Delays and advances in the onset of instability in the Shishkova equation

The phenomenon of slow passage through Hopf bifurcations arises in many applications, and it is of interest to understand the resulting onset of instability in the solutions, which often shows a counterintuitive delay. Here, we study a model system known as the Shishkova equation, introduced in M. A. Shishkova [ <italic>Examination of a system of differential equations with a small parameter in the highest derivatives</italic> , Dokl. Akad. Nauk SSSR 209 (1973) 576–579]. It exhibits many—though not all—of the features of slow passage through Hopf bifurcations. First, we show how the original proof of the phenomenon of delayed onset of instability in the Shishkova equation can be cast in a modern framework using an extension into the complex plane that differs from the original. We use a reflection operator that preserves analyticity. In addition, we show how some steps in the convergence proof for the iterative scheme can be simplified, including the decomposition of the key domain in the complex time plane and the piecewise constant function used to bound the nonlinear terms, so that the method can be applied to equations with other nonlinearities. The analysis of the delayed onset applies to the parameter regime of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> natural frequencies of rotation at the instantaneous Hopf point. Second, we present the results of numerical continuation for the solutions of the Shishkova equation. For the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}(1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> natural frequencies commonly studied in the Shishkova equation, the numerics confirm the theory. Then, in the regime where the natural frequency at the instantaneous Hopf bifurcation is small, the numerical results reveal that there is an advance in the onset of instability, rather than a delay. We present analysis to demonstrate this advance.