Yu.A. Kuznetsov (Utrecht University, The Netherlands)

Joint work with V. De Witte and W. Govaerts (Gent, Belgium), and M. Friedman (Huntsville, Alabama)

Abstract:

New functionalities of MatCont (a MATLAB continuation package for the interactive numerical study of parameterized systems of smooth ODEs) related to the codimension-one orbits homoclinic to hyperbolic and non-hyperbolic equilibria will be presented. The continuation of such orbits is implemented using the CIS-algorithm, with the Ricatti equation included in the defining system. It is possible to initiate the homoclinic orbits of both types interactively, within a GUI, starting from an equilibrium point and using a homotopy method.

All known codimension-two homoclinic bifurcations can be detected and located during the continuation. Test functions for the inclination-flip bifurcations are implemented in a new and more efficient way.

Examples illustrating the homotopy method in the cases of 1D- and 2D-unstable invariant manifolds will be presented.

Jorge Galán

(Joint work with F. J Muñoz Almaraz, E. Freire, A. Vanderbauwhede, and E. J. Doedel)

Abstract:

The usual procedure to study the dynamical behaviour of systems with conserved quantities and/or symmetries is to reduce the dimension of the problem and include the integrals of motion as explicit parameters. Some years ago, the authors proposed an alternative method that instead adds an additional "unfolding" term to the equations and treats the conserved quantities as internal parameters, i.e. not present in the equations. In this talk we review some of the applications of the method paying special attention to
the numerical implementation for the continuation of periodic orbits.

Hinke Osinga

Abstract:

The Lorenz system still fascinates many people because of the simplicity of the equations that generate such complicated dynamics on the famous butterfly attractor. This talk addresses the role of the stable and unstable manifolds in organising the dynamics more globally. A main object of interest is the stable manifold of the origin of the Lorenz system, also known as the Lorenz manifold. This two-dimensional manifold and associated manifolds of saddle periodic orbits can be computed accurately with numerical methods based on the continuation of orbit segments, defined as solutions of suitable boundary value problems. This allows us to study bifurcations of global manifolds as the Rayleigh parameter is changed. We show how the entire phase space of the Lorenz system is organised and how the manifolds change dramatically during the transition to chaotic dynamics.

Bill Kalies

Abstract:

A generally applicable method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by fast combinatorial algorithms and is classified via Morse decompositions. These Morse decompositions are compared at adjacent parameter sets via continuation to detect possible changes in the dynamics. The Conley index is used to study the structure of isolated invariant sets associated with the computed Morse decompositions and to detect the existence of certain types of dynamics. The method is illustrated with an application to the two-dimensional, density-dependent, Leslie population model.

Renato Calleja, McGill University

Abstract:

We formulate and justify rigorously a numerically efficient criterion for the computation of the analyticity breakdown of quasi-periodic solutions in Symplectic maps and 1-D Statistical Mechanics models. Depending on the physical interpretation of the model, the analyticity breakdown may correspond to the onset of mobility of dislocations, or of spin waves (in the 1-D models) and to the onset of global transport in symplectic twist maps. The criterion we propose here is based on the blow-up of Sobolev norms of the hull functions. The justification of the criterion suggests fast numerical algorithms that we have implemented in several examples.

Roberto Barrio, Departamento de Matemática Aplicada, Universidad de Zaragoza, E-50009 Zaragoza, Spain

(Joint work with Fernando Blesa, Marcos Rodríguez, and Sergio Serrano)

Abstract:

In the numerical study of Dynamical Systems we have to deal with Differential Equations, and we have to solve them with different goals. Sometimes we have to integrate very long time and we need speed, or we need to study the evolution of partial derivatives along the solution or we deal with chaotic problems and we need high precision to follow the orbit, 100 digits, 1000 digits, or more. In all these cases we can use the new integrator tool TIDES [1,3], acronym of Taylor Series Integrator of Differential EquationS. The Taylor series method applied to the integration of Ordinary Differential Equations can be adapted easily to these special demands which hardly can be obtained using other standard methods such as Runge-Kutta.

In this new software we have implemented all these features and we have created the program providing easy-to-use interfaces. A simple line in Mathematica generates automatically the necessary files in C or FORTRAN to solve the problem. Besides, changing a simple option and linking with the library -lTIDES also provided in this software we can integrate without difficulties partial derivatives and/or arbitrary high precision via the multiple precision library MPFR [6].

