@article{geng_lin_liu_2019, title={Chaotic Traveling Wave Solutions in Coupled Chua's Circuits}, volume={31}, ISSN={["1572-9222"]}, DOI={10.1007/s10884-017-9631-1}, number={3}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Geng, Fengjie and Lin, Xiao-Biao and Liu, Xingbo}, year={2019}, month={Sep}, pages={1373–1396} }
@article{lin_zhu_2017, title={Codiagonalization of matrices and existence of multiple homoclinic solutions}, volume={7}, number={1}, journal={Journal of Applied Analysis and Computation}, author={Lin, X. B. and Zhu, C. R.}, year={2017}, pages={172–188} }
@article{lin_zhu_2017, title={SADDLE-NODE BIFURCATIONS OF MULTIPLE HOMOCLINIC SOLUTIONS IN ODES}, volume={22}, ISSN={["1553-524X"]}, DOI={10.3934/dcdsb.2017069}, abstractNote={We study codimension 3 degenerate homoclinic bifurcations under periodic perturbations. Assume that among the 3 bifurcation equations, one is due to the homoclinic tangecy along the orbital direction. To the lowest order, the bifurcation equations become 3 quadratic equations. Under generic conditions on perturbations of the normal and tangential directions of the homoclinic orbit, up to 8 homoclinic orbits can be created through saddle-node bifurcations. Our results generate the homoclinic tangency bifurcation in Guckenheimer and Holmes [8].}, number={4}, journal={DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B}, author={Lin, Xiao-Bin and Zhu, Changrong}, year={2017}, month={Jun}, pages={1435–1460} }
@article{lin_schecter_2016, title={Stability of Concatenated Traveling Waves}, volume={28}, ISSN={["1572-9222"]}, DOI={10.1007/s10884-015-9428-z}, number={3-4}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao and Schecter, Stephen}, year={2016}, month={Sep}, pages={867–896} }
@article{lin_long_zhu_2015, title={Multiple transverse homoclinic solutions near a degenerate homoclinic orbit}, volume={259}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2015.01.046}, abstractNote={Consider an autonomous ordinary differential equation in Rn that has a homoclinic solution asymptotic to a hyperbolic equilibrium. The homoclinic solution is degenerate in the sense that the linear variational equation has 2 bounded, linearly independent solutions. We study bifurcation of the homoclinic solution under periodic perturbations. Using exponential dichotomies and Lyapunov–Schmidt reduction, we obtain general conditions under which the perturbed system can have transverse homoclinic solutions and nearby periodic or chaotic solutions.}, number={1}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao and Long, Bin and Zhu, Changrong}, year={2015}, month={Jul}, pages={1–24} }
@article{lin_long_zhu_2015, title={Multiple transverse homoclinic solutions near a degenerate homoclinic orbit (vol 259, pg 1, 2015)}, volume={259}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2015.07.032}, abstractNote={The authors sincerely apologize for the error in counting the codimension of bifurcations in the above published paper, which will be called [LLZ] for short. First, to fix the paper [LLZ], we look for bifurcations of heteroclinic solutions x(t) = γμ, rather than the homoclinic solutions as in [LLZ]. And the bifurcated heteroclinic solutions x(t) are close to γ (t) as a function to another function, without a phase shift. That is x(t) = γ (t) +z(t) where z(0) ⊥ γ (0). Second, we assume the heteroclinic solution connects two hyperbolic equilibria u = 0 to u = A, i.e., γ (−∞) = 0, γ (∞) = A, with Re{σ(Df (0))} = 0, Re{σ(Df (A))} = 0. Let Lu := u′ −Df (γ (t))u. Let N(L) and N(L∗) be the null spaces of L and its adjoint operator L∗. If d is the dimension of N(L) and d∗ is the dimension of N(L∗), then the index of the Fredholm operator L is I (L) := d − d∗. The basic change in this corrigendum is the following hypothesis: (H) d = 3, d∗ = 2, and I (L) = 1. Let N(L) be spanned by (u1, u2, u3) with u3 = γ (t), and N(L∗) be spanned by (ψ1, ψ2). Let K : R(L) → N(L)⊥ be a particular solution map to the equation L(z) = h, where h ∈ R(L). With the phase condition z(0) ⊥ γ (0), the general solution to L(z) = h, h ∈ R(L) is z=∑p=1,2 βpup +Kh.}, number={11}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao and Long, Bin and Zhu, Changrong}, year={2015}, month={Dec}, pages={6885–6886} }
