ISSN:
 2158-2491

eISSN:
 2158-2505

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Volume 3, 2016

Volume 2, 2015

Volume 1, 2014

JCD is focused on the intersection of computation with deterministic and stochastic dynamics. The mission of the journal is to publish papers that explore new computational methods for analyzing dynamic problems or use novel dynamical methods to improve computation. The subject matter of JCD includes both fundamental mathematical contributions and applications to problems from science and engineering. A non-exhaustive list of topics includes

  *  Computation of phase-space structures and bifurcations
  *  Multi-time-scale methods
  *  Structure-preserving integration
  *  Nonlinear and stochastic model reduction
  *  Set-valued numerical techniques
  *  Network and distributed dynamics

JCD includes both original research and survey papers that give a detailed and illuminating treatment of an important area of current interest.

The editorial board of JCD consists of world-leading researchers from mathematics, engineering, and science, all of whom are experts in both computational methods and the theory of dynamical systems.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 2 issues a year in June and December.
  • Publishes online only.
  • Indexed in Emerging Sources Citation Index, MathSciNet and Zentralblatt MATH.
  • Archived in Portico and CLOCKSS.
  • JCD is a publication of the American Institute of Mathematical Sciences. All rights reserved.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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Kernel methods for the approximation of some key quantities of nonlinear systems
Jake Bouvrie and Boumediene Hamzi
2017, 4(1&2) : 1-19 doi: 10.3934/jcd.2017001 +[Abstract](450) +[HTML](180) +[PDF](440.88KB)
Abstract:

We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success -once the nonlinear system has been mapped into a high or infinite dimensional feature space. In particular, we embed a nonlinear system in a reproducing kernel Hilbert space where linear theory can be used to develop computable, non-parametric estimators approximating controllability and observability energy functions for nonlinear systems. In all cases the relevant quantities are estimated from simulated or observed data. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system.

Parameterization method for unstable manifolds of delay differential equations
C. M. Groothedde and J. D. Mireles James
2017, 4(1&2) : 21-70 doi: 10.3934/jcd.2017002 +[Abstract](804) +[HTML](204) +[PDF](20182.49KB)
Abstract:

This work is concerned with efficient numerical methods for computing high order Taylor and Fourier-Taylor approximations of unstable manifolds attached to equilibrium and periodic solutions of delay differential equations. In our approach we first reformulate the delay differential equation as an ordinary differential equation on an appropriate Banach space. Then we extend the Parameterization Method for ordinary differential equations so that we can define operator equations whose solutions are charts or covering maps for the desired invariant manifolds of the delay system. Finally we develop formal series solutions of the operator equations. Order-by-order calculations lead to linear recurrence equations for the coefficients of the formal series solutions. These recurrence equations are solved numerically to any desired degree.

The method lends itself to a-posteriori error analysis, and recovers the dynamics on the manifold in addition to the embedding. Moreover, the manifold is not required to be a graph, hence the method is able to follow folds in the embedding. In order to demonstrate the utility of our approach we numerically implement the method for some 1, 2, 3 and 4 dimensional unstable manifolds in problems with constant, and (briefly) state dependent delays.

Rigorous continuation of bifurcation points in the diblock copolymer equation
Jean-Philippe Lessard, Evelyn Sander and Thomas Wanner
2017, 4(1&2) : 71-118 doi: 10.3934/jcd.2017003 +[Abstract](488) +[HTML](176) +[PDF](1800.09KB)
Abstract:

We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through \begin{document}${\mathbb{Z}}_2$\end{document} symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.

Set-oriented numerical computation of rotation sets
Katja Polotzek, Kathrin Padberg-Gehle and Tobias Jäger
2017, 4(1&2) : 119-141 doi: 10.3934/jcd.2017004 +[Abstract](328) +[HTML](199) +[PDF](1845.96KB)
Abstract:

We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of \begin{document}$\varepsilon$\end{document}-rotation sets. These are obtained by replacing orbits with \begin{document}$\varepsilon$\end{document}-pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as \begin{document}$\varepsilon$\end{document} decreases to zero. Based on this result, we prove the convergence of the numerical approximations as precision and iteration time tend to infinity. Further, we provide analytic error estimates for the algorithm under an additional boundedness assumption, which is known to hold in many relevant cases and in particular for non-empty interior rotation sets.

