arXiv subjects

Andrew P. Bassom

Publications and source records attributed to Andrew P. Bassom.

6 recordsLinked to original sources

arXiv2024

Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model

We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called "crises" that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviours are quite generic and robust.

nlin.CD↗

arXiv2017

On Integrable Ermakov-Painlevé IV Systems

Novel hybrid Ermakov-Painlevé IV systems are introduced and an associated Ermakov invariant is used in establishing their integrability. Bäcklund transformations are then employed to generate classes of exact solutions via the linked canonical Painlevé IV equation.

nlin.SI↗

arXiv2017

Conservation laws and integral relations for the Boussinesq equation

We are concerned with conservation laws and integral relations associated with rational solutions of the Boussinesq equation, a soliton equation solvable by inverse scattering which was first introduced by Boussinesq in 1871. The rational solutions are logarithmic derivatives of a polynomial, are algebraically decaying and have a similar appearance to rogue-wave solutions of the focusing nonlinear Schrödinger equation. For these rational solutions the constants of motion associated with the conserved quantities are zero and they have some interesting integral relations which depend on the total degree of the associated polynomial.

nlin.SI↗

arXiv2016

On the interaction of uni-directional and bi-directional buckling of a plate supported by an elastic foundation

A thin flat rectangular plate supported on its edges and subjected to in-plane loading exhibits stable post-buckling behaviour. However, the introduction of a nonlinear (softening) elastic foundation may cause the response to become unstable. Here the post-buckling of such a structure is investigated and several important phenomena are identified, including the transition of patterns from stripes to spots and back again. The interaction between these forms is of importance for understanding the possible post-buckling behaviours of this structural system. In addition, both periodic and some localized responses are found to exist as the dimensions of the plate are increased and this becomes relevant when the characteristic wavelengths of the buckle pattern are small compared to the size of the plate. Potential application of the model range from macroscopic industrial manufacturing of structural elements to the understanding of micro- and nano-scale deformations in materials.

nlin.PS↗

arXiv1994

Backlund Transformations and Hierarchies of Exact Solutions for the Fourth Painleve Equation and their Application to Discrete Equations

In this paper we describe Bäcklund transformations and hierarchies of exact solutions for the fourth Painlevé equation (PIV) $${\d^2 w\over\d z^2}={1\over2w}\left(\d w\over\d z\right)^2 + {3\over2}w^3 + 4zw^2 + 2(z^2-α)w+{β\over w},\eqno(1){\hbox to 16pt{\hfill}}$$ with $α$, $β$ constants. Specifically, a nonlinear superposition principle for PIV, hierarchies of solutions expressible in terms of complementary error or parabolic cylinder functions as well as rational solutions will be derived. Included amongst these hierarchies are solutions of (1) for which $α=\pm\tfr12n$ and $β=-\tfr12n^2$, with $n$ an integer. These particular forms arise in quantum gravity and also satisfy a discrete analogue of the first Painlevé equation. We also obtain a number of exact solutions of the discrete fourth Painlevé equation $$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})= {-2z_nx_n^3+(η-3δ^{-2}-z_n^2)x_n^2+μ^2\over (x_n+z_n+γ)(x_n+z_n-γ)},\eqno(2){\hbox to 16pt{\hfill}}$$}% {\narrower\noindent\baselineskip=12pt where $z_n=nδ$ and $η$, $δ$, $μ$ and $γ$ are constants, which, in an appropriate limit, reduces to PIV (1). A suitable factorisation of (2) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable $z_n$ and the limits of these solutions yield rational solutions of (1).

solv-int↗

arXiv1994

New exact solutions for the discrete fourth Painlevé equation

In this paper we derive a number of exact solutions of the discrete equation $$x_{n+1}x_{n-1}+x_n(x_{n+1}+x_{n-1})= {-2z_nx_n^3+(η-3δ^{-2}-z_n^2)x_n^2+μ^2\over (x_n+z_n+γ)(x_n+z_n-γ)},\eqno(1)$$ where $z_n=nδ$ and $η$, $δ$, $μ$ and $γ$ are constants. In an appropriate limit (1) reduces to the fourth \p\ (PIV) equation $${\d^2w\over\d z^2} = {1\over2w}\left({\d w\over\d z}\right)^2+\tfr32w^3 + 4zw^2 + 2(z^2-α)w +{β\over w},\eqno(2)$$ where $α$ and $β$ are constants and (1) is commonly referred to as the discretised fourth Painlevé equation. A suitable factorisation of (1) facilitates the identification of a number of solutions which take the form of ratios of two polynomials in the variable $z_n$. Limits of these solutions yield rational solutions of PIV (2). It is also known that there exist exact solutions of PIV (2) that are expressible in terms of the complementary error function and in this article we show that a discrete analogue of this function can be obtained by analysis of (1).

solv-int↗