamics that was proposed by Saltzman and Maasch (1988) (from this point referred to as SM88). We show that it is...

This paper addresses testing of compressed structures, such as shells, that exhibit catastrophic buckling and notorious imperfection sensitivity. The central concept is the probing of a loaded structural specimen by a controlled lateral displacement to gain quantitative insight into its buckling behavior and to measure the energy barrier against bu...

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold (in a run-away positive feedback loop). We study how rapidly one needs to turn around once one has crossed the threshold. We derive a simple criterion that relates the peak and curvature of the parameter path in an inverse-square law to easi...

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algo...

A common approach to studying high-dimensional systems with emergent low-dimensional behavior is based on lift-evolve-restrict maps (called equation-free methods): first, a user-defined lifting operator maps a set of low-dimensional coordinates into the high-dimensional phase space, then the high-dimensional (microscopic) evolution is applied for s...

Early-warning indicators (increase of autocorrelation and variance) are commonly applied to time series data to try and detect tipping points of real-world systems. The theory behind these indicators originates from approximating the fluctuations around an equilibrium observed in time series data by a linear stationary (Ornstein-Uhlenbeck) process....

A dynamical system is said to undergo rate-induced tipping when it fails to track its quasi-equilibrium state due to an above-critical-rate change of system parameters. We study a prototypical model for rate-induced tipping, the saddle-node normal form subject to time-varying equilibrium drift and noise. We find that both most commonly used early-w...

We propose a formula to approximate the probability of rate-induced tipping with additive white noise occurring for small to moderate equilibrium drift speeds. Early-warning indicators have generally been used on historical tipping events as a form of verification. This can lead to false-positives (false alarms) and false-negatives (missed alarms)...

Under increasing compression, an unbuckled shell is in a metastable state which becomes increasingly precarious as the buckling load is approached. So to induce premature buckling a lateral disturbance will have to overcome a decreasing energy barrier which reaches zero at buckling. Two archetypal problems that exhibit a severe form of this behavio...

Under increasing compression, an unbuckled shell is in a metastable state which becomes increasingly precarious as the buckling load is approached. So to induce premature buckling a lateral disturbance will have to overcome a decreasing energy barrier which reaches zero at buckling. Two archetypal problems that exhibit a severe form of this behavio...

A non-autonomous system is defined to pass a tipping point when gradual
changes in input levels cause the output to change suddenly. We study a
prototypical model for rate-induced tipping, the saddle-node normal form
subject to parameter drift and noise. We determine the most likely time of
escape by finding the optimal path of escape. This is a va...

Time delayed feedback control is one of the most successful methods to
discover dynamically unstable features of a dynamical system in an experiment.
This approach feeds back only terms that depend on the difference between the
current output and the output from a fixed time T ago. Thus, any periodic orbit
of period T in the feedback controlled sys...

We study a system of phase oscillators with non-local coupling in a ring that
supports self-organized patterns of coherence and incoherence, called chimera
states. Introducing a global feedback loop, connecting the phase lag to the
order parameter, we can observe chimera states also for systems with a small
number of oscillators. Numerical simulati...

DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation
analysis of systems of delay differential equations with discrete constant and
state-dependent delays. The package supports continuation and stability
analysis of steady state solutions and periodic solutions. Further one can
compute and continue several local and global bifurc...

We present a control scheme that is able to find and stabilize an unstable chaotic regime in a system with a large number of interacting particles. This allows us to track a high dimensional chaotic attractor through a bifurcation where it loses its attractivity. Similar to classical delayed feedback control, the scheme is noninvasive, however only...

We use the numerical continuation package AUTO to investigate families of periodic orbits in the solar sail circular restricted three-body problem. For a sail orientated perpendicular to the Sun-line we find significant differences to the classical case for some families near the Earth, including the L Halo family and retrograde satellite family. S...

Equation-free methods make possible an analysis of the evolution of a few
coarse-grained or macroscopic quantities for a detailed and realistic model
with a large number of fine-grained or microscopic variables, even though no
equations are explicitly given on the macroscopic level. This will facilitate a
study of how the model behavior depends on...

For an elastic system that is non-conservative but autonomous, subjected for
example to time-independent loading by a steadily flowing fluid (air or water),
a dangerous bifurcation, such as a sub-critical bifurcation, or a cyclic fold,
will trigger a dynamic jump to one or more remote stable attractors. When there
is more than one candidate attract...

We present a scheme of analysis for predicting the approach to a fold in the noisy time series of a slowly evolving system. It provides estimates of the evolution rate of the control parameter, the variation of the stability coefficient, the path itself, and the level of noise in the time series. Finally, it gives probability estimates of the futur...

We use the numerical continuation package AUTO (Doedel et al. 1991) to
investigate the L1 Halo family, and associated branching families, for a solar
sail orientated perpendicular to the Sun-line in the Earth-Sun
photo-gravitational circular restricted three-body problem (CRTBP). This
problem is parameterized by the sail lightness number \beta, the...

