If we encase the strut between two elastic media (`strut on a foundation') then under certain conditions the deformation remains restricted to a small section of the length of the strut:

Other well-known examples of such localization are the axially loaded cylinder and a strut (or rod) under torsion; you also see it if you replace the elastic foundation by a viscous one.

The strut-on-foundation model leads to the ODE

where F is a single- or multiple-well potential. This equation, known
also as the stationary Extended Fisher-Kolmogorov Equation or
the stationary Swift-Hohenberg equation, has many solutions that
are bounded on the real line.

All activity described on this page is in collaboration with the Center for Nonlinear Mechanics in Bath, and most importantly Giles Hunt and Chris Budd.

- Mark A. Peletier,
Sequential Buckling: A Variational Analysis (160 kB),
SIAM Journal on Mathematical Analysis, Vol. 32, pp. 1142-1168 (2001);
link to the paper on the SIAM web site.
The model of an elastic strut on an elastic foundation produces the equation above as the Euler-Lagrange equation associated with the constrained minimization problem

where we minimize the strain energy associated with a deformation u under a constraint of prescribed total shortening of the strut. The parameter p, which arises as a Lagrange multiplier in this minimization, is the load that is necessary to keep the deformed strut in equilibrium. In this paper we show that this minimization problem is well-posed, i.e. that a minimizing sequence always converges to a `nice' limit. Since we consider struts of infinite length, we can also ask the question what happens when the shortening becomes large; it turns out that for large lambda the minimizers of this problem have a long periodic section, flanked by decaying tails. - Mark A. Peletier, Generalized Monotonicity from Global Minimization in
Fourth-Order ODE's (120 kB), Nonlinearity, Vol. 14, pp. 1221-1238 (2001).
As the next step in characterizing the selection of solutions by the minimization process we show in this paper that a minimizer necessarily has the aspect of a periodic function multiplied by an amplitude function that has a single maximum. Or more precisely, the sequence of local maxima of these oscillating solutions is first increasing, then decreasing. We call this

*generalized monotonicity*. This result severely limits the set of potential solutions. - Rob Beardmore,
Mark A. Peletier,
Chris Budd, and
M. Ahmer Wadee,
Bifurcation of
Periodic Solutions Satisfying the Zero-Hamiltonian Constraint in
Fourth-Order Differential Equations (pdf, 675 kB),
SIAM J. Math. Anal., 36, 1461-1488 (2005).
Rob Beardmore saw how the Lyapunov-Schmidt reduction could be used to construct branches of periodic solutions which have zero Hamiltonian. Together with some global analysis like in the Nonlinearity paper below and with numerical results by Chris Budd and Ahmer Wadee this gives a nice description of the bifurcation picture.

- Giles W. Hunt, Mark A. Peletier,
Alan R. Champneys,
Patrick D. Woods, M. Ahmer Wadee,
Chris J. Budd,
and Gabriel Lord,
Cellular Buckling in Long Structures,
Nonlinear Dynamics, Vol. 21, pp. 3-29 (2000).
This paper arose as the result of the authors spending an afternoon together, finding out that from different points of view we had been working on very similar issues. The binding element is the combination of localized buckling (which can be viewed as a property related to small-amplitude behaviour) with subsequent large-amplitude

*restiffening*. All three papers above fit in this class. Although slightly heterogeneous in nature, the paper gives a good feeling for the different aspects of these types of systems. Hint: don't try to print Fig. 6. - Mark A. Peletier, Non-existence and Uniqueness
Results for Fourth-Order Hamiltonian Systems (126 kB), Nonlinearity, Vol. 12, pp. 1555-1570
(1999); PDF version at the IOP web site (295 kB).
The combination of a monotonicity result for this type of Hamiltonian systems due to Toland, and an a priori estimate of bounded solutions due to L. A. Peletier & Troy, allows us in some cases to rule out, in a simple manner, solutions with a specific value of the Hamiltonian, based on the form of the potential F.

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