Nonlinear Bifurcation Problems Research Articles

Transverse Dynamics of Two-Dimensional Gravity–Capillary Periodic Water Waves

We study the transverse dynamics of two-dimensional gravity–capillary periodic water waves in the case of large surface tension. In this parameter regime, predictions based on model equations suggest that periodic traveling waves are stable with respect to two-dimensional perturbations, and unstable with respect to three-dimensional perturbations which are periodic in the direction transverse to the direction of propagation. In this paper, we confirm the second prediction. We show that, as solutions of the full water-wave equations, the periodic traveling waves are linearly unstable under such three-dimensional perturbations. In addition, we study the nonlinear bifurcation problem near these transversely unstable two-dimensional periodic waves. We show that a one-parameter family of three-dimensional doubly periodic waves is generated in a dimension-breaking bifurcation. The key step of this approach is the analysis of the purely imaginary spectrum of the linear operator obtained by linearizing the water-wave equations at a periodic traveling wave. Transverse linear instability is then obtained by a perturbation argument for linear operators, and the nonlinear bifurcation problem is studied with the help of a center-manifold reduction and the classical Lyapunov center theorem.

Pseudospectral methods for computing the multiple solutions of the Lane–Emden equation

Based on the Liapunov–Schmidt reduction and symmetry-breaking bifurcation theory, we compute and visualize multiple solutions of the Lane–Emden equation on a square and a disc, using Legendre and Fourier–Legendre pseudospectral methods. Starting from the nontrivial solution branches of the corresponding nonlinear bifurcation problem, we obtain multiple solutions of Lane–Emden equation with various symmetries numerically. Numerical results demonstrate the effectiveness of these approaches.

Local structure of bifurcation curves for nonlinear Sturm–Liouville problems

We consider the nonlinear bifurcation problem arising in population dynamics and nonlinear Schrödinger equation: − u ″ ( t ) = f ( λ , u ( t ) ) , u ( t ) > 0 , t ∈ I : = ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 , where λ > 0 is a parameter. We mainly treat the case where f ( u ) = λ u ± u p ( p > 1 ) and establish the precise asymptotic expansion formulas for the bifurcation curve near the bifurcation point λ = π 2 in L q -framework. Together with the result of the global behavior of the bifurcation curve, we understand completely the structure of the bifurcation curve. We also consider the nodal solution u n , λ of the equation − u ″ ( t ) = λ ( u ( t ) + | u ( t ) | p − 1 u ( t ) ) with u ( 0 ) = u ( π ) = 0 and establish an asymptotic expansion formula for λ with respect to the gradient norm of the solution associated with λ as λ → n 2 .

Symmetry and secondary bifurcation of elastic structures

We consider non-linear bifurcation problems for elastic structures modeled by the operator equation F[ w; α]=0 where F: X× R k → Y, X, Y are Banach spaces and X⊂ Y. We focus attention on problems whose bifurcation equations are of the form f i(α 1,α 2;λ,μ)=(a iμ+b iλ)α i+p iα i 3+q iα i∑ j=1,j≠i kα j+1 2+α ih i(λ,μ;α 1,α 2,…α k) i=1,2,…k which emanates from bifurcation problems for which the linearization of F is Fredholm operators of index 0. Under the assumption of F being odd we prove an important theorem of existence of secondary bifurcation. Under this same assumption we prove a symmetry condition for the reduced equations and consequently we got an existence result for secondary bifurcation. We also include a stability analysis of the bifurcating solutions.

Dual Extremum Principles in Finite Deformation Theory With Applications to Post-Buckling Analysis of Extended Nonlinear Beam Model

The critical points of the generalized complementary energy variation principles are clarified. An open problem left by Hellinger and Reissner is solved completely. A pure complementary energy (involving the Kirchhoff type stress only) is constructed, and a complete duality theory in geometric nonlinear system is established. We prove that the well-known generalized Hellenger-Reissner’s energy L(u,s) is a saddle point functional if and only if the Gao-Strang gap function is positive. In this case, the system is stable and the minimum potential energy principle is equivalent to a unique maximum dual variational principle. However, if this gap function is negative, then L(u,s) is a so-called ∂+-critical point functional. In this case, the system has two extremum complementary principles. An interesting trinity theorem for nonconvex variational problem is discovered, which can be used to study nonlinear bifurcation problems, phase transitions, variational inequality, and other things. In order to study the shear effects in frictional post-buckling problems, a new second order 2-D nonlinear beam model is developed. Its total potential is a double-well energy. A stability criterion for post-buckling analysis is proposed, which shows that the minimax complementary principle controls a stable buckling state. The unilaterial buckling state is controled by a minimum complementary principle. However, the maximum complementary principle controls the phase transitions.

Duality Theory in Nonlinear Buckling Analysis for von Kármán Equations

The nonlinear eigenvalue problem in buckling analysis is studied for von Kármán plates. By using the general duality theory developed by Gao‐Strang [1, 2] it is proved that the stability criterion for the bifurcated state depends on a reduced complementary gap function. The duality theory is established for nonlinear bifurcation problems. This theory shows that the nonlinear eigenvalue problem is eventually equivalent to a coupled quadratic dual optimization problem. A series of equivalent variational principles are constructed and a lower bound theorem for the first eigenvalue of the buckling factor is proved.

Existence and uniqueness of equilibrium states of a rotating elastic rod

A flexible rod is rotated from one end. The equilibrium equation is a fourth order nonlinear two‐point boundary value problem which depends on two parameters λ and α representing the importance of centrifugal effects to flexural rigidity and the angle between the rotation axis and the clamped end, respectively. Previous studies on the existence and uniqueness of solution of the equilibrium equation assumed α = 0. Among the findings of these studies is the existence of a critical value λc beyond which the uniqueness of the trivial solution is lost. The computations of λc required the solution of a nonlinear bifurcation problem. On the other hand, this work is concerned with the existence and uniqueness of solution of the equilibrium equation when α ≠ 0 and in particular in the computations of a critical value λc such that the equilibrium equation has a unique solution for each α ≠ 0 provided λ < λc. For small α ≠ 0 this requires the solution of a nonlinear perturbed bifurcation problem.

Singular Perturbations of Bifurcations with Multiple Independent Bifurcation Parameters

There are several nonlinear bifurcation problems which involve multiple bifurcation parameters (“specified inputs”). Often there is a corresponding “perturbed” problem associated with the bifurcation problem which models imperfections. Instead of bifurcation, the perturbed problem exhibits a smooth but rapid transition in the critical range of the bifurcation parameter. A new generalization of the single bifurcation parameter method of Matkowsky and Reiss, SPB (“Singular Perturbations of Bifurcations”) [SIAM J. Appl. Math., 33 (1977), pp. 230–255] is employed to solve the problem of the slightly crooked rotating elastica (shaft) subject to a dead load applied at the ends.

Iterations, perturbations and multiplicities for nonlinear bifurcation problems

Iterations, perturbations and multiplicities for nonlinear bifurcation problems