George Haller

Abstract:

We review the fundamental physical motivation behind the definition of Lagrangian Coherent Structures (LCS) and show how it leads to the concept of finite-time hyperbolicity in non-autonomous dynamical systems. Using this concept of hyperbolicity, we obtain a self-consistent criterion for the existence of attracting and repelling material surfaces in unsteady fluid flows and in other non-autonomous dynamical systems. The existence of LCS is often postulated in terms of features of the Finite-Time Lyapunov Exponent (FTLE) field associated with the system. As simple examples show, however, the FTLE field does not necessarily highlight LCS, or may highlight structures that are not LCS. Under appropriate nondegeneracy conditions, we show that ridges of the FTLE field indeed coincide with LCS in volume-preserving flows. For general flows, we obtain a more general scalar field whose ridges correspond to LCS. We illustrate the results on applications to atmospheric and oceanic flow data.

Wolf-Jürgen Beyn

Abstract:

We consider semilinear parabolic systems in one and two space dimensions which exhibit localized moving patterns, such as traveling, rotating and spiral waves. These patterns may be viewed as relative equilibria of an equivariant time-dependent PDE. We discuss nonlinear stability and instability of such relative equilibria in terms of (generally continuous) spectra of linearizations.

The results are closely related to a numerical approach, called the freezing method or the method of slices. It uses equivariance of the PDE and transforms the given system into a partial differential algebraic equation (PDAE) which is subsequently discretized by truncation to a bounded domain with finite boundary conditions. The approach allows to freeze the target pattern in a comoving frame while simultaneously providing information on the motion of the group variables. For several examples we show how Lyapunov stability of the resulting PDAE and its numerical analog relate to stability with asymptotic phase for the original pattern.

Greg Lewis

(Joint work with Matt Hennessy)

Abstract:

A preliminary investigation of the secondary fluid flow transitions that occur in an O(2)-symmetric differentially heated rotating periodic channel is presented. In the channel, the primary transition occurs when a time-independent flow that is uniform along the channel bifurcates to a stationary wave flow. Using numerical continuation, we search for the parameter values at which transitions occur from this wave solution.

We discuss a strategy for computing steady solutions of the three dimensional Navier-Stokes equations in the Boussinesq approximation and for determining flow transitions. In particular, the Navier-Stokes equations are solved using a velocity-vorticity method in combination with a second-order centered finite-difference scheme on an unstaggered and nonuniform grid. The resulting nonlinear system of algebraic equations is solved using Newton's method. GMRES with block Jacobi preconditioning is used to iteratively solve the large sparse linear systems that result from the discretization.

Tobias Schneider

(Joint work with John F. Gibson and John Burke)

Abstract:

Inducing turbulence in linearly stable shear flows such as pipe and plane Couette flow requires perturbations of finite amplitude. The shape and size of critical perturbations can be determined using the edge tracking algorithm (PRL 96, 174101 (2006); PRL 99, 034502 (2007)). For small, periodically continued domains both critical perturbations and the turbulent dynamics itself have recently been linked to exact equilibrium and traveling wave solutions of the Navier-Stokes equations. In large domains, where localized perturbations are observed to induce spatially localized patches of turbulence which slowly invade the surrounding laminar flow, these periodic solutions can however not capture the full spatio-temporal dynamics but suggest the existence of their localized counterparts. We here examine a new class of localized solutions to plane Couette flow that under continuation in Reynolds number exhibit a sequence of saddle-node bifurcations strikingly similar to the "homoclinic snaking" phenomenon observed the Swift-Hohenberg equation. These localized solutions originate from bifurcations off the spatially periodic equilibria discovered by Nagata and others and retain their physical structure, demonstrating the relevance of exact solutions to turbulent flows in spatially extended domains.

Rouslan Krechetnikov

Abstract:

In this work, motivated by the problem of model-based predictive control of separated flows, we identify the key variables and the requirements on a model-based observer and construct a prototype low-dimensional model to be embedded in control applications.

