BAL: a library for the brute-force analysis of dynamical systems. (English) Zbl 1348.37001

Summary: This paper describes the functionality and usage of bal, a C/C++ library with a Python front-end for the brute-force analysis of continuous-time dynamical systems described by ordinary differential equations (ODEs). bal provides an easy-to-use wrapper for the efficient numerical integration of ODEs and, by detecting intersections of the trajectory with appropriate Poincaré sections, allows to classify the asymptotic trajectory of a dynamical system for bifurcation analysis. Some examples of application are discussed, concerning two-dimensional bifurcation diagrams, Lyapunov exponents and finite-time Lyapunov exponents, basins of attraction, simulation of switching ODE systems, and integration with AUTO, a software package for continuation analysis.


37-04 Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory
37M20 Computational methods for bifurcation problems in dynamical systems
Full Text: DOI


[1] Ermentrout, B., Simulating, analyzing, and animating dynamical systems: A guide to XPPAUT for researchers and students, (2002), Society for Industrial and Applied Mathematics · Zbl 1003.68738
[2] Back, A.; Guckenheimer, J.; Myers, M.; Wicklin, F.; Worfolk, P., Dstool: computer assisted exploration of dynamical systems, Notices Amer. Math. Soc., 39, 4, 303-309, (1992)
[3] Bordeianu, C.; Beşliu, C.; Jipa, A.; Felea, D.; Grossu, I., Scilab software package for the study of dynamical systems, Comput. Phys. Comm., 178, 10, 788-793, (2008) · Zbl 1196.37003
[4] Bordeianu, C.; Felea, D.; Beşliu, C.; Jipa, A.; Grossu, I., A new version of scilab software package for the study of dynamical systems, Computer Physics Communications, 180, 11, 2398-2399, (2009) · Zbl 1197.37002
[5] Abad, A.; Barrio, R.; Blesa, F.; Rodríguez, M., Algorithm 924: TIDES, a Taylor series integrator for differential equations, ACM Trans. Math. Software, 39, 1, 5, (2012) · Zbl 1295.65138
[6] Kuznetsov, Y., Elements of applied bifurcation theory, (2004), Springer-Verlag New York · Zbl 1082.37002
[7] Allgower, E.; Georg, K., Numerical continuation methods: an introduction, (1990), Springer Verlag · Zbl 0717.65030
[8] Doedel, E.; Oldeman, B., AUTO-07P: continuation and bifurcation software for ordinary differential equations, (Oct. 2009), Computer Science Department, Concordia University Montreal, Quebec, Canada
[9] Dhooge, A.; Govaerts, W.; Kuznetsov, Y., MATCONT: A MATLAB package for numerical bifurcation analysis of odes, ACM Trans. Math. Software, 29, 141-164, (2003) · Zbl 1070.65574
[10] Champneys, A.; Kuznetsov, Y.; Sandstede, B., A numerical toolbox for homoclinic bifurcation analysis, Int. J. Bifurcation Chaos, 6, 5, 867-887, (1996) · Zbl 0877.65058
[11] Dercole, F.; Kuznetsov, Y., SLIDECONT: an AUTO 97 driver for bifurcation analysis of Filippov systems, ACM Trans. Math. Software, 31, 95-119, (2005) · Zbl 1073.65070
[12] Thota, P.; Dankowicz, H., TC-HAT (\(\hat{T C}\)): A novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems, SIAM J. Appl. Dyn. Syst., 7, 4, 1283-1322, (2008) · Zbl 1192.34004
[13] Guckenheimer, J., Computer simulation and beyond-for the 21st century, Notices Amer. Math. Soc., 45, 1120-1123, (1998)
[14] González-Miranda, J. M., Complex bifurcation structures in the hindmarsh-rose neuron model, Int. J. Bifurcation Chaos, 17, 9, 3071-3083, (2007) · Zbl 1185.37189
[15] Gallas, J., The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows, Int. J. Bifurcation Chaos, 20, 2, 197-211, (2010) · Zbl 1188.34057
