Narrow Results

We consider two-parameter families of reversible vector fields having (at the critical parameter value) a homoclinic orbit to a nonhyperbolic fixed point. The nonhyperbolicity is due to a pair of purely imaginary eigenvalues. We give a complete description of the bifurcating one-homoclinic orbits to the center manifold. For that purpose we adapt Lin's method.

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

In this paper, we present four limit cycles in quadratic near-integrable polynomial systems. It is shown that when a quadratic integrable system has two centers and is perturbed by quadratic polynomials, it can generate at least four limit cycles with (3,1)-distribution. This result provides a positive answer to an open question in this research area.

We prove the existence of symmetric homoclinic orbits to a saddle-focus symmetric periodic orbit that appears in a generic family of reversible three degrees of freedom Hamiltonian system due to periodic Hamiltonian Hopf bifurcation, if some coefficient A of the normal form of the fourth order is positive. If this coefficient is negative, then for the opposite side of the bifurcation parameter value, we prove the existence of symmetric homoclinic orbits to saddle invariant 2-tori.

A codimension-three Takens–Bogdanov bifurcation in reversible systems has been very recently analyzed in the literature. In this paper, we study with the help of the nonlinear time transformation method, the codimension-one and -two homoclinic and heteroclinic connections present in the corresponding unfolding. The algorithm developed allows to obtain high-order approximations for the global connections, in such a way that it supplies in a very efficient manner the coefficients that would be obtained with high-order Melnikov functions. As we show, all our analytical predictions have excellent agreement with the numerical results. In particular we remark that, for the two different codimension-two points, the theoretical approximation coincides in six decimal digits with the numerical continuation, even being quite far from the codimension-three point. The better approximations we provide in this work will help in the study of reversible systems that exhibit this codimension-three Takens–Bogdanov bifurcation.

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder $O(2)\times \mathrm{\Gamma }\times {\mathbb{Z}}_2$-equivariant degree theory. Here, $O(2)$ is related to the reversal symmetry combined with the autonomous form of the system, $\mathrm{\Gamma }$ reflects extra spacial symmetries of the system, and ${\mathbb{Z}}_2$ is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with $\mathrm{\Gamma }=D_6$ — the dihedral group of order 12.

Stability of a linear autonomous non-conservative system in the presence of potential, gyroscopic, dissipative, and non-conservative positional forces is studied. The cases when the non-conservative system is close either to a gyroscopic system or to a circulatory one, are examined. It is known that marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. We show that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle”, “Break of an edge” and “Whitney's umbrella” that govern stabilization and destabilization as well as are responsible for the imperfect merging of modes. Sensitivity analysis of the critical parameters is performed with the use of the perturbation theory for eigenvalues and eigenvectors of non-self-adjoint operators. In case of two degrees of freedom, stability boundary is found in terms of the invariants of matrices of the system. Bifurcation of the stability domain due to change of the structure of the damping matrix is described. As a mechanical example, the Hauger gyropendulum is analyzed in detail; an instability mechanism in a general mechanical system with two degrees of freedom, which originates after discretization of models of a rotating disc in frictional contact and possesses the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is analytically described and its role in the excitation of vibrations in the squealing disc brake and in the singing wine glass is discussed.

We show that universal maps (i.e. such whose iterations approximate every possible dynamics arbitrarily well) form a residual subset in an open set in the space of smooth dynamical systems. The result implies that many dynamical systems emerging in natural applications may, on a very long time scale, have quite unexpected dynamical properties, like coexistence of many non-trivial hyperbolic attractors and repellers and attractors with all zero Lyapunov exponents. Applications to reversible and symplectic maps are also considered.