Namely, using a phenomenological physics-based approach and dynamical systems and singularity theories, we uncover the low-dimensional nature of the complex dynamics of actuated separated flows and capture the crucial bifurcation and hysteresis inherent in separation phenomena. This new look at the problem naturally leads to several important implications, such as, firstly, uncovering the physical mechanisms for hysteresis, secondly, predicting a finite amplitude instability of the bubble, and, thirdly, to new issues to be studied theoretically and tested experimentally.

Juan Sánchez (Joint work with M. Net and C. Simó)

Abstract:

Invariant tori arise in general continuous or discrete dynamical systems, usually, when a branch of periodic orbits loses stability at a Neimark-Sacker bifurcation, although in Hamiltonian systems they appear typically around a (totally or partially) elliptic fixed point, periodic orbit or lower dimensional torus.

A method to compute invariant tori in high-dimensional systems, obtained as discretizations of PDEs, by continuation and Newton-Krylov methods will be described. Invariant tori are found as fixed points of a generalized Poincare map so that the dimension of the system of equations to be solved is that of the original system of ODEs. Therefore there is no prohibitive increase in the size of the problem. Due to the dissipative nature of the systems studied, the convergence of the linear solvers is extremely fast. The computation of periodic orbits inside the Arnold's tongues is also considered.

The thermal convection of a binary mixture of fluids, in a rectangular cavity, has been used to test the method. The branch of tori obtained, starts at a Neimark-Sacker bifurcation, and ends at a pitchfork bifurcation. Later on, the stable branches of invariant tori undergo a cascade of period-doubling bifurcations leading to chaos. In this case, as typically happens, only a finite number of period doublings is found before the chaotic range is reached. Although the method is suitable for stable tori, we have been able to compute an small branch of unstable invariant tori.

Robert Hölzel

Abstract:

The nonlocal complex Ginzburg-Landau equation (NCGLE), an extension of the complex Ginzburg-Landau equation (CGLE), is a normal-form approach modelling the nonlocal migration coupling that arises in electrochemical systems [V. Garcia-Morales and K. Krischer, Phys. Rev. Lett. 100, 054101 (2008)]. A peculiarity of migration coupling is that the range of the coupling can be tuned experimentally. The limit of negligible coupling range corresponds to diffusion coupling, and in this limit, the NCGLE becomes the CGLE.

We show with numerical simulations of the NCGLE that with increasing range of the nonlocal coupling coherent patterns develop in the turbulent parameter region of the CGLE, and WE present a bifurcation analysis of the NCGLE with the coupling range as main bifurcation parameter. We discuss in detail a series of bifurcations involving heteroclinic connections of stationary states and of limit cycles.

Andrew Hazel

Abstract:

I shall demonstrate the use of spatial adaptivity, continuation and bifurcation detection within OOMPH-LIB, an open-source, C++ software library (developed jointly with Matthias Heil) designed for the robust solution of multi-physics problems, see http://www.oomph-lib.org. Examples will be presented from an ongoing project (joint with Rich Hewitt and Phil Haines) to investigate the relationship between the nonlinear behaviour of similarity solutions to the Navier--Stokes equations and the corresponding full solutions in finite domains.

Fluid flows often exhibit features on many different length-scales. A uniform numerical discretisation based on the smallest length-scale in the problem can be extremely inefficient if the small-scale features are localised in space. One solution, known as spatial adaptivity, is to solve the problem with a relatively coarse discretisation and then to use an error estimator to select the regions that contain fine-scale features. The discretisation in the selected regions is refined and the problem is solved again. The process continues iteratively until the error estimate is below a prescribed tolerance, or the computer runs out of memory! If the error estimator is "good", the final discretisation will be "adapted" to the particular problem and usually contains far fewer degrees of freedom than the uniform discretisation required to achieve equivalent accuracy. When assessing the stability of a fluid flow and/or seeking bifurcations the associated eigenfunctions of the linearised system must also be accurately resolved. In many cases, particularly near Hopf bifurcations of reasonably large frequency, the flow and the eigenfunction can have quite different spatial structures and using a discretisation based on the flow alone can lead to large errors in the location of the bifurcation.

