Activity for Harry Dankowicz
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Harry Dankowicz modified ticket #51
Frequency Response Plots using COLL/PO toolbox
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Updated po utilities and demos to handle tangent vectors correctly. Fixed bug in mvdP files.
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Updated coll restart constructors, utilities, and demos to handle tangent vectors correctly.
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Updated ep_construct_tst with event handler to store SN eigenvector also when not detected as fold point: Modified ep_read_solution accordingly. Updated constructors to properly handle tangent vector stored with solution file. Tentatively updated restart constructors to allow for a -no-pars option in the event that one wishes to reconstruct without adding continuation parameters. This is necessary if constructing multiple instances from stored solutions.
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Hi Jonathan, The tbp array contains the time stamps for the basepoint mesh. Assuming that you use adaptation (i.e., with NAdapt greater than 0), this is not uniformly distributed, as the algorithms redistribute the mesh to capture areas of greater variability in the solution. To obtain a uniformly distributed time array, use interpolation. I recommend interpolation using the continuous piecewise-smooth polynomial approximant that COCO assumes when discretizing the trajectory segment. Such interpolation...
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Hello Jonathan, There are two ways to have user-defined data appear in the bd array: either by appending such data using a slot function that responds to the bddat signal or by adding continuation parameters. For example, the po toolbox defines several slot functions for the bddat signal that add the Floquet multipliers and norms of the solution to the bd array. The function po_add_bddat (as explained in the GettingStartedWithCOCO tutorial) can be used to create such a slot function. The function...
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Hi Marc, By design, the function defined in the fourth argument of ep_add_bddattakes three input arguments, the last two of which correspond to the state vector and vector of problem parameters for an equilibrium point. The first argument is not used in the example that you are referring to (but is required syntactically). We could have written prob = ep_add_bddat(prob, '', 'svds', @(~,x,p) min(svds(feval(F('x'),x,p)))); to make that clear. By analogy, the function defined in the fourth argument...
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coll and po: corrected use of VAR for restarting from branch of Hopf bifurcations.
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CurveSegment: support for branch switching
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covering/toolbox/@atlas_1d: introduced support for branch switching
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Hi Bo, Continuation proceeds as long as solutions remain inside the computational domain AND as long as the maximum number of points (PtMX) has not been reached. The default value for PtMX is 100 in both directions along a one-dimensional solution manifold. You can change this to other values, including different values in each of the two directions, by using a command like prob = coco_set(prob, 'cont', 'PtMX', [200 50]); As to how much of the solution manifold that is covered by a particular setting...
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Hi Bo, As I read your questions, I think it is important to separate out different components of the code. Let's start with coco_plot_bd('freq_resp', 'T', @(T) 2*pi./T, '||po.orb.x||_{L_2[0,T]}') This command is executed once a run of the coco entry-point function is complete. It assumes that the run was named 'freq_resp' and that there was a continuation parameter named 'T' (or that a user-defined function responding to the bddat signal created a column with this header). It also assumes that the...
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Dear Bo, Thank you for your question. The MX label indicates a failure of the algorithm to converge to a solution within the desired tolerance. This is a warning to you, not necessarily that there is anything wrong with your continuation settings, but perhaps that there may be something amiss or at least needing attention with your problem. For example, I note that the initial value for the first problem parameter is (line 4 of demo_po.m) 2*pi/7. From line 5 of gen_sym_po.m, I conclude that p1 is...
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Hi Marc, I am not sure what you mean by "one of the period-doubled branches." If I understand the situation correctly, there is only one branch that connects the two bifurcation points. It may show up as having two parts because of sampling at the original period, but it's the same orbit sampled twice (as when we plot intersections with a Poincare section). If you would like to sample twice per period, you would need to create an appropriate monitor function. For example, such a function can use...