A Lin's method approach for detecting all canard orbits arising from a folded node
José Mujica, Bernd Krauskopf and Hinke M. Osinga
2017, 4(1&2) : 143-165 doi: 10.3934/jcd.2017005 +[Abstract](129) +[HTML](65) +[PDF](1853.05KB)
Abstract:

Canard orbits are relevant objects in slow-fast dynamical systems that organize the spiraling of orbits nearby. In three-dimensional vector fields with two slow and one fast variables, canard orbits arise from the intersection between an attracting and a repelling two-dimensional slow manifold. Special points called folded nodes generate such intersections: in a suitable transverse two-dimensional section Σ, the attracting and repelling slow manifolds are counter-rotating spirals that intersect in a finite number of points. We present an implementation of Lin's method that is able to detect all of these intersection points and, hence, all of the canard orbits arising from a folded node. With a boundary-value-problem setup we compute orbit segments on each slow manifold up to Σ, where we require that the corresponding end points in Σ lie in a one-dimensional subspace known as the Lin space Z. The Lin space Z must be transverse to the slow manifolds and it remains fixed during the detection of canard orbits as zeros of the signed distance along Z. During the computation, a tangency of Z with one of the intersection curves in Σ may arise. To overcome this, we update the Lin space at an intermediate continuation step to detect a double tangency of Z to both curves in Σ, after which the canard detection is able to continue. Our method is demonstrated with the examples of the normal form for a folded node and of the Koper model.

