We classify the local bifurcations of quasi-periodic d-dimensional tori in maps (abbr. MTd) and in flows (abbr. FTd) for d ≥ 1. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MTd can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark-Sacker bifurcations. Furthermore, normal bifurcations of FTd can be classified into three classes: saddle-node, double covering, and Neimark-Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MTd and FTd: namely, saddle-node cycle and heteroclinic cycle bifurcations of the (d − 1)-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST1) and zero-dimensional tori in sections (abbr. ST0). The bifurcations of ST1 can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and NeimarkSacker bifurcations. The bifurcations of ST0 can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark-Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST1/ST0 and the bifurcations of MTd/FTd. We present examples of all of these bifurcations.

In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincaré section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability of the periodic orbit. Analogously, a quasi-periodic orbit (2-torus) becomes an invariant closed curve (ICC) on a Poincaré section. From the Lyapunov exponents of an ICC, we can determine the time average of the exponential divergence rate of the orbit, which corresponds to the eigenvalues of a fixed point. We denote the Lyapunov exponent with the smallest nonzero absolute value as the Dominant Lyapunov Exponent (DLE). A local bifurcation manifests as a crossing or touch of the DLE locus with zero. However, the type of bifurcation cannot be determined from the DLE. To overcome this problem, we define the Dominant Lyapunov Bundle (DLB), which corresponds to the dominant eigenvectors of a fixed point. We prove that the DLB of a 1-torus in a map can be classified into four types: A+ (annulus and orientation preserving), A- (annulus and orientation reversing), M (Möbius band), and F (focus). The DLB of a 2-torus in a flow can be classified into three types: A+ × A+, A- × M (equivalently M × A- and M × M), and F × F. From the results, we conjecture the possible local bifurcations in both cases. For the 1-torus in a map, we conjecture that type A+ and A- DLBs correspond to a saddle-node and period-doubling bifurcations, respectively, whereas a type M DLB denotes a double-covering bifurcation, and type F relates to a Neimark–Sacker bifurcation. Similarly, for the 2-torus in a flow, we conjecture that type A+ × A+ DLBs correspond to saddle-node bifurcations, type A- × M DLBs to double-covering bifurcations, and type F × F DLBs to the Neimark–Sacker bifurcations. After introducing the mathematical concepts, we provide a DLB-calculating algorithm and illustrate all of the above bifurcations by examples.

This study analyzes an Arnold resonance web, which includes complicated quasi-periodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a two-dimensional invariant torus (IT), which corresponds to a three-dimensional torus in vector fields. Numerous one-dimensional invariant closed curves (ICCs), which correspond to two-dimensional tori in vector fields, exist in a very complicated but reasonable manner inside an IT-generating region. Periodic solutions emerge at the intersections of two different thin ICC-generating regions, which we call ICC-Arnold tongues, because all three independent-frequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICC-Arnold tongues through a Neimark-Sacker bifurcation in the neighborhood of a quasi-periodic Hopf bifurcation (or a quasi-periodic Neimark-Sacker bifurcation) boundary.

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We develop two algorithms for obtaining an index 1 saddle torus between two attractors of which basin boundary is smooth by using the bisection method. In the first algorithm, which we name "overall template method", we make a template approximating a whole attractor and use it for pattern matching. The attractor can be a periodic orbit, a torus and even a chaos. In the second algorithm, which we name "partial template method", we make a template approximating only a part of an attractor. In this algorithm, the attractor is restricted to a periodic orbit and a two-torus, but calculation time is much smaller than that of the overall template method. We demonstrate these two algorithms for two solutions, observed in a ring of six-coupled bistable oscillators. One is a saddle torus in the basin boundary of the two switching solutions, and the other is of the two quasi-periodic propagating wave solutions. We calculate the Lyapunov spectrum of the obtained saddle torus, and confirm one positive Lyapunov exponent.