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coco_plot_bd many points
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Indeed, Marc, there are surely (as you point out) other ways to address the problem of too many data files. I am encouraging of users developing alternatives to, or modifications of, the routines that ship with the release. I do want to stress, however, that the situation you describe is quite unusual, and very possibly a symptom of something amiss. I am assuming that you are tracking this down in your code. Best, Harry
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Hi Marc, If you find yourself adjusting the minimum stepsize to 10^-16 in order to achieve convergence, then there is likely something amiss in your implementation (or the problem is highly unusual). Does convergence appear to be quadratic? Generally speaking, I would recommend against adjusting continuation or nonlinear solver settings until you are sure that the need is motivated by unusual aspects of your problem. I am inclined to suspect an error in your Jacobians. Harry
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Hi Marc, Let me begin by asking for the reason for such a large number of points (I assume this is from continuation along a one-dimensional manifold). Is this necessary or a result of setting the step size maximum to a small value? Best, Harry
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Hello NK, Sorry for the delayed response. I notice a couple of things when running your code: The periodic orbit found using ode15s includes a couple of variables that are of order 10^-7 and 10^-5 while all the other variables are of order 1. I still think that could cause numerical difficulties as the numerical routines use relative tolerances. The third component of your vector field equals 0. That seems to suggest that the periodic orbit that you compute is not isolated (since any value of y3...
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Hello NK. My recommendation is that you try to rebalance your variables so that they are of similar magnitude and vary on comparable scales. When I graph the solutions that COCO finds, the variables are of very different orders of magnitude and vary over very different orders of magnitude. This is not good practice when doing numerics. Since COCO uses various tolerances for identifying convergence and estimating errors, significant imbalance between variables renders the results insensitive to some...
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Hi Jonathan, The phase lives on S^1, but since Matlab works in R^1, I map S^1 onto [-pi,pi] and identify the end points. The jump from pi to -pi just brings the phase back to the beginning of the interval. There is no relationship to directions. The phase always increases at a constant rate. I hope that helps. Harry
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Thank you for your question. The po toolbox constructors build problems associated with periodic trajectory segments, i.e., those for which the two end points coincide. The hspo toolbox constructors can handle other types of relationships between the end points of successive segments, including jumps. In the non-autonomous encoding of the linear oscillator, the two state variables are identical at the two end points, and so po works fine. But if you augment the state to include a phase variable,...
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Hi Jonathan. The phase reset event triggers when x(3)-pi=0 and at that point resets x(3) to x(3)-2*pi, in other words so that while the phase is pi at the terminal point of the incoming segment, it is -pi at the initial point of the outgoing segment. Similarly, the impact event triggers when x(1)-p(5)=0 and at that point resets x(2) to -p(6)*x(2), in other words so that while the displacement is p(5) at the terminal point of the incoming segment, the outgoing velocity is -p(6)*x(2). Here, p(6) is...
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Hi Jonathan. The phrase reset event triggers when x(3)-pi=0 and at that point resets x(3) to x(3)-2*pi, in other words so that while the phase is pi at the terminal point of the incoming segment, it is -pi at the initial point of the outgoing segment. Similarly, the impact event triggers when x(1)-p(5)=0 and at that point resets x(2) to -p(6)*x(2), in other words so that while the displacement is p(5) at the terminal point of the incoming segments, the outgoing velocity is -p(6)*x(2). Here, p(6)...
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Oh, and since the hspo constructors assume an autonomous vector field, you can either replace your harmonic forcing with a scaled version of one of the components of the Hopf normal form dynamics or introduce a phase variable that gets reset at a phase reset event (like in the impact demo.
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Hi Jonathan, The hspo constructors were designed to handle problems with piecewise-defined vector fields, so that discontinuities would not affect the accuracy of the results. There are two issues here: the lack of smoothness (or even continuity) of the vector field and the lack of smoothness of the solution. The former means that your problem Jacobian will change discontinuously as mesh points move across the discontinuity. It is still possible (perhaps even likely, but surely problem-dependent)...
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Error when following TR
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fixes
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Thank you, Marc. I very much appreciate you bringing this to my attention. You have attempted to do something very logical, namely restart continuation of torus bifurcations from a point on a previously computed branch of torus bifurcations. As you note, the same functionality appears to work for period-doubling bifurcations, and it seems very reasonable that po constructors should support this. Indeed, I see no reason why they shouldn't. Obviously, I did not test this functionality or else I would...
