We all know that that the curve given by $\gamma (t)=(t,\sin t)$ has a repeated pattern, even though it's not a periodic curve. Can we generalize somehow the definition of a periodic curve so that curves like the one above are included?
When can we say that $\gamma (t)=(x(t), y(t))$ is 'periodic'? I found it good for $x'$ and $y$ to be periodic, with the same period. Is that the most general result that can be foud?
On
$\newcommand{\Reals}{\mathbf{R}}$The example $t \mapsto (t, \sin t)$ falls under the umbrella of equivariant mappings, along the lines mentioned by Taladris.
To start with an overly-general definition: Let $G$ be a group, and let $X$ and $Y$ be topological spaces equipped with left actions of $G$. (Concretely, for each group element $g$, there is a homeomorphism $\phi_{g}:X \to X$, and $\phi_{gh} = \phi_{g} \circ \phi_{h}$ for all $g$ and $h$ in $G$. Particularly, $\phi_{e}$ is the identity map of $X$, and $\phi_{g^{-1}} = (\phi_{g})^{-1}$ for all $g$ in $G$. The space $Y$ is equipped similarly with homeomorphisms $\psi_{g}$.)
A continuous mapping $f:X \to Y$ is equivariant (with respect to the group actions) if $f \circ \phi_{g} = \psi_{g} \circ f$ for all $g$ in $G$, namely, if $$ f\bigl(\phi_{g}(x)\bigr) = \psi_{g}\bigl(f(x)\bigr) \quad\text{for all $x$ in $X$ and all $g$ in $G$.} \tag{1} $$
Let $G$ be the additive group of integers, $X = \Reals$ the Euclidean line, and $Y = \Reals^{n}$ for some $n \geq 1$. Fix a positive real number $\ell$. Let $G$ act on $X = \Reals$ through translation $\phi_{n}(x) = x + n\ell$, and on $Y = \Reals^{n}$ by the identity transformation. An equivariant mapping $f:\Reals \to \Reals^{n}$ is precisely an $\ell$-periodic mapping; condition (1) says $f(x + n\ell) = f(x)$ for all $x$ in $\Reals$.
If in the preceding item $G$ acts on $\Reals^{n}$ by translation $\psi_{n}(y) = y + n\bigl(f(\ell) - f(0)\bigr)$, then $f$ is equivariant if and only if $$ \underbrace{f(x + n\ell)}_{(f \circ \phi_{n})(x)} = \underbrace{f(x) + n\bigl(f(\ell) - f(0)\bigr)}_{(\psi_{n} \circ f)(x)} \quad\text{for all real $x$.} \tag{2} $$ The mapping $f:\Reals \to \Reals^{2}$ defined by $f(x) = (x, \sin x)$ satisfies (2) for $\ell = 2\pi$, since $f(2\pi) - f(0) = (2\pi, 0)$, and $$ f(x + 2\pi n) = f(x) + (2\pi n, 0)\quad\text{for all real $x$.} $$
The action on the target space $Y$ need not be through translation. Fix an integer $k \geq 2$. If the additive group of integers acts on $X = \Reals$ by translation through multiples of $\ell = 2\pi/k$ and acts on $Y = \Reals^{2}$ by rotations through multiples of $2\pi/k$, i.e., by $\psi_{n}(r, \theta) = (r, \theta + 2\pi n/k)$ in polar coordinates, then $f(x) = \cos(kx)(\cos x, \sin x)$ (which traces a rose curve) is equivariant: $$ f(x + \ell) = \cos(kx)\bigl(\cos(x + \ell), \sin(x + \ell)\bigr) \quad\text{for all real $x$,} $$ and the right-hand side is the result of rotating the plane through an angle $\ell = 2\pi/k$.
Let $k > 1$ be a real number. If the additive group of integers acts on $X = \Reals$ through translation by multiples of $2\pi$ and acts of $Y = \Reals^{2}$ by radial scaling with factor $k$, i.e., $\psi_{n}(y_{1}, y_{2}) = k^{n}(y_{1}, y_{2})$, then the logarithmic spiral $f(x) = k^{x/(2\pi)}(\cos x, \sin x)$ is equivariant: $$ f(x + 2\pi) = k^{(x + 2\pi)/(2\pi)}\bigl(\cos(x + 2\pi), \sin(x + 2\pi)\bigr) = kf(x)\quad\text{for all real $x$.} $$
The group $G$ need not be discrete. If $k$ is a real number, and if $G$ is the additive group of reals, acting on itself by translation $\phi_{t}(x) = x + t$, and acting on $Y = \Reals^{3}$ in cylindrical coordinates by $$ \psi_{t}(r, \theta, z) = (r, \theta + t, z + kt), $$ then the circular helix $f(x) = (\cos x, \sin x, kx)$ is equivariant. (This example is closely related to why threaded screws and bolts work. The threads "enforce" a particular group action of the preceding type. A screwdriver or wrench impels rotation about an axis, and the group action forces the fastener to undergo translational motion along its axis.)
If $G$ is the multiplicative group $\{1, -1\}$, and if $X = Y = \Reals$, we may define a $G$-action $\phi = \psi$ on $X = Y$ by $\phi_{g}(x) = gx$. An equivariant function $f:\Reals \to \Reals$ is precisely an odd function, satisfying $$ f(-x) = -f(x)\quad\text{for all real $x$.} $$ If instead $G$ acts "trivially" on $Y$, i.e., by the identity, then an equivariant function is even. (At this point it should be noted how easy it is to make the simplest idea complicated, or conversely, to point out that the most sophisticated mathematical notions grow from elementary roots.)
These examples barely scratch the surface of equivariance. (In each, the group $G$ is Abelian, and the domain and target are Euclidean spaces.)
You may say that a curve is "periodic with respect to a vector $v$" if the function $t \mapsto \langle \gamma (t), v \rangle$ is periodic. In your example, choose $v = (0, 1)$.
A periodic curve (in the usual sense), then, would be one with the following property: there exist $T$ (the principal period) such that for every $v$ there exist $n_v \in \mathbb N$ making the curve $\gamma$ periodic with respect to $v$ with period $n_v T$. (Just use the vectors of the basis to get that all the components are periodic.)