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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.

We classify the local bifurcations of quasi-periodic $d$-dimensional tori in maps (abbr. MT$d$) and in flows (abbr. FT$d$) for $d\ge 1$. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT$d$ can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FT$d$ 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 MT$d$ and FT$d$: 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. ST$1$) and zero-dimensional tori in sections (abbr. ST$0$). The bifurcations of ST$1$ can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST$0$ 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 ST$1$/ST$0$ and the bifurcations of MT$d$/FT$d$. We present examples of all of these bifurcations.