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Different types of lattice spin systems with competing interactions have rich and interesting phase diagrams. In this study we present some new results for such systems involving the Ising spin system (i.e. sigma = +/- 1) using a generalization of the Cayley tree-like lattice approximation. We study the phase diagrams for the Ising model on a Cayley tree-like lattice, a new lattice type called pentagonal chandelier, with competing nearest-neighbor interactions J(1), prolonged next-nearest-neighbor interactions J(p) and one-level next-nearest-neighbor senary interactions Jt(2)((6)). The colored phase diagrams contain some multicritical Lifshitz points that are at nonzero temperature and many modulated new phases. We also investigate the variation of the wave vector with temperature in the modulated phase and the Lyapunov exponent associated with the trajectory of the system.

The Competing Interactions on a Cayley Tree-Like Lattice: Pentagonal Chandelier