@article{lin_schecter_2015, title={Stability of concatenated traveling waves: Alternate approaches}, volume={259}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2015.04.015}, abstractNote={We consider a reaction–diffusion equation in one space dimension whose initial condition is approximately a sequence of widely separated traveling waves with increasing velocity, each of which is asymptotically stable. As in [14], [24], [25], we show that the sequence of traveling waves is itself asymptotically stable: as t→∞, the solution approaches the concatenated wave pattern, with different shifts of each wave allowed. Our proof is similar to that of [14] in that it is based on spatial dynamics, Laplace transform, and exponential dichotomies, but it incorporates a number of modifications.}, number={7}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao and Schecter, Stephen}, year={2015}, month={Oct}, pages={3144–3177} }
@article{lin_wechselberger_2014, title={TRANSONIC EVAPORATION WAVES IN A SPHERICALLY SYMMETRIC NOZZLE}, volume={46}, ISSN={["1095-7154"]}, DOI={10.1137/120875363}, abstractNote={This paper studies the liquid-to-vapor phase transition in a cone-shaped nozzle. Using the geometric method presented in [P. Szmolyan and M. Wechselberger, J. Differential Equations, 200 (2004), pp. 69--104], [M. Wechselberger and G. Pettet, Nonlinearity, 23 (2010), pp. 1949--1969], we further develop results on subsonic and supersonic evaporation waves in [H. Fan and X.-B. Lin, SIAM J. Math. Anal., 44 (2012), pp. 405--436] to transonic waves. It is known that transonic waves do not exist if restricted solely to the slow system on the slow manifolds [H. Fan and X.-B. Lin, SIAM J. Math. Anal., 44 (2012), pp. 405--436]. Thus we consider the existence of transonic waves that include layer solutions of the fast system that cross or connect to the sonic surface. In particular, we are able to show the existence and uniqueness of evaporation waves that cross from supersonic to subsonic regions and evaporation waves that connect from the subsonic region to the sonic surface and then continue onto the supersonic branch via the slow flow.}, number={2}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Lin, Xiaobiao and Wechselberger, Martin}, year={2014}, pages={1472–1504} }
@article{lin_2013, title={Stability of standing waves for monostable reaction-convection equations in a large bounded domain with boundary conditions}, volume={255}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2013.03.010}, abstractNote={It is well known that the standing wave u 0 for the KPP type convection–diffusion equation is stable if the perturbations of the initial data are in the weighted function spaces proposed by Sattinger. We study boundary conditions so that in a large finite domain, there is a stable standing wave u ˜ near u 0 . The standing wave u ˜ may not be monotone, and the stability is proved by pseudo exponential dichotomies that are weighted both in the spatial variable ξ and in the dual variable s to the time t .}, number={1}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao}, year={2013}, month={Jul}, pages={58–84} }
@article{chow_jiang_lin_2013, title={Traveling wave solutions in coupled Chua's circuits, part I: Periodic solutions}, volume={3}, number={3}, journal={Journal of Applied Analysis and Computation}, author={Chow, S. N. and Jiang, M. and Lin, X. B.}, year={2013}, pages={213–237} }
@article{fan_lin_2012, title={STANDING WAVES FOR PHASE TRANSITIONS IN A SPHERICALLY SYMMETRIC NOZZLE}, volume={44}, ISSN={["1095-7154"]}, DOI={10.1137/11082213x}, abstractNote={We study the existence of standing waves for liquid/vapor phase transition in a spherically symmetric nozzle. The system is singularly perturbed and the solution consists of an internal layer where the liquid quickly becomes vapor. Using methods from dynamical systems theory, we prove the existence of the internal layer as a heteroclinic orbit connecting the liquid state to the vapor state. The heteroclinic orbit is reproduced numerically and is also shown numerically to be a transversal heteroclinic orbit. The proof of the existence of an exact standing wave solution near the singular limit is based on the geometric singular perturbation theory and is outlined in the paper.}, number={1}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Fan, Haitao and Lin, Xiao-Biao}, year={2012}, pages={405–436} }
