求ℝ⁴中存在非奇异非周期轨道γ且γ⊂ω(γ)的线性场实例

Great question! Let's break this down step by step to see why, and construct an explicit example that meets all your criteria.

First, let's align on key definitions to avoid confusion:

• Non-singular orbit: The orbit never touches the origin (i.e., ( \gamma(t) \neq 0 ) for all ( t \in \mathbb{R} ))
• Non-periodic orbit: There’s no non-zero ( T ) where ( \gamma(T) = \gamma(0) ) (the orbit never repeats itself)
• ω-limit set ( \omega(\gamma) ): The set of all points in ( \mathbb{R}^4 ) that are limits of some subsequence of ( \gamma(t) ) as ( t \to \infty )

Explicit Construction #

We can build this linear field using a matrix ( A ) that produces quasi-periodic orbits—these orbits fill out a torus without repeating, and their ω-limit set is the entire torus (which includes the orbit itself).

Take ( A ) as a block-diagonal 4x4 matrix with two 2x2 rotation blocks, where the rotation angles have an irrational ratio:

A = [[0, 1, 0, 0],
     [-1, 0, 0, 0],
     [0, 0, 0, π],
     [0, 0, -π, 0]]

This corresponds to the linear field ( F(x) = Ax ).

Now consider the orbit starting at ( x_0 = (1, 0, 1, 0) ):
[
\gamma(t) = (\cos t, \sin t, \cos(\pi t), \sin(\pi t))
]

Verifying the Conditions #

1. Non-singular: Every point on ( \gamma(t) ) has a norm of ( \sqrt{\cos^2 t + \sin^2 t + \cos^2(\pi t) + \sin^2(\pi t)} = \sqrt{2} \neq 0 ), so the orbit never hits the origin.
2. Non-periodic: Suppose there exists ( T \neq 0 ) where ( \gamma(T) = \gamma(0) ). This would require:
   • ( \cos T = 1 ) and ( \sin T = 0 ) → ( T = 2k\pi ) for some integer ( k )
   • ( \cos(\pi T) = 1 ) and ( \sin(\pi T) = 0 ) → ( \pi T = 2m\pi ) → ( T = 2m ) for some integer ( m )
     Combining these gives ( 2k\pi = 2m ), which implies ( \pi = m/k )—impossible since ( \pi ) is irrational. No such ( T ) exists.
3. ( \gamma \subset \omega(\gamma) ): The ω-limit set of ( \gamma(t) ) is the entire 2-torus ( \mathbb{T}^2 = { (\cos t, \sin t, \cos s, \sin s) \mid t, s \in \mathbb{R} } ). Since ( \gamma(t) ) lies entirely on this torus, every point of ( \gamma ) is contained in ( \omega(\gamma) ).

Why This Works #

Quasi-periodic orbits emerge when a linear system has purely imaginary eigenvalues with irrational frequency ratios. In this case, the orbit never repeats (non-periodic) and densely fills the torus—meaning every point on the torus is a limit point of the orbit, so the ω-limit set is the full torus. Since the orbit is part of that torus, it’s trivially a subset of its own ω-limit set.

Other cases (like eigenvalues with non-zero real part) fail: positive real parts send orbits to infinity (empty ω-limit set), negative real parts send orbits to the origin (ω-limit set is just {0}, which doesn’t contain the non-singular orbit). Only the quasi-periodic case meets all your requirements.

内容的提问来源于stack exchange，提问作者Pires Dankan