Addendum to "Optimal control of multiscale systems using reduced-order models"
Carsten Hartmann, Juan C. Latorre, Wei Zhang and Grigorios A. Pavliotis
2017, 4(1&2) : 167-167 doi: 10.3934/jcd.2017006 +[Abstract](101) +[HTML](54) +[PDF](140.81KB)
Abstract:
On dynamic mode decomposition: Theory and applications
Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton and J. Nathan Kutz
2014, 1(2) : 391-421 doi: 10.3934/jcd.2014.1.391 +[Abstract](1041) +[PDF](1657.5KB) Cited By(94)
Attraction-based computation of hyperbolic Lagrangian coherent structures
Daniel Karrasch, Mohammad Farazmand and George Haller
2015, 2(1) : 83-93 doi: 10.3934/jcd.2015.2.83 +[Abstract](317) +[PDF](2835.1KB) Cited By(10)
Optimal control of multiscale systems using reduced-order models
Carsten Hartmann, Juan C. Latorre, Wei Zhang and Grigorios A. Pavliotis
2014, 1(2) : 279-306 doi: 10.3934/jcd.2014.1.279 +[Abstract](303) +[PDF](1246.3KB) Cited By(8)
Modularity revisited: A novel dynamics-based concept for decomposing complex networks
Marco Sarich, Natasa Djurdjevac Conrad, Sharon Bruckner, Tim O. F. Conrad and Christof Schütte
2014, 1(1) : 191-212 doi: 10.3934/jcd.2014.1.191 +[Abstract](345) +[PDF](1083.3KB) Cited By(8)
Global invariant manifolds near a Shilnikov homoclinic bifurcation
Pablo Aguirre, Bernd Krauskopf and Hinke M. Osinga
2014, 1(1) : 1-38 doi: 10.3934/jcd.2014.1.1 +[Abstract](385) +[PDF](6559.1KB) Cited By(7)
Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools
Gary Froyland, Cecilia González-Tokman and Anthony Quas
2014, 1(2) : 249-278 doi: 10.3934/jcd.2014.1.249 +[Abstract](335) +[PDF](8600.4KB) Cited By(6)
Modularity of directed networks: Cycle decomposition approach
Nataša Djurdjevac Conrad, Ralf Banisch and Christof Schütte
2015, 2(1) : 1-24 doi: 10.3934/jcd.2015.2.1 +[Abstract](358) +[PDF](1434.0KB) Cited By(4)
An elementary way to rigorously estimate convergence to equilibrium and escape rates
Stefano Galatolo, Isaia Nisoli and Benoît Saussol
2015, 2(1) : 51-64 doi: 10.3934/jcd.2015.2.51 +[Abstract](255) +[PDF](505.8KB) Cited By(4)
Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof
Piotr Zgliczyński
2015, 2(1) : 95-142 doi: 10.3934/jcd.2015.2.95 +[Abstract](328) +[PDF](623.1KB) Cited By(4)
A kernel-based method for data-driven koopman spectral analysis
Matthew O. Williams, Clarence W. Rowley and Ioannis G. Kevrekidis
2015, 2(2) : 247-265 doi: 10.3934/jcd.2015005 +[Abstract](631) +[PDF](1682.6KB) Cited By(4)
Rigorous continuation of bifurcation points in the diblock copolymer equation
Jean-Philippe Lessard, Evelyn Sander and Thomas Wanner
2017, 4(1&2) : 71-118 doi: 10.3934/jcd.2017003 +[Abstract](488) +[HTML](176) +[PDF](1800.09KB) PDF Downloads(85)
Kernel methods for the approximation of some key quantities of nonlinear systems
Jake Bouvrie and Boumediene Hamzi
2017, 4(1&2) : 1-19 doi: 10.3934/jcd.2017001 +[Abstract](450) +[HTML](180) +[PDF](440.88KB) PDF Downloads(45)
Set-oriented numerical computation of rotation sets
Katja Polotzek, Kathrin Padberg-Gehle and Tobias Jäger
2017, 4(1&2) : 119-141 doi: 10.3934/jcd.2017004 +[Abstract](328) +[HTML](199) +[PDF](1845.96KB) PDF Downloads(28)
Compressed sensing and dynamic mode decomposition
Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu and J. Nathan Kutz
2015, 2(2) : 165-191 doi: 10.3934/jcd.2015002 +[Abstract](831) +[PDF](9556.1KB) PDF Downloads(28)
Parameterization method for unstable manifolds of delay differential equations
C. M. Groothedde and J. D. Mireles James
2017, 4(1&2) : 21-70 doi: 10.3934/jcd.2017002 +[Abstract](804) +[HTML](204) +[PDF](20182.49KB) PDF Downloads(24)
Towards a formal tie between combinatorial and classical vector field dynamics
Tomasz Kaczynski, Marian Mrozek and Thomas Wanner
2016, 3(1) : 17-50 doi: 10.3934/jcd.2016002 +[Abstract](547) +[PDF](562.5KB) PDF Downloads(19)
A Lin's method approach for detecting all canard orbits arising from a folded node
José Mujica, Bernd Krauskopf and Hinke M. Osinga
2017, 4(1&2) : 143-165 doi: 10.3934/jcd.2017005 +[Abstract](129) +[HTML](65) +[PDF](1853.05KB) PDF Downloads(17)
On dynamic mode decomposition: Theory and applications
Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton and J. Nathan Kutz
2014, 1(2) : 391-421 doi: 10.3934/jcd.2014.1.391 +[Abstract](1041) +[PDF](1657.5KB) PDF Downloads(11)
Addendum to "Optimal control of multiscale systems using reduced-order models"
Carsten Hartmann, Juan C. Latorre, Wei Zhang and Grigorios A. Pavliotis
2017, 4(1&2) : 167-167 doi: 10.3934/jcd.2017006 +[Abstract](101) +[HTML](54) +[PDF](140.81KB) PDF Downloads(7)
On the numerical approximation of the Perron-Frobenius and Koopman operator
Stefan Klus, Péter Koltai and Christof Schütte
2016, 3(1) : 51-79 doi: 10.3934/jcd.2016003 +[Abstract](605) +[PDF](2004.7KB) PDF Downloads(7)

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