@article{lin_weng_wu_2011, title={Traveling Wave Solutions for a Predator-Prey System With Sigmoidal Response Function}, volume={23}, ISSN={["1040-7294"]}, DOI={10.1007/s10884-011-9220-7}, abstractNote={We study the existence of traveling wave solutions for a diffusive predator-prey system. The system considered in this paper is governed by a Sigmoidal response function which is more general than those studied previously. Our method is an improvement to the original method introduced in the work of Dunbar \cite{Dunbar1,Dunbar2}. A bounded Wazewski set is used in this work while unbounded Wazewski sets were used in \cite{Dunbar1,Dunbar2}. The existence of traveling wave solutions connecting two equilibria is established by using the original Wazewski's theorem which is much simpler than the extended version in Dunbar's work.}, number={4}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Lin, Xiaobiao and Weng, Peixuan and Wu, Chufen}, year={2011}, month={Dec}, pages={903–921} }
@article{fan_lin_2010, title={Collapsing and Explosion Waves in Phase Transitions with Metastability, Existence, Stability and Related Riemann Problems}, volume={22}, ISSN={["1572-9222"]}, DOI={10.1007/s10884-009-9150-9}, number={2}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Fan, Haitao and Lin, Xiao-Biao}, year={2010}, month={Jun}, pages={163–191} }
@article{lin_2009, title={Algebraic dichotomies with an application to the stability of Riemann solutions of conservation laws}, volume={247}, ISSN={["1090-2732"]}, DOI={10.1016/j.jde.2009.07.002}, abstractNote={Recently, there has been some interest on the stability of waves where the functions involved grow or decay at an algebraic rate |x|m. In this paper we define the so-called algebraic dichotomy that may aid in treating such problems. We discuss the basic properties of the algebraic dichotomy, methods of detecting it, and calculating the power of the weight function. We present several examples: (1) The Bessel equation. (2) The n-degree Fisher type equation. (3) Hyperbolic conservation laws in similarity coordinates. (4) A system of conservation laws with a Dafermos type viscous regularization. We show that the linearized system generates an analytic semigroup in the space of algebraic decay functions. This example motivates our work on algebraic dichotomies.}, number={11}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao}, year={2009}, month={Dec}, pages={2924–2965} }
@article{lin_2007, title={Gearhart-Pruss theorem and linear stability for riemann solutions of conservation laws}, volume={19}, ISSN={["1040-7294"]}, DOI={10.1007/s10884-007-9098-6}, number={4}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao}, year={2007}, month={Dec}, pages={1037–1074} }
@article{lai_li_lin_2006, title={Fast solvers for 3D Poisson equations involving interfaces in a finite or the infinite domain}, volume={191}, ISSN={["1879-1778"]}, DOI={10.1016/j.cam.2005.04.025}, abstractNote={In this paper, numerical methods are proposed for Poisson equations defined in a finite or infinite domain in three dimensions. In the domain, there can exists an interface across which the source term, the flux, and therefore the solution may be discontinuous. The existence and uniqueness of the solution are also discussed. To deal with the discontinuity in the source term and in the flux, the original problem is transformed to a new one with a smooth solution. Such a transformation can be carried out easily through an extension of the jumps along the normal direction if the interface is expressed as the zero level set of a three-dimensional function. An auxiliary sphere is used to separate the infinite region into an interior and exterior domain. The Kelvin's inversion is used to map the exterior domain into an interior domain. The two Poisson equations defined in the interior and the exterior written in spherical coordinates are solved simultaneously. By choosing the mesh size carefully and exploiting the fast Fourier transform, the resulting finite difference equations can be solved efficiently. The approach in dealing with the interface has also been used with the artificial boundary condition technique which truncates the infinite domain. Numerical results demonstrate second order accuracy of our algorithms.}, number={1}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Lai, MC and Li, ZL and Lin, XB}, year={2006}, month={Jun}, pages={106–125} }