Coupled wave equations are a popular tool for investigating longitudinal dynamical effects in semiconductor lasers, for example, sensitivity to delayed optical feedback. We study a model that consists of a hyperbolic linear system of partial differential equations with one spatial dimension, which is nonlinearly coupled with a slow subsystem of ord...

We study properties of basic solutions in systems with dime delays and
$S^1$-symmetry. Such basic solutions are relative equilibria (CW solutions) and
relative periodic solutions (MW solutions). It follows from the previous theory
that the number of CW solutions grows generically linearly with time delay
$\tau$. Here we show, in particular, that th...

We present a general method for systematically investigating the dynamics and bifurcations of a physical nonlinear experiment. In particular, we show how the odd-number limitation inherent in popular noninvasive control schemes, such as (Pyragas) time-delayed or washout-filtered feedback control, can be overcome for tracking equilibria or forced pe...

We consider the chemostat model with the substrate concentration as the
single measurement. We propose a control strategy that drives the system at a
steady state maximizing the gas production without the knowledge of the
specific growth rate. Our approach separates the extremum seeking problem from
the feedback control problem such that each of th...

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how one can use this control law to trace out (reconstruct) the whole graph of the growth function. The process of tracing out the graph can be...

First, we give a rigorous convergence result for equation-free analysis in
the setting of slow-fast systems using implicit lifting. Second, we apply this
result to study the idealized traffic modeling problem of phantom jams
generated by cars with uniform behavior on a circular road. It is shown, that
the implicitly defined coarse-level time steppe...

We propose an adaptive control law that allows one to identify unstable steady states of the open-loop system in the single-species chemostat model without the knowledge of the growth function. We then show how to use a continuation method to reconstruct the whole graph of the growth function. Two variants, in continuous and discrete time, are pres...

The current threat of global warming and the public demand for confident projections of climate change pose the ultimate challenge to science: predicting the future behaviour of a system of such overwhelming complexity as the Earth's climate. This Theme Issue addresses two practical problems that make even prediction of the statistical properties o...

Approaching a dangerous bifurcation, from which a dynamical system such as the Earth's climate will jump (tip) to a different state, the current stable state lies within a shrinking basin of attraction. Persistence of the state becomes increasingly precarious in the presence of noisy disturbances. We argue that one needs to extract information abou...

J. Sieber, B. Krauskopf, D. Wagg, S. Neild, A. Gonzalez-Buelga, Control-based continuation of unstable periodic orbits.ASME Journal of Computational and Nonlinear Dynamics 6(1) 011005, 2011.

There is currently much interest in examining climatic tipping points, to see if it is feasible to predict them in advance. Using techniques from bifurcation theory, recent work looks for a slowing down of the intrinsic transient responses, which is predicted to occur before an instability is encountered. This is done, for example, by determining t...

This paper presents a method that is able to continue periodic orbits in systems where only output of the evolution over a given time period is available, which is the typical situation in an experiment. The starting point of our paper is an analysis of time-delayed feedback control, a method to stabilize periodic orbits experimentally that is popu...

Models of global climate phenomena of low to intermediate complexity are very
useful for providing an understanding at a conceptual level. An important
aspect of such models is the presence of a number of feedback loops that
feature considerable delay times, usually due to the time it takes to transport
energy (for example, in the form of hot/cold...

We study a chain of $N+1$ phase oscillators with asymmetric but uniform
coupling. This type of chain possesses $2^{N}$ ways to synchronize in so-called
travelling wave states, i.e. states where the phases of the single oscillators
are in relative equilibrium. We show that the number of unstable dimensions of
a travelling wave equals the number of o...

Dynamically stable periodic rotations of a driven pendulum provide a unique mechanism for generating a uniform rotation from bounded excitations. This paper studies the effects of a small ellipticity of the driving, perturbing the classical parametric pendulum. The first finding is that the region in the parameter plane of amplitude and frequency o...

We prove a necessary and sufficient criterion for the exponential stability
of periodic solutions of delay differential equations with large delay. We show
that for sufficiently large delay the Floquet spectrum near criticality is
characterized by a set of curves, which we call asymptotic continuous spectrum,
that is independent on the delay.

We present an experimental procedure to track pe- riodic orbits through a fold (saddle-node) bifurcation, and demonstrate it with a parametrically excited pendulum ex- periment where the control parameter is the amplitude of the excitation. Specifically, we track the initially stable period- one rotation of the pendulum through its fold bifurcation...

J.M.T. Thompson & J. Sieber, Climate tipping predictions: noisy folds and nonlinear softening,
Proc. 7th European Nonlinear Dynamics Conf. (ENOC 2011), 24-29 July 2011, Rome. (Eds: D. Bernardini, G. Rega and F. Romeo) ISBN: 978-88-906234-2-4, DOI: 10.3267/ENOC2011Rome.