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While equilibria, periodic orbits, and invariant tori can be computed using existing methods, the numerical identification of heteroclinic and homoclinic connections between them remains challenging. We propose a robust matrix-free variational method for computing connecting orbits between equilibrium solutions. Instead of a common shooting-based approach, we view the identification of a connecting orbit as a minimization problem in the space of smooth curves in the state space that connect the two equilibria. In this approach, the deviation of a connecting curve from an integral curve of the vector field is penalized by a non-negative cost function. Minimization of the cost function deforms a trial curve until, at a global minimum, a connecting orbit is obtained. The method has no limitation on the dimension of the unstable manifold at the origin equilibrium and does not suffer from exponential error amplification associated with time-marching a chaotic system. Owing to adjoint-based minimization techniques, no Jacobian matrices need to be constructed. Therefore, the memory requirement scales linearly with the size of the problem, allowing the method to be applied to high-dimensional dynamical systems. The robustness of the method is demonstrated for the one-dimensional Kuramoto\u2013Sivashinsky equation.<\/jats:p>","DOI":"10.1063\/5.0143923","type":"journal-article","created":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T15:13:00Z","timestamp":1689606780000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":5,"title":["Jacobian-free variational method for computing connecting orbits in nonlinear dynamical systems"],"prefix":"10.1063","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5684-4681","authenticated-orcid":false,"given":"Omid","family":"Ashtari","sequence":"first","affiliation":[{"name":"Emergent Complexity in Physical Systems Laboratory (ECPS), \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) , CH-1015 Lausanne, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8617-8998","authenticated-orcid":false,"given":"Tobias M.","family":"Schneider","sequence":"additional","affiliation":[{"name":"Emergent Complexity in Physical Systems Laboratory (ECPS), \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne (EPFL) , CH-1015 Lausanne, Switzerland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2023,7,17]]},"reference":[{"key":"2023071715125277100_c1","doi-asserted-by":"publisher","first-page":"P1","DOI":"10.1017\/jfm.2018.144","article-title":"Coherent structures in wall-bounded turbulence","volume":"842","year":"2018","journal-title":"J. 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