[16] Freire, J. G.; Gallas, J. A.C., Non-shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback, Phys. Rev. E, 82, (2010)
[17] Vitolo, R.; Glendinning, P.; Gallas, J. A.C., Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows, Phys. Rev. E, 84, (2011)
[18] Barrio, R.; Blesa, F.; Serrano, S.; Shilnikov, A., Global organization of spiral structures in biparameter space of dissipative systems with shilnikov saddle-foci, Phys. Rev. E, 84, (2011)
[19] De Feo, O.; Maggio, G.; Kennedy, M., The colpitts oscillator: families of periodic solutions and their bifurcations, Int. J. Bifurcation Chaos, 10, 5, 935-958, (2000) · Zbl 1090.34540
[20] Green, K.; Champneys, A.; Lieven, N., Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors, J. Sound Vib., 291, 861-881, (2006)
[21] Stellardo, D.; Bizzarri, F.; Storace, M.; De Feo, O., On the complexity of periodic and non-periodic behaviors of a hysteresis-based electronic oscillator, Chaos, 17, 4, (2007), 043108(1-13) · Zbl 1163.37370
[22] Storace, M.; Linaro, D.; de Lange, E., The hindmarsh-rose neuron model: bifurcation analysis and piecewise-linear approximations, Chaos, 18, 3, (2008)
[23] Coombes, S.; Laing, C., Delays in activity-based neural networks, Phil. Trans. R. Soc. A, 367, 1117-1129, (2009) · Zbl 1185.92003
[24] Cohen, S. D.; Hindmarsh, A. C., CVODE, a stiff/nonstiff ODE solver in C, Comput. Phys., 10, 2, 138-143, (1996)
[25] Hindmarsh, A. C.; Brown, P. N.; Grant, K. E.; Lee, S. L.; Serban, R.; Shumaker, D. E.; Woodward, C. S., SUNDIALS: suite of nonlinear and differential/algebraic equation solvers, ACM Trans. Math. Software, 31, 3, 363-396, (2005) · Zbl 1136.65329
[26] Karlsson, B., Beyond the C++ standard library: an introduction to boost, (2005), Addison-Wesley
[27] Folk, M.; Cheng, A.; McGrath, R. E., HDF5: A new file format and I/O library for scientific data management, (Mehringer, D. M.; Plante, R. L.; Roberts, D. A., Astronomical Data Analysis software and Systems VIII Proceedings, vol. 172, (1999), Astronomical Society of the Pacific)
[28] D.M. Beazley, SWIG: an easy to use tool for integrating scripting languages with C and C++, in: Proceedings of the 4th conference on USENIX Tcl/Tk Workshop, vol. 4, Monterey, California, 1996.
[29] Abrahams, D.; Grosse-Kunstleve, R., Building hybrid systems with boost python, C/C++ Users J., 21, 29-36, (2003)
[30] Wolf, A.; Swift, J.; Swinney, H.; Vastano, J., Determining Lyapunov exponents from a time series, Physica D, 16, 285-317, (1985) · Zbl 0585.58037
[31] Gamma, E.; Helm, R.; Johnson, R.; Vlissides, J. M., Design patterns: elements of reusable object-oriented software, (1994), Addison-Wesley Professional
[32] Hindmarsh, J.; Rose, R., A model of neuronal bursting using three coupled first order differential equations, Proc. R. Soc. B, 221, 87-102, (1984)
[33] Innocenti, G.; Morelli, A.; Genesio, R.; Torcini, A., Dynamical phases of the hindmarsh-rose neuronal model: studies of the transition from bursting to spiking chaos, Chaos, 17, 4, (2007), 043128(1-11) · Zbl 1163.37336
[34] Shilnikov, A.; Kolomiets, M., Methods of the qualitative theory for the hindmarshrose model: a case study a tutorial, Int. J. Bifurcation Chaos, 18, 2141-2168, (2008) · Zbl 1165.34364
[35] Shadden, S.; Lekien, F.; Marsden, J., Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212, 271-304, (2005) · Zbl 1161.76487
[36] Brambilla, A.; Linaro, D.; Storace, M., Nonlinear behavioural model of charge pump plls, Int. J. Circuit Theory Appl., 41, 10, 1027-1046, (2013)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.