Cusp bifurcations: Numerical detection via two-parameter continuation and computer-assisted proofs of existence

Jean-Philippe Lessard; Alessandro Pugliese

doi:10.3934/dcdsb.2024181

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2024. Doi: 10.3934/dcdsb.2024181

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Cusp bifurcations: Numerical detection via two-parameter continuation and computer-assisted proofs of existence

Jean-Philippe Lessard^1, and
Alessandro Pugliese^2, ,

1.
McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A 0B9, Canada
2.
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

^*Corresponding author: Alessandro Pugliese
^*Corresponding author: Alessandro Pugliese

Received: March 2024

Revised: December 2024

Early access: December 2024

The first author is supported by NSERC. The second author gratefully acknowledges the hospitality of the School of Mathematics of Georgia Tech and the support provided by Università degli Studi di Bari Aldo Moro and by INdAM–GNCS. The second author has been partially funded by PRIN2022PNRR n. P2022M7JZW "SAFER MESH - Sustainable mAnagement oF watEr Resources ModEls and numerical MetHods" research grant, funded by the Italian Ministry of Universities and Research (MUR) and by the European Union through Next Generation EU, M4.C2.1.1, CUP H53D23008930001

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Abstract

This paper introduces a novel computer-assisted method for detecting and constructively proving the existence of cusp bifurcations in differential equations. The approach begins with a two-parameter continuation along which a tool based on the theory of Poincaré index is employed to identify the presence of a cusp bifurcation. Using the approximate cusp location, Newton's method is then applied to a given augmented system (the cusp map), yielding a more precise numerical approximation of the cusp. Through a successful application of a Newton-Kantorovich type theorem, we establish the existence of a non-degenerate zero of the cusp map in the vicinity of the numerical approximation. Employing a Gershgorin circles argument, we then prove that exactly one eigenvalue of the Jacobian matrix at the cusp candidate has zero real part, thus rigorously confirming the presence of a cusp bifurcation. Finally, by incorporating explicit control over the cusp's location, a rigorous enclosure for the normal form coefficient is obtained, providing the explicit dynamics on the center manifold at the cusp. We show the effectiveness of this method by applying it to four distinct models.

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Mathematics Subject Classification: Primary: 37G99, 65P30, 65G40, 34C23, 37M20.

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Figure 1. Visual sketch of the continuation procedure: in blue the first hexagonal patch, in yellow an incomplete patch centered at the "current point" $ \mathbf{x}^{({k})} $, in red/yellow the simplices which are on the boundary of the (un)explored portion of $ \mathcal{M} $

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Figure 2. On the left the projection of the triangulation for the manifold of equilibria of (31) on the $ (\lambda_1,\lambda_2,x_1) $-space; on the right a close up of the projection on the $ (\lambda_1,\lambda_2) $-plane near the second cusp point (black circular mark). The darkest shade of grey identifies the region where three equilibria exist for the same value of the parameter

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Figure 3. On the left the projection of the triangulation of the manifold of equilibria of (33) on the $ (\lambda_1,\lambda_2,\Vert x\Vert_2) $-space; on the right the projection on the $ (\lambda_1,\lambda_2) $-plane; a black circular mark signals the location of the cusp point

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Figure 4. On the left the projection of the triangulation of the manifold of equilibria of (35) on the $ (\lambda_1,\lambda_2,\Vert x\Vert_2) $-space; on the right the projection on the $ (\lambda_1,\lambda_2) $-plane; a black circular mark signals the location of the cusp point

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Figure 5. On the left the projection of the triangulation of the manifold of equilibria of (36) on the $ (\lambda_1,\lambda_2,x_2) $-space; on the right the projection on the $ (\lambda_1,\lambda_2) $-plane; a black circular mark signals the location of the cusp point

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