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Hi Marc, The tor demo in the po toolbox folder shows the continuation of families of saddle-node, period-doubling, and torus bifurcations of periodic orbits. The script syntax you have shown above seems correct, so I am inclined to assume a coding error. Feel free to share your full set of files for debugging, or consider comparing your code against the demo (which I hope runs without difficulty for you). Best, Harry
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Hi Jonathan, Your question is both general to numerical methods and specific to COCO. For the general answer, I refer you to a multitude of other sources that explore the computational cost and benefits of explicit derivatives. For COCO, and given the use of derivatives in detecting special points (e.g., bifurcations), my general practice is to encode derivatives of vector fields, at least to first order and in many cases to higher order, including directional derivatives, as appropriate. The symcoco...
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Hello, Jonathan. This is difficult to debug in the absence of your actual code. My gut guess is that you have made errors in the encoding of derivatives. That's where I usually start, rather than to assume that the error is due to something inherently remarkable about the problem at hand. In case you are able to make use of the symcoco routines for automatic generation of vector fields and their derivatives (see symcoco-doc.pdf), consider comparing the results. Best, Harry
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Hello, Jonathan. This is difficult to debug in the absence of your actual code. My gut guess is that you have made errors in the encoding of derivatives. That's where I usually start, rather than to assume that the error is due to something inherently remarkable about the problem at hand. In case you are able to make use of the symcoco routines for automatic generation of vector fields and their derivatives (see symcoco-doc.pdf, consider comparing the results. Best, Harry
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Dear Joseph, I see that your problem involves use of the ode_isol2hspo constructor. To construct the corresponding adjoint contributions, use the adjt_isol2hspo constructor, as described in Section 12.3 of the PO_Tutorial.pdf file. For the monitor function, I note that it uses the trapz function to perform numerical integration using the trapezoid rule. Since optimization will require derivatives of your monitor function, you will need to provide these explicitly, as first-order numerical differentiation...
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Hi Joseph, This kind of functionality is made available in COCO using the staged construction of adjoint conditions. You can find several demos in the release that rely on this functionality., e.g., duffing_optim. The methodology is described in several publications, e.g., "Staged Construction of Adjoints for Constrained Optimization of Integro-Differential Boundary-Value Problems" and "Methods of continuation and their implementation in the COCO software platform with application to delay differential...
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When lambda equals 0, the two equilibrium branches are x=0 and mu=x^2, which intersect at x=mu=0 at a branch point (not an SN). This is a singular point of the continuation problem. When you include explicit derivatives, points at some distance away from the singular point can be located using Newton iterations within the allowed number of iterates and the continuation algorithm typically jumps across the singular point along either of the two branches. When you omit explicit derivatives, convergence...
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Issues continuing simple two-parameter ODE
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Thank you, David, for reaching out. First of all, I recommend that you have a look at the EP_Tutorial.pdf documentation, which includes a demo that concerns precisely the dynamical system you are investigating. Secondly, it appears to me that there may be an error in the encoding of the Jacobian with respect to the problem parameters. Finally, I recommend against using coco_set to change default options, especially of the corrector, as more than likely unexpected behavior is not due to their settings...
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If you are looking to perform bifurcation analysis of periodic orbits in piecewise-smooth or hybrid dynamical systems, please see the tutorial documentation for the 'po' toolbox. If you are looking to perform bifurcation analysis of equilibria in piecewise-smooth systems, with emphasis on border collision bifurcations, you will need to extend the 'ep' toolbox with your preferred monitor functions and events, per the methodological guidance in Recipes for Continuation. I hope that helps.
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Please see the responses in ticket #51. Kind regards, Harry
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The function I had in mind you using was po_add_func and I suspect that's what you, in fact, did. This function is a wrapper to coco_add_func that also includes a preceding call to coco_get_func_data. Together these commands extract required information from the associated 'coll' instance and then impose a further constraint on the corresponding continuation variables. The reason for introducing po_add_func as part of the GettingStartedwithCOCO tutorial was to avoid the need to require i) explicit...