Steven Baer

Abstract:

When a parameter slowly ramps through a Hopf bifurcation, stability loss is delayed considerably when compared to classical static theory. Inherent to biological, chemical, and physical systems, but often overlooked or misunderstood in the literature are nonlinear ramp problems where a parameter slowly accelerates or de-accelerates through the bifurcation point. In this talk I will briefly review, from a neuroscience perspective, the importance of the dynamic bifurcation problem. I will then present recent results that show, numerically and analytically, how slow nonlinear ramps can significantly increase or decrease the onset threshold, changing profoundly our understanding of stability loss delay in dynamic bifurcation problems. I will apply the results to membrane accommodation in nerves, the formation of pacemakers in the Belousov-Zhabotinsky reaction, and neuronal elliptic bursting. At the end to the talk I will discuss ongoing research and several open problem areas.

Federico Frascoli

(Joint work with L. van Veen, D. T. J. Liley, and I. Bojak)

Abstract:

Mean field models (MFMs) of cortical tissues focus on salient, average features of populations of neurons, with the scope of understanding the foundations of electrical brain activity. One of the common aspects of MFMs is the presence of a high dimensional parameter space, which tries to capture the essential neurobiological attributes of the cortex.

The parameter space of a MFMs of brain electrocortical activity is here investigated, in relation to the effects in brain dynamics induced by general anesthetics (GAs). It is possible to discover meaningful correlations through the study of bifurcations of the anesthetized cortex. The generality of our approach is also well suited to probe the parameter spaces of other existing MFMs, in search of correlations between physiological attributes and dynamical responses.

In particular, we show the presence of correlates among measurable attributes of the brain, EEG spectral powers and dynamical patterns in the model. These links are not accessible by standard linear or nonlinear parameter analyses, but emerge when the parameter space is partitioned according to the dynamical responses to GAs. These responses belong to two archetypal categories or "families", which are investigated and characterized in depth. It also emerges that families can be influenced dramatically by the actions of exhogeneous stimuli, like those driven by the thalamic input.

The complexity of oscillatory activity within each family is not equivalent, with diverse repertoires of sustained orbits. Distributions of some cortical attributes are also antithetic, with some parameters exerting a strong control on the dynamics of the reaction to GAs. Finally, the role of inhibition in affecting the nature of the overall cortical activity appears to be very strong.

Andrey Shilnikov

Abstract:

A Central Pattern Generator is a neural network controlling various vital repetitive locomotive functions including respiration and walking of humans, swimming and crawling of leeches etc. This talk is focused on modeling polyrhythmic dynamics in a multi-functional CPG, that is able to generate multiple rhythms associated with distinct types of locomotive activity. We describe synergetic mechanisms of the emergence of several synchronous behaviors in in mutually inhibitory motifs being a network's building block.

Leon Glass

Joint work with Michael Guevara, Alvin Shrier, Art Winfree, Thomas Gedeon, Mark Josephson, Trine Krogh-Madsen, Eusebius Doedel, Bart Borek and Bart Oldeman

Abstract:

One of the classic ways to characterize biological or other oscillators is to measure the resetting of the oscillation as a function of the phase of the stimulus. To do this, we define the phase transition curve which gives the new phase of the oscillation as a function of the phase of the stimulus.  Provided the stimulus leaves the oscillator in the same basin of attraction for all stimuli phases, using topological arguments, the phase transition curve should be a continuous map of the unit circle onto itself. Careful experimental tests of the predictions are rare. However, in the 1980s, Michael Guevara did an experiment that appeared to contradict the topological result. I will consider mathematical and experimental issues involved with the resetting of oscillations associated with limit cycles in ordinary differential equations representing biological oscillations, partial differential equations representing the propagation of an excitation wave in a one dimensional ring of excitable medium, and partial differential equations representing a one dimension excitable medium in which a pacemaker is embedded. These problems are relevant to problems associated with the initiation and termination of serious cardiac arrhythmias.

Willy Govaerts

(Joint work with Charlotte Sonck)

Abstract:

Basic knowledge of the cell cycle of various types of cells consists of the substances involved in the process (cyclins, APC's etcetera) and the regulatory network.

Dynamical systems models based on this regulatory network are fairly recent, and they are not the only possible approach. Competing models use logical dynamic modeling, a simpler approach that in some cases is quite successful by showing that most trajectories funnel into a path which steps through the cell cycle in a robust way.

On the other hand, models based on differential equations allow to use mathematical tools such as bifurcation theory and numerical continuation under parameter variation. In this way, multistability, periodic behaviour etc. can be derived and studied in a fairly standard way.