@article{lin_2006, title={Slow eigenvalues of self-similar solutions of the Dafermos regularization of a system of conservation laws: An analytic approach}, volume={18}, ISSN={["1040-7294"]}, DOI={10.1007/s10884-005-9001-2}, abstractNote={The Dafermos regularization of a system of n hyperbolic conservation laws in one space dimension has, near a Riemann solution consisting of n Lax shock waves, a self-similar solution u = u ε(X/T). In Lin and Schecter (2003, SIAM J. Math. Anal. 35, 884–921) it is shown that the linearized Dafermos operator at such a solution may have two kinds of eigenvalues: fast eigenvalues of order 1/ε and slow eigenvalues of order one. The fast eigenvalues represent motion in an initial time layer, where near the shock waves solutions quickly converge to traveling-wave-like motion. The slow eigenvalues represent motion after the initial time layer, where motion between the shock waves is dominant. In this paper we use tools from dynamical systems and singular perturbation theory to study the slow eigenvalues. We show how to construct asymptotic expansions of eigenvalue-eigenfunction pairs to any order in ε. We also prove the existence of true eigenvalue-eigenfunction pairs near the asymptotic expansions.}, number={1}, journal={JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS}, author={Lin, Xiao-Biao}, year={2006}, month={Jan}, pages={1–52} }
@article{lin_2005, title={L-2 semigroup and linear stability for Riemann solutions of conservation laws}, volume={2}, DOI={10.4310/dpde.2005.v2.n4.a2}, abstractNote={Riemann solutions for the systems of conservation laws u +f(u) = 0 are self-similar solutions of the form u = u( = ). Using the change of vari- ables x = = ; t = ln( ), Riemann solutions become stationary to the system ut + (Df(u) xI)ux = 0. For the linear variational system around the Rie- mann solution with n-Lax shocks, we introduce a semigroup in the Hilbert space with weighted L 2 norm. We show that (A) the region consists of normal points only. (B) Eigenvalues of the linear system correspond to zeros of the determinant of a transcendental matrix. They lie on vertical lines in the complex plane. There can be resonance values where the response of the system to forcing terms can be arbitrarily large, see Denition 6.2. Resonance values also lie on vertical lines in the complex plane. (C) Solutions of the linear system are O(e t) for any constant that is greater than the largest real parts of the eigenvalues and the coordinates of resonance lines. This work can be applied to the linear and nonlinear stability of Riemann solutions of conserva- tion laws and the stability of nearby solutions of the Dafermos regularizations ut + (Df(u) xI)ux = u xx.}, number={4}, journal={Dynamics of Partial Differential Equations}, author={Lin, X. B.}, year={2005}, pages={301–333} }
@article{lin_schecter_2003, title={Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws}, volume={35}, ISSN={["1095-7154"]}, DOI={10.1137/S0036141002405029}, abstractNote={In contrast to a viscous regularization of a system of n conservation laws, a Dafermos regularization admits many self-similar solutions of the form \( u=u(\frac{X}{T})\). In particular, it is known in many cases that Riemann solutions of a system of conservation laws have nearby self-similar smooth solutions of an associated Dafermos regularization. We refer to these smooth solutions as {\em Riemann--Dafermos solutions}. In the coordinates $x=\frac{X}{T}$, $t=\ln T$, Riemann--Dafermos solutions become stationary, and their time-asymptotic stability as solutions of the Dafermos regularization can be studied by linearization. We study the stability of Riemann--Dafermos solutions near Riemann solutions consisting of n Lax shock waves. We show, by studying the essential spectrum of the linearized system in a weighted function space, that stability is determined by eigenvalues only. We then use asymptotic methods to study the eigenvalues and eigenfunctions. We find there are fast eigenvalues of order $\frac{1}{\epsilon}$ and slow eigenvalues of order 1. The fast eigenvalues correspond to eigenvalues of the viscous profiles for the individual shock waves in the Riemann solution; these have been studied by other authors using Evans function methods. The slow eigenvalues are related to inviscid stability conditions that have been obtained by various authors for the underlying Riemann solution.}, number={4}, journal={SIAM JOURNAL ON MATHEMATICAL ANALYSIS}, author={Lin, XB and Schecter, S}, year={2003}, pages={884–921} }
