# American Institute of Mathematical Sciences

June  2007, 6(2): 311-333. doi: 10.3934/cpaa.2007.6.311

## Ackerberg-O'Malley resonance in boundary value problems with a turning point of any order

 1 Hasselt University, Campus Diepenbeek, Agoralaan-Gebouw D, B-3590 Diepenbeek, Belgium

Received  January 2006 Revised  October 2006 Published  March 2007

We consider an Ackerberg-O'Malley singular perturbation problem $\epsilon y'' + f(x,\epsilon)y' + g(x,\epsilon)y=0, y(a)=A, y(b)=B$ with a single turning point and study the nature of resonant solutions $y=\varphi(x,\epsilon)$, i.e. solutions for which $\varphi(x,\epsilon)$ tends to a nontrivial solution of $f(x,0)y'+ g(x,0)y=0$ as $\epsilon\to 0$. Many techniques have been applied to the study of this problem (WKBJ, invariant manifolds, asymptotic methods, spectral methods, variational techniques) and they have been successful in characterizing these resonant solutions when $f(x,0)$ has a simple zero at the origin. When the order of zero is higher the increase in complexity of the problem is significant. The existence of a nonzero formal power series solution is no longer necessary for resonance and resonant solutions are in general not smooth at the origin. We apply the method of blow up to study the nature of resonant solutions in this setting, using techniques from invariant manifold theory and planar singular perturbation theory. The main result is the sufficiency of the Matkowsky condition for turning points of arbitrary order (based on Gevrey-asymptotics), but we also give a characterization of the location of the boundary layer in resonant solutions.
Citation: P. De Maesschalck. Ackerberg-O'Malley resonance in boundary value problems with a turning point of any order. Communications on Pure & Applied Analysis, 2007, 6 (2) : 311-333. doi: 10.3934/cpaa.2007.6.311
 [1] P. De Maesschalck, Freddy Dumortier. Detectable canard cycles with singular slow dynamics of any order at the turning point. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 109-140. doi: 10.3934/dcds.2011.29.109 [2] Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485 [3] Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701 [4] Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29 [5] Shanshan Wang, Yanxia Chen, Taohui Xiao, Lei Zhang, Xin Liu, Hairong Zheng. LANTERN: Learn analysis transform network for dynamic magnetic resonance imaging. Inverse Problems & Imaging, 2021, 15 (6) : 1363-1379. doi: 10.3934/ipi.2020051 [6] Juan Sánchez, Marta Net, José M. Vega. Amplitude equations close to a triple-(+1) bifurcation point of D4-symmetric periodic orbits in O(2)-equivariant systems. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1357-1380. doi: 10.3934/dcdsb.2006.6.1357 [7] Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275 [8] Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971 [9] Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635 [10] Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $O(n)$ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021082 [11] Qi An, Chuncheng Wang, Hao Wang. Analysis of a spatial memory model with nonlocal maturation delay and hostile boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (10) : 5845-5868. doi: 10.3934/dcds.2020249 [12] Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048 [13] Urszula Foryś, Yuri Kheifetz, Yuri Kogan. Critical-Point Analysis For Three-Variable Cancer Angiogenesis Models. Mathematical Biosciences & Engineering, 2005, 2 (3) : 511-525. doi: 10.3934/mbe.2005.2.511 [14] Shoya Kawakami. Two notes on the O'Hara energies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 953-970. doi: 10.3934/dcdss.2020384 [15] Weishi Liu. Geometric approach to a singular boundary value problem with turning points. Conference Publications, 2005, 2005 (Special) : 624-633. doi: 10.3934/proc.2005.2005.624 [16] Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167 [17] Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103 [18] Heather Hannah, A. Alexandrou Himonas, Gerson Petronilho. Anisotropic Gevrey regularity for mKdV on the circle. Conference Publications, 2011, 2011 (Special) : 634-642. doi: 10.3934/proc.2011.2011.634 [19] Jian Zhao, Fang Deng, Jian Jia, Chunmeng Wu, Haibo Li, Yuan Shi, Shunli Zhang. A new face feature point matrix based on geometric features and illumination models for facial attraction analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1065-1072. doi: 10.3934/dcdss.2019073 [20] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006

2020 Impact Factor: 1.916