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Dear Ali, Thank you for reaching out and for contributing to this thread. I am wondering if the information in the Getting StartedwithCOCO tutorial document and demos (new with the August release of COCO) may be helpful to you. For example, Section 7.3 shows how you can monitor properties of periodic orbits during continuation and Section 7.4 shows how to track and constraint orbit maxima. There are other ways to do this as well, but perhaps this material is a good start. Best, Harry
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As described in Section 2 of ATLAS-Tutorial.pdf, you can see the settings associated with the 'atlas_1d' atlas algorithm by typing atlas_1d_settings on the command line. When you write 'cont' in the call to coco_set, you are specifying that the following name-value pair defines a setting associated with the atlas algorithm. In your example, you are setting the 'NAdapt' setting to 1, implying that adaptive updates will be made to the zero problem before each new continuation step. Individual toolboxes...
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As described in Section 2 of ATLAS-Tutorial.pdf, you can see the settings associated with the 'atlas_1d' atlas algorithm by typing atlas_1d_settings on the command line. When you write 'cont' in the call to coco_set, you are specifying that the following name-value pair defines a setting associated with the atlas algorithm. In your example, you are setting the 'Nadapt' setting to 1, implying that adaptive updates will be made to the zero problem before each new continuation step. Individual toolboxes...
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Hello, Marc. Continuation may terminate because of a lack of convergence of the nonlinear solver or because of the detection of a user- or toolbox-defined terminal event. In the former case, the label is defined by the atlas algorithm. In the latter case, the label used to describe the event is also user- or toolbox-defined. For example, the MXCL label describes a terminal event that is triggered when the 'coll' toolbox estimates a truncation error of the discretization of a trajectory segment that...
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required function not found' with '-var' option in ode_isol2po
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Thank you, Jacob, also for the greetings! Although option #2 provides a fix, the behavior you have identified should count as a bug, since a construction error is thrown for a syntactically correct construction. I will aim to encode a transparent and well-documented solution to this in a future COCO update. Kind regards, Harry
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Thank you, Jacob, for this question. I am very happy to see that you are exploring COCO and attempting to reproduce existing results using independently developed code. There are several parts to this that I will address below. 1. Even if the error you encountered hadn't been thrown, your code would not have executed, since the introduction of the variational problem added 9 to the dimensional deficit. To prevent this from happening, you would need to also introduce 9 constraints, e.g., using the...
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Here's a longer answer: For an arbitrary COCO-compatible continuation problem, columns may be added to the output cell array by defining slot functions that respond to the pre-defined 'bddat' signal and perform calculations using abstract and numerical data associated with each solution. In the October 26, 2023, release of COCO, one such slot function is defined in the ep_add constructor in the 'ep' toolbox and is shown below. function [data, res] = bddat(prob, data, command, varargin) %BDDAT Append...
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Hello Marc, Thank you for exploring ways to extend COCO for your own needs. That is exactly what the platform was built to support. I am glad that you are feeling inspired to push the envelope. The short answer to your question is as follows. When constructing a continuation problem for bifurcation analysis of equilibria using the 'ep' toolbox, the constructor ep_add_bddat can be invoked to append a column to the output cell array consisting of scalars computed from the values of the state variable...
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Hopf bifurcation continuation with 1 parameter only
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Hi Jonathan, This is Matlab syntax for an anonymous function of three arguments (p, d, and u) that returns two arguments, one of which evaluates to the input argument d and one of which evaluates to the expression u(1)-2*pi/u(2). Look at the Matlab manual for more information about anonymous functions. The reason to use this particular syntax here is the requirement that zero and monitor functions added using coco_add_func must be of this form. An alternative is to provide a function handle to a...
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coco_close_efunc: Added text to warning message
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One further observation: however you choose to define your vector field, you can always perform the conversion between excitation frequency and excitation period in your post-continuation analysis, e.g., when graphing the frequency-response diagram using coco_plot_bd or your own plotting routines.