@article{lin_vivancos_2002, title={Heteroclinic and periodic cycles in a perturbed convection model}, volume={182}, ISSN={["0022-0396"]}, DOI={10.1006/jdeq.2001.4090}, abstractNote={Vivancos and Minzoni (New Choatic behaviour in a singularly perturbed model, preprint) proposed a singularly perturbed rotating convection system to model the Earth's dynamo process. Numerical simulation shows that the perturbed system is rich in chaotic and periodic solutions. In this paper, we show that if the perturbation is sufficiently small, the system can only have simple heteroclinic solutions and two types of periodic solutions near the simple heteroclinic solutions. One looks like a figure “Delta” and the other looks like a figure “Eight”. Due to the fast – slow characteristic of the system, the reduced slow system has a relay nonlinearity (“Asymptotic Method in Singularly Perturbed Systems,” Consultants Bureau, New York and London, 1994) – solutions to the slow system are continuous but their derivative changes abruptly at certain junction surfaces. We develop new types of Melnikov integral and Lyapunov–Schmidt reduction methods which are suitable to study heteroclinic and periodic solutions for systems with relay nonlinearity.}, number={1}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, XB and Vivancos, IB}, year={2002}, month={Jun}, pages={219–265} }
@article{lin_2001, title={Construction and asymptotic stability of structurally stable internal layer solutions}, volume={353}, ISSN={["0002-9947"]}, DOI={10.1090/S0002-9947-01-02769-6}, abstractNote={We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined. Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo PoincarÃ© mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues. An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichelâs transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection. We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.}, number={8}, journal={TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY}, author={Lin, XB}, year={2001}, pages={2983–3043} }
@article{lin_2000, title={Generalized Rankine-Hugoniot condition and shock solutions for quasilinear hyperbolic systems}, volume={168}, ISSN={["0022-0396"]}, DOI={10.1006/jdeq.2000.3889}, abstractNote={For a quasilinear hyperbolic system, we use the method of vanishing viscosity to construct shock solutions. The solution consists of two regular regions separated by a free boundary (shock). We use Melnikov's integral to obtain a system of differential/algebraic equations that governs the motion of the shock. For Lax shocks in conservation laws, these equations are equivalent to the Rankine–Hugoniot condition. For under compressive shocks in conservation laws, or shocks in non-conservation systems, the Melnikov-type integral obtained in this paper generalizes the Rankine–Hugoniot condition. Under some generic conditions, we show that the initial value problem of shock solutions can be solved as a free boundary problem by the method of characteristics.}, number={2}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Lin, XB}, year={2000}, month={Dec}, pages={321–354} }
@article{hale_lin_1999, title={Multiple internal layer solutions generated by spatially oscillatory perturbations}, volume={154}, ISSN={["0022-0396"]}, DOI={10.1006/jdeq.1998.3566}, abstractNote={For a singularly perturbed system of reaction-diffusion equations, we study the bifurcation of internal layer solutions due to the addition of a spatially oscillatory term. In the singular limit, the existence and stability of internal layer solutions are determined by the intersection of a fast jump surface Γ 1 and a slow switching curve C . The case when the intersection is transverse was studied by X.-B. Lin (Construction and asymptotic stability of structurally stable internal layer solutions, preprint). In this paper, we show that when Γ 1 intersects with C tangentially, saddle-node or cusp type bifurcation may occur. Higher order expansions of internal layer solutions and eigenvalue–eigenfunctions are also presented. To find a true internal layer solution and true eigenvalue-eigenfunctions, we use a Newton's method in functions spaces that is suitable for numerical computations.}, number={2}, journal={JOURNAL OF DIFFERENTIAL EQUATIONS}, author={Hale, JK and Lin, XB}, year={1999}, month={May}, pages={364–418} }