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This paper studies the Neumann problem of a nonlocal Allen–Cahn equation in an interval. A main result finds a symmetry breaking (secondary) bifurcation point on the bifurcation curve of solutions with odd-symmetry. Our proof is based on a level set analysis for the associated integral map. A method using the complete elliptic integrals proves the uniqueness of secondary bifurcation point. We also show some numerical simulations concerning the global bifurcation structure. [ABSTRACT FROM AUTHOR]

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection — the saddle-foci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n -pulse heteroclinic tangencies — trajectories that follow the original cycle n times around before they arrive at the other node. Each n -pulse heteroclinic tangency is accumulated by a sequence of ( n + 1 ) -pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that any continuum of nontrivial solutions of a non-cooperative system of elliptic equations on the sphere S n − 1 , bifurcating from the set of trivial solutions, is unbounded. Moreover, we characterize bifurcation points of this system at which the global symmetry-breaking phenomenon occurs. As the main tool we use the degree for SO ( 2 ) -invariant strongly indefinite functionals defined in [13] . [ABSTRACT FROM AUTHOR]

Abstract: In this article we study the symmetry breaking phenomenon of solutions of non-cooperative elliptic systems. We apply the degree for -invariant strongly indefinite functionals to obtain simultaneously a symmetry breaking and a global bifurcation phenomenon. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: Our object of study is the dynamics that arises in generic perturbations of an asymptotically stable heteroclinic cycle in . The cycle involves two saddle-foci of different type and is structurally stable within the class of -symmetric vector fields. The cycle contains a two-dimensional connection that persists as a transverse intersection of invariant surfaces under symmetry-breaking perturbations. Gradually breaking the symmetry in a two-parameter family we get a wide range of dynamical behaviour: an attracting periodic trajectory; other heteroclinic trajectories; homoclinic orbits; n-pulses; suspended horseshoes and cascades of bifurcations of periodic trajectories near an unstable homoclinic cycle of Shilnikov type. We also show that, generically, the coexistence of linked homoclinic orbits at the two saddle-foci has codimension 2 and takes place arbitrarily close to the symmetric cycle. [Copyright &y& Elsevier]

Abstract: We demonstrate that the recent paper by Abhinav and Panigrahi entitled ‘Supersymmetry, PT-symmetry and spectral bifurcation’ [K. Abhinav, P.K. Panigrahi, Ann. Phys. 325 (2010) 1198], which considers two different types of superpotentials for the PT-symmetric complexified Scarf II potential, fails to take into account the invariance under the exchange of its coupling parameters. As a result, they miss the important point that for unbroken PT-symmetry this potential indeed has two series of real energy eigenvalues, to which one can associate two different superpotentials. This fact was first pointed out by the present authors during the study of complex potentials having a complex sl(2) potential algebra. [Copyright &y& Elsevier]

Abstract: In this paper we show the existence and bifurcation of T-periodic solutions of a special form for an autonomous Newtonian system with symmetry. If the phase-space is equipped with the structure of an orthogonal representation and the potential is invariant, then for every such a solution the set of indices of nonvanishing Fourier coefficients is finite and depends on only. If the potential V depends on the squares of complex coordinates, then for every such a solution T is the minimal period. [Copyright &y& Elsevier]

In this article we study global and symmetry-breaking bifurcations of solutions of Emden–Fowler-type elliptic differential equations. As the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals. Namely, we will apply degree for S1-equivariant orthogonal maps defined in Rybicki (1994, Nonlinear Anal TMA 23, 83–102). [Copyright &y& Elsevier]

Abstract: In “Comment on Supersymmetry, PT-symmetry and spectral bifurcation”, Bagchi and Quesne correctly show the presence of a class of states for the complex Scarf-II potential in the unbroken PT-symmetry regime, which were absent in . However, in the spontaneously broken PT-symmetry case, their argument is incorrect since it fails to implement the condition for the potential to be PT-symmetric: C PT [2(A − B)+ α]=0. It needs to be emphasized that in the models considered in , PT is spontaneously broken, implying that the potential is PT-symmetric, whereas the ground state is not. Furthermore, our supersymmetry (SUSY)-based ‘spectral bifurcation’ holds independent of the sl(2) symmetry consideration for a large class of PT-symmetric potentials. [Copyright &y& Elsevier]