I am looking for simple examples of structural stability, I read the definition of structural stability but couldn't figure out a concrete example of a system, its perturbated version and its conjugacy. The definition that I am using is from Wikipedia.
"Structural stability means that the qualitative behavior of the trajectories is unaffected by small perturbations ($C^1$-small)."
Moreover, what does it mean a $C^1$-small perturbation?
Anosov diffeomorphisms (= diffeomorphisms whose hyperbolic set is the whole state space) are the classical examples of structurally stable diffeomorphisms (the reason why such diffeomorphisms are named after Anosov is precisely because he was able to prove that indeed they are structurally stable (Anosov himself called Anosov diffeomorphisms $\mathcal{C}$-diffeomorphisms or $\Upsilon$-diffeomorphisms)). This means that if $M$ is a compact $C^\infty$ manifold and $f:M\to M$ is a $C^1$ diffeomorphism that is Anosov, then for any other $C^1$ diffeomorphism $g:M\to M$ with the property that
$$\forall p\in M: g(p)\approx f(p), g'(p)\approx f'(p),$$
there is a homeomorphism $\Phi_g:M\to M$ (called a topological conjugacy, see Examples of conjugate-like structures across mathematics) with the property that
$$\forall p\in M:\Phi_g(p)\approx p \text{ and } \Phi_g\circ f(p)=g\circ\Phi_g(p).$$
(See the discussion at Hirsch's Differential Topology vs Rudin Functional analysis definition of weak and strong topology. for a more rigorous account of the foundations of $C^1$-small perturbations.)
It is somewhat hard to write down perturbations explicitly, but here is a somewhat explicit example. Consider the toral automorphism $f:\mathbb{T}^2\to \mathbb{T}^2, (x,y)\mapsto (2x+y,x+y)$. Note that the derivative of $f$ at any point $(x,y)$ is
$$ f'(x,y)=T_{(x,y)}f=\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}. $$
As I said in my answer to another question of yours (Examples of hyperbolic sets of dynamical systems) $f$ is Anosov. Say $g:\mathbb{T}^2\to \mathbb{T}^2$ is a sufficiently $C^1$-small perturbation of $f$. This means that $g$ is of the form
$$g(x,y)=(2x+y+\alpha(x,y),x+y+\beta(x,y))$$
for some functions $\alpha,\beta: \mathbb{R}^2\to \mathbb{R}$, where the additions are interpreted as modulo integers. Since $g$ is on the torus and is a homeomorphism, $\alpha$ and $\beta$ have to be periodic. The derivative of $g$ at any point $(x,y)$ is of the form
$$ g'(x,y)=T_{(x,y)}g= \begin{pmatrix} 2+\dfrac{\partial \alpha}{\partial x}(x,y) & 1+\dfrac{\partial \alpha}{\partial y}(x,y) \\ 1+\dfrac{\partial\beta}{\partial x}(x,y) & 1+\dfrac{\partial\beta}{\partial y}(x,y) \end{pmatrix}. $$
Note that $f(x,y)-g(x,y)=(\alpha(x,y),\beta(x,y))$ and
$$ f'(x,y)-g'(x,y) =\begin{pmatrix} \dfrac{\partial \alpha}{\partial x}(x,y) & \dfrac{\partial \alpha}{\partial y}(x,y) \\ \dfrac{\partial\beta}{\partial x}(x,y) & \dfrac{\partial\beta}{\partial y}(x,y ) \end{pmatrix}. $$
Thus since $g$ is a small enough $C^1$-perturbation we must have that
\begin{align*} \forall (x,y)\in\mathbb{T}^2: &|\alpha(x,y)|\approx0,\\ &|\beta(x,y)|\approx0,\\ &|\alpha_x(x,y)|\approx0,\\ &|\alpha_y(x,y)|\approx0,\\ &|\beta_x(x,y)|\approx0,\\ &|\beta_y(x,y)|\approx0. \end{align*}
One (but not the only) way to write such $\alpha$ and $\beta$ explicitly would be to take a a small disk $D$ around a point, e.g. $(x_0,y_0)=(1/2,1/2)\in\mathbb{T}^2$ with radius $r_0<10^{-10^{10^{10^{10}}}}$ , and define $\alpha,\beta$ as functions that vanish outside $D$. For a $C^1$ small perturbation, the heights of $\alpha$ and $\beta$ can't be too large, and the slopes of their graphs can't be too steep; thus their graphs should look like pleasant bumps. (In this case since our Anosov diffeomorphism is algebