## Involve: A Journal of Mathematics

• Involve
• Volume 2, Number 5 (2009), 559-580.

### Ineffective perturbations in a planar elastica

#### Abstract

An elastica is a bendable one-dimensional continuum, or idealized elastic rod. If such a rod is subjected to compression while its ends are constrained to remain tangent to a single straight line, buckling can occur: the elastic material gives way at a certain point, snapping to a lower-energy configuration.

The bifurcation diagram for the buckling of a planar elastica under a load $λ$ is made up of a trivial branch of unbuckled configurations for all $λ$ and a sequence of branches of buckled configurations that are connected to the trivial branch at pitchfork bifurcation points. We use several perturbation expansions to determine how this diagram perturbs with the addition of a small intrinsic shape in the elastica, focusing in particular on the effect near the bifurcation points.

We find that for almost all intrinsic shapes $ϵf(s)$, the difference between the buckled solution and the trivial solution is $O(ϵ1∕3)$, but for some ineffective $f$, this difference is $O(ϵ)$, and we find functions $uj(s)$ so that $f$ is ineffective at bifurcation point number $j$ when $〈f,uj〉=0$. These ineffective perturbations have important consequences in numerical simulations, in that the perturbed bifurcation diagram has sharper corners near the former bifurcation points, and there is a higher risk of a numerical simulation inadvertently hopping between branches near these corners.

#### Article information

Source
Involve, Volume 2, Number 5 (2009), 559-580.

Dates
Received: 12 February 2009
Accepted: 2 May 2009
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513799211

Digital Object Identifier
doi:10.2140/involve.2009.2.559

Mathematical Reviews number (MathSciNet)
MR2601577

Zentralblatt MATH identifier
1197.34079

#### Citation

Peterson, Kaitlyn; Manning, Robert. Ineffective perturbations in a planar elastica. Involve 2 (2009), no. 5, 559--580. doi:10.2140/involve.2009.2.559. https://projecteuclid.org/euclid.involve/1513799211

#### References

• A. Bolshoy, P. McNamara, R. E. Harrington, and E. N. Trifonov, “Curved DNA without A-A: Experimental estimation of all 16 wedge angles”, Proc. Natl. Acad. Sci. USA 88:6 (1991), 2312–2316.
• P. De Santis, A. Palleschi, M. Savino, and A. Scipioni, “Theoretical prediction of the gel electrophoretic retardation changers due to point mutations in a tract of Sv40 DNA”, Biophys. Chem. 42:2 (1992), 147–152.
• S. B. Dixit, D. L. Beveridge, D. A. Case, T. E. C. III, E. Giudice, F. Lankas, R. Lavery, J. H. Maddocks, R. Osman, H. Sklenar, K. M. Thayer, and P. Varnai, “Molecular dynamics simulations of the 136 unique tetranucleotide sequences of DNA oligonucleotides, II: Sequence context effects on the dynamical structures of the 10 unique dinucleotide steps”, Biophys. J. 89:6 (2005), 3721–3740.
• E. Doedel, H. B. Keller, and J.-P. Kernévez, “Numerical analysis and control of bifurcation problems, I: Bifurcation in finite dimensions”, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1:3 (1991), 493–520.
• E. Doedel, H. B. Keller, and J.-P. Kernévez, “Numerical analysis and control of bifurcation problems, II: Bifurcation in infinite dimensions”, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 1:4 (1991), 745–772.
• P. Glendinning, Stability, instability and chaos: an introduction to the theory of nonlinear differential equations, Cambridge University Press, Cambridge, 1994.
• S. Goyal, T. Lillian, S. Blumberg, J.-C. Meiners, E. Meyhöfer, and N. C. Perkins, “Intrinsic curvature of DNA influences LacR-mediated looping”, Biophys. J. 93:12 (2007), 4342–4359.
• G. Iooss and D. D. Joseph, Elementary stability and bifurcation theory, Springer, New York, 1980.
• J. D. Kahn and D. M. Crothers, “Measurement of the DNA bend angle induced by the catabolite activator protein using Monte Carlo simulation of cyclization kinetics”, J. Mol. Biol. 276:1 (1998), 287–309.
• R. S. Manning, J. H. Maddocks, and J. D. Kahn, “A continuum rod model of sequence-dependent DNA structure”, J. Chem. Phys. 105:13 (1996), 5626–5646.
• J. Marko and E. D. Siggia, “Stretching DNA”, Macromolecules 28:26 (1995), 8759–8770.
• J. A. Murdock, Perturbations: Theory and methods, Classics in Applied Mathematics 27, Soc. Industrial Appl. Math., Philadelphia, 1999.
• W. K. Olson, A. A. Gorin, X. J. Lu, L. M. Hock, and V. B. Zhurkin, “DNA sequence-dependent deformability deduced from protein-DNA crystal complexes”, Proc. Natl. Acad. Sci. USA 95:19 (1998), 11163–11168.
• Y. Seol, J. Li, P. C. Nelson, T. T. Perkins, and M. D. Betterton, “Elasticity of short DNA molecules: Theory and experiment for contour lengths of 0.6–7,$\mu$m”, Biophys. J. 93:12 (2007), 4360–4359.
• T. Shifrin and M. R. Adams, Linear algebra: a geometric approach, W. H. Freeman, New York, 2002.
• D. Swigon, B. D. Coleman, and W. K. Olson, “Modeling the Lac repressor-operator assembly, I: The influence of DNA looping on Lac repressor conformation”, Proc. Natl. Acad. Sci. USA 103:26 (2006), 9879–9884.