A survey of the already impressive literature was recently published by A. Csikasz-Nagy ("Computational systems biology of the cell cycle", Briefings in Bioinformatics, Vol. 10 no 4, 424-434). The leading group appears to be that of John. J. Tyson and Bela Novak. Among other things, they built the most detailed model so far of cell-cycle regulation by describing the control network of budding yeast /Saccharomyces cerevisiae./

In most cases, these models exhibit an even richer bifurcation structure than is usually taken into account, e.g. Takens-Bogdanov points, periodic oscillations internally in the G1 state of the system etc. It is not always obvious how much of this is relevant for the applications.

We discuss a computational and bifurcation study of the budding yeast model that includes a few phenomena that were so far not discussed in this context. This includes the existence of different limit cycles, born at different Hopf points that (approximately) merge in a single bigger limit cycle and a rather unexpected relation between the growth rate of the cell and the mass increase after DNA-replication.

We further discuss the implications of the funneling effect for the cell cycle as a boundary value problem, and the computation of this cycle as the fixed point of a map.

Markus Kirkilionis

Abstract:

Mass-action kinetics is a powerful tool to describe events created by collision of molecules or individuals in a well-mixed environment giving them locally the same probability to meet each other. Moreover this probability is only dependent on the concentration of the mutual partners. Mass action systems can be found in chemistry, cell biology, but also game theory and economics. Mathematically this gives rise to dynamical systems of a special type, more specific of polynomial type. I will give an overview how this property can be used to determine different types of bifurcations, for example the occurrence of bistability, or oscillations via a Hopf bifurcation. All tools will be borrowing methods from algebraic geometry. Finally I will give an outlook what usually goes wrong in the modelling part while using mass-action kinetics if biochemical or cellular molecular events are considered.

Mark Kramer

Abstract:

Neurons can exhibit a variety of dynamical states, including rapid spiking and bursting.  We describe these two states --- and their transitions --- in a reduced mathematical model of a cerebellar Purkinje cell.  We find at the transition a canard phenomena that follows temporarily a branch of repelling limit cycles.  To explore these dynamics, we have recently developed a simpler mathematical model of the canard phenomena, which we will also discuss.

Bart Oldeman

Abstract:

The method of successive continuation of orbit segments (also known as the homotopy method) is a very powerful method for determining how a trajectory varies as initial conditions change. Instead of shooting from a set of initial conditions we pose a boundary value problem (BVP). First we compute one orbit segment that satisfies one initial condition, which can actually be done by continuation in its 'period' T, from 0 until the orbit satisfies a certain end condition, for instance, for its length, time, or its intersection with a plane. Next the initial condition of the orbit can be varied, keeping the end condition from the first step.

Keeping the initial and end conditions well-posed the orbit is then continued as a whole, which means that all points on the orbit are taken into account, and not just the initial conditions. For example, sometimes the initial conditions might not vary numerically but the segment changes a lot at its end or elsewhere, and this method has no problems with that.

This method can be implemented in standard continuation software such as AUTO, and has been used to compute, for example, the Lorenz manifold. I show here its application to compute a simple unstable manifold and a phase resetting curve for a Van der Pol-style system.

Harry Dankowicz

Abstract:

A central theme in the recent development of the continuation package COCO by Frank Schilder and the speaker is the construction of extended continuation problems that afford run-time flexibility in deploying a particular covering algorithm and selectively constraining the continuation. The corresponding mathematical formulation naturally supports the idea of task embedding, a shared responsibility for constructing the extended continuation problem across several toolboxes. According to this notion, functionality afforded by distinct toolboxes should be formulated in such a way that it can be combined to solve composite continuation problems without code modification or reimplementation. As argued in this talk, such an extended formulation enables innovative computations that are not supported in a similar way as 'built-in' functionality by any existing core implementations.

Specifically, this talk illustrates the notion of task embedding through the implementation of a COCO-compatible collocation toolbox for multi-point, boundary-value problems, which supports the general-purpose parameter continuation of sets of constrained orbits segments, such as i) segmented trajectories in hybrid dynamical systems, for example, mechanical systems with impacts, friction, and switching control, ii) homoclinic orbits represented by an equilibrium point and a finite-time trajectory that starts and ends near this equilibrium point, and iii) collections of trajectories that represent quasi-periodic invariant tori. The collocation algorithm allows for segment-dependent meshing and non-trivial boundary conditions involving internal mesh points and includes a full discretization of the corresponding variational equations, making it straightforward to track solution branches associated with critical combinations of eigenvalues.

