Global Continua Research Articles

Bifurcation from zero or infinity in nonlinearizable Sturm–Liouville problems with indefinite weight

In this paper, we consider bifurcation from zero or infinity of nontrivial solutions of the nonlinear Sturm–Liouville problem with indefinite weight. This problem is mainly important because of it is related with a selection-migration model in genetic population. We show the existence of four families of unbounded continua of nontrivial solutions to this problem bifurcating from intervals of the line of trivial solutions or the line R × {∞} (these intervals are called bifurcation intervals). Moreover, these global continua have the usual nodal properties in some neighborhoods of bifurcation intervals.

Path-connectedness in global bifurcation theory

<p style='text-indent:20px;'>A celebrated result in bifurcation theory is that, when the operators involved are compact, global connected sets of non-trivial solutions bifurcate from trivial solutions at non-zero eigenvalues of odd algebraic multiplicity of the linearized problem. This paper presents a simple example in which the hypotheses of the global bifurcation theorem are satisfied, yet all the path-connected components of the connected sets that bifurcate are singletons. Another example shows that even when the operators are everywhere infinitely differentiable and classical bifurcation occurs locally at a simple eigenvalue, the global continua may not be path-connected away from the bifurcation point. A third example shows that the non-trivial solutions which bifurcate at non-zero eigenvalues, irrespective of multiplicity when the problem has gradient structure, may not be connected and may not contain any paths except singletons.</p>

Some global results for nonlinear fourth order eigenvalue problems

AbstractIn this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.

Continua of Cycles of Higher-Order Equations

where μj ≥ 0 and μ0 + · · · + μ −1 > 0. This estimate is a natural analog of the ordinary two-sided sector estimate |f(x)| ≤ q|x|. By virtue of (3), zero is an equilibrium of Eq. (1) for each λ. Our main theorems guarantee the existence of a continuum of cycles issuing from zero and going to infinity. (A rigorous definition is given in the following section.) The theorems contain assumptions about the polynomial (2) that guarantee the existence of such a continuum of cycles for Eq. (1) with an arbitrary continuous nonlinearity satisfying the estimate (3) for q < q0, where q0 is determined by the polynomial (2) and the numbers μj. We suggest a constructive method for the computation of q0, which can readily be implemented on a computer. In the examples given in the paper, MAPLE has been used to perform the computations. We point out that the theorems on global continua of cycles apply simultaneously to the entire class of equations (1) with common linear part and common coefficients q and μj in the estimate (3) of the continuous nonlinearity. No information on the differentiability of the nonlinearity at any point is required. Our method can also be used in the analysis of local continua of cycles. The main classical example of the existence of such continua is the theorems on Andronov–Hopf bifurcations (e.g., see [1] and the bibliography therein). The known theorems are based on the linearization of the equation around the equilibrium; information on the linearization permits one to prove the existence of a continuum of cycles in a small neighborhood of this point. Theorems based on the differentiability of the nonlinearity only at the equilibrium were stated for the first time in [2]. Similar theorems are valid for bifurcations at infinity [3]. In the present paper, we neither assume the differentiability nor use linearizations. The main results of [2, 3] for higher-order equations readily follow from the theorems given here. In conclusion, we present results that can be treated as conditions for the existence of cycles in situations where the nonlinearity is known only approximately. Here we prove the existence of a continuum of cycles in the domain lying between two concentric spheres; the equilibrium, whose exact position is not known, is localized in the ball bounded by the inner sphere. Although the stability of cycles is important, related issues are not discussed in the present paper. Their analysis requires additional information about the nonlinearities.

Existence of Continua of Cycles in Hamiltonian Control Systems

Simple conditions for the existence of global continua of cycles in autonomous Hamiltonian single-loop control systems are derived from ordinary linear two-sided nonlinearity estimates. Limiting bounds of sector estimates are determined by the properties of the linear part. The designed methods are useful in determining two-sided estimates for the period of a cycle in a sector.

Unbounded Branches of Classical Injective Solutions to the Forced Displacement Problem in Nonlinear Elastostatics

We consider classical solutions of the equilibrium equations of nonlinear elastostatics under prescribed boundary displacements and body forces, both of which depend on a loading parameter. We employ recent results on global continuation [HS] that establish general existence of global continua of solutions in nonlinear elastostatics. Under reasonable hypotheses on the stored energy function we prove the existence of an unbounded branch of globally injective solutions to the boundary value problem.

Positive solutions of systems of nonlinear eigenvalue problems

the real-valued functions a:’ = a$), a$, rz,) 2 0, 1 5 i, j 5 N, 1 I k 5 r are Lipschitz continuous on 0 and the matrix (a?‘), 1 I k 5 r, is positive definite on a. The operators Lk are uniformly elliptic. In the case of a single equation, r = 1, Ambrosetti and Hess, [2], have proved that the bifurcation from infinity occurs when the nonlinearity, f, is asymptotically linear and its asymptotic derivative positive. Peitgen and Schmitt, [8, 91, have obtained similar results for nonlinearities that satisfy more general asymptotic boundedness conditions. We extend these results to the system (1. l), r L 1. In Section 3 we prove that the bifurcation from infinity occurs at some A*, when f is asymptotically linear and its asymptotic derivative, M(x) = (m,(X)), satisfies the following conditions: mij 2 0, i # j, and for at least one i E [ 1, . . . , r), rn: # 0. (If we look at the linearized system (at co), these are the conditions that guarantee the existence of a positive eigenvalue with a positive eigenfunction, a result due to Hess, [5].) Our result is analogous to that of Hess, [5, theorem 21, treating the case of bifurcation from the trivial solution, and we obtain the same kind of estimates for A *. We use global perturbation techniques, employed by Peitgen and Schmitt, [8, 91, for the case of a single equation. This method enables us to also trace these bifurcations via global continua of solutions of a perturbed problem.