A scalar nonlinear heat equation is investigated where the bifurcation parameter is assumed close to its first critical value and varies slowly in time. The long time evolution of small amplitude solutions is investigated and the somewhat surprising result is exhibited that when $| {\int_0^t {(\lambda - \lambda _c )dt} } |$ is bounded for all t , where ${\lambda - \lambda _c }$ is the deviation of the bifurcation parameter $\lambda $ from its critical value $\lambda _c $, the solution will decay to its equilibrium. A special case of this is when $\lambda $ oscillates about its critical value. Other situations, when the bifurcation parameter remains below critical, or reaches an equilibrium value above critical, yield expected results.

A formal asymptotic procedure, for large $\lambda $, is developed to construct a uniform approximation to solutions of $y'' + \lambda ^2 p( {x,\lambda } )y = 0,x \in ( {a,b} )$, in regions containing finitely many zeros of $p( {x,\lambda } )$ of finite orders. The procedure used is that of constructing a simpler related equation which retains the local turning point properties of the original one.

An asymptotic procedure for deriving equations governing the passage of a weakly coupled nonlinear system of oscillators is discussed. The procedure avoids an inner‐outer‐matching technique and is valid when the small coupling and detuning parameters are arbitrary. Resonance is permitted to occur at one or several instances of time or to last for a finite length of time. Numerical results are discussed.

The dynamical behavior of small-norm solutions to a simple fluid-loop model describing convection in a two-constituent fluid is discussed. Asymptotic techniques are developed, to describe the dynamic behavior of solutions near those parts of the linear stability boundary where one eigenvalue passes through zero (exchange of stabilities), where two eigenvalues become purely imaginary conjugates (Hopf bifurcation), and where two eigenvalues coalesce to zero.

Communications on Pure and Applied MathematicsVolume 29, Issue 4 p. 343-367 Article Uniform asymptotic solutiion for a linear ordinary differential equation with one μ-th order turning point: Analytic theory B. Willner, B. Willner Rensselaer Polytechnic InstituteSearch for more papers by this authorL. A. Rubenfeld, L. A. Rubenfeld Rensselaer Polytechnic InstituteSearch for more papers by this author B. Willner, B. Willner Rensselaer Polytechnic InstituteSearch for more papers by this authorL. A. Rubenfeld, L. A. Rubenfeld Rensselaer Polytechnic InstituteSearch for more papers by this author First published: July 1976 https://doi.org/10.1002/cpa.3160290402Citations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume29, Issue4July 1976Pages 343-367 RelatedInformation

We consider a simple fluid-loop model describing convection in a two-constituent fluid. The model permits explicit construction of linear stability and global stability boundaries in parameter space. A rigorous proof of global stability within the appropriate region is provided.