Alan Champneys

(Joint work with Arne Nordmark and Harry Dankowicz)

Abstract:

Newtonian impact and coulomb friction, when treated seperately are known to lead to consistent formulations in terms of piecewise-smooth dynamical systems. Nevertheless, so-called discontinuity-induced bifurcations such as grazing and sliding bifurcations can occur. This talk considers what happens when friction is taken into account during an impact event. The analysis is restricted to 2D, where the canonical example is that of a slender body allowed to contact a rigid frictional surface (a piece of chalk on a blackboard). First it is shown how to derive consistent impact laws that generalise the coefficient of resitution law. These lead to the possibilities of discontinuity-induced bifurcations as the sequences of sticking or slipping changes during an impact event. Next it is shown that for contacting motion the so-called Painleve paradox of non-uniqueness can be resolved by smoothing and passing to the limit. However, there remains the possibility of reverse chatter, where infinite numbers of impacts accumulate in reverse time, that cannot be ruled out at transition points between stick and slip. The existence of such motion shows the fundamental non-uniqueness in forward time simulations of rigid formulations of impact with Coulomb friction.

Bernd Krauskopf, University of Bristol

(Joint work with James Rankin, Mark Lowenberg, and Mathieu Desroches)

Abstract:

Aircraft are meant to fly, but they also need to operate efficiently and safely on the ground. As ground vehicles, commercial aircraft are quite special; in particular, they feature a tricycle configuration and very strong deformations of the tyres.

The talk will demonstrate how the standard ground manoeuvre of aircraft turning can be studied by means of a bifurcation analysis of a nonlinear aircraft model. In particular, it is shown that the phenomenon of a sudden loss of tyre holding force when entering a spin is mathematically a canard explosion.

Gabor Domokos

(Joint work with A. A. Sipos, Gy Szabo, and P. Varkonyi)

Abstract:

While the number of asteroids with known shapes has drastically increased over the past few years, little is known on the the time-evolution of shapes and the underlying physical processes. One apparent common feature of these shapes is the existence of large, relatively flat areas separated by edges. Here we propose a simple abrasion model based on multiple collisions with small objects, which accounts for the emergence of such polyhedral shapes. We show that the model is realistic, since the bulk of the collisions falls into this category. We point out other effects, not captured by our model as well as an analogy to pebble shapes.

Pablo Aguirre

(Joint work with Bernd Krauskopf and Hinke Osinga)

Abstract:

Invariant manifolds of saddle-type equilibria and periodic orbits are a key ingredient to understand the global dynamics in many applied vector fields. They act as boundaries that organize the overall dynamics in phase space [1]. While it is fairly easy to study the related one-dimensional objects, the same is not true for (un)stable manifolds of higher dimension. These manifolds can now be computed with high accuracy with numerical methods based on continuation of orbit segments, defined as solutions of suitable boundary value problems [1, 4]. As an example we consider a model for a laser with optical injection [6]. Under certain conditions, the laser dynamics presents excitable behavior just before the creation of a homoclinic orbit to a saddle-focus p (also known as Shilnikov homoclinic [3, 5]). The two-dimensional stable manifold Ws(p) of this equilibrium becomes an excitability threshold for the laser. A small perturbation above W^s(p) makes an excursion following the unstable manifold W^u(p) before converging to an attractor q, originating a pulse-like response in the state variables. Multipulse behavior is also possible, meaning several pulses or responses are generated from a single perturbation above this boundary. Implementing our calculations with AUTO [2] we show the topological and geometrical changes the two-dimensional stable manifold W^s(p) undergoes through a Shilnikov homoclinic bifurcation. In this way, we are able to understand how the phase space from the laser dynamics is organized.

Harry Dankowicz

Abstract:

This demonstration illustrates the fundamental elements of the constructor formalism of the continuation package COCO and its support of task embedding -- the shared responsibility for formulating and processing an extended continuation problem across several toolboxes. Examples demonstrate a wide range of functionality and adaptability, including flexible event handling, continuation of sets of constrained trajectory segments and the associated variational problem, and multi-dimensional manifold covering.