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A longer answer is as follows: Since COCO is an object-oriented development platform, the software is agnostic to the way you choose to construct your zero functions and monitor functions. The COCO toolboxes package a limited set of problem constructors for problems that are common in basic bifurcation analysis. These are each wrapped around one or several calls to the COCO core constructor coco_add_func. As you work with particular classes of problems, you may realize that you repeat the same actions...
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Frequency Response Plots using COLL/PO toolbox
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Hello, Jonathan. Thank you for your interest in using COCO. The short answer to your question is as follows: Preferred toolbox For the continuation of periodic orbits in a smooth dynamical system, the 'po' toolbox is preferred to the 'coll' toolbox, since the former makes use of the latter and then appends additional constraints imposing periodicity and, as appropriate, non-degeneracy using a phase condition. The demo in po/examples/bistable shows how to use the 'po' toolbox for continuation of periodic...
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symcoco: syntax error in Matlab 2023b
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Thank you Alois for flagging this and Jan for fixing it! An updated release (coco_2023October26) has been posted with this update.
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fixes
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Updated sco_sym2funcs to ensure proper function endings for both Unix and Windows-based Matlab systems.
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Dear Marc, Thank you for your query. I'm happy to hear that you are giving COCO a try. Learning by modifying demo code is a great way to explore a package like COCO. The short answer to your question (but see below) is as follows: In the demo, I first detect a Hopf bifurcation under variations in one parameter ('B') and then continue along a curve of such bifurcations under simultaneous variations in two parameters ('A' and 'B'). If you don't allow 'A' to also vary, then you will at best detect a...
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fixes
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The proposed change in the nullspace function is included as commented code as of today. Speed test suggest minimal difference for the po/examples/tor demo.
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This took many years, but a recent change to atlas_1d scales the step size by norm(x,inf) which appears to produce consistent behavior.
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Potential bug in coco_remesh
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Add function for adding integral constraints to COLL.
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See po/examples/int_optim
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Enable adaptation by default.
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Due to backward compatibility, mesh adaptation is disabled by default and must be enabled by the use of the coco_set command. Error estimation for trajectory segments and termination when the error estimate exceeds a threshold is enabled by default. The latter may be disabled by the use of the coco_set command. As a consequence, continuation of solutions to boundary-value problems, e.g., periodic orbits from a Hopf bifurcation, may result in premature termination at an MXCL point, corresponding to...
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Classification of HB points
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The 'ep' tutorial documentation and demos includes an example (chemosc) that accomplishes this for a quadratic vector field. The GettingStartedwithCOCO tutorial documentation and demos shows a complete encoding for an arbitrary vector field.
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Check if PD->PO restarter adapts mesh.
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More time steps or lower time step size for time domain
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lsol_splu: Reorganized conditional statements to check for NV field.
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November 17, 2017 release of COCO
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March 22, 2020 release of COCO
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August 14, 2023, release of COCO
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August 14, 2023, release of COCO
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August 14, 2023, release of COCO
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fixes
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fixes
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Bug in subtract_half_space: when vertices lie on the half space. This happens exceedingly rarely, but it occurred in an example where the solution manifold is flat. Also update plot_atlas_kd to plot this case, and corrected minor issue in demo_atlas2d
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lsol_splu: disuse of coco_func_data for slot function. This was done previously to allow other functions (set_mode) to update content accessible to lsol_splu. instead, introduce get_slot_data function, so that this change to data is accessible without artificially encapsulating in pointer.
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New functionality to support overconstrained problems. Addition of regularizing vectors, when possible and appropriate counting of variables.
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COCO for delay-coupled multi-segment boundary-value problems
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August 14, 2023, release of COCO
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August 14, 2023, release of COCO
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August 14, 2023, release of COCO
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COCO for delay-coupled multi-segment boundary-value problems
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August 14, 2023, release of COCO
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March 22, 2020 release of COCO
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fixes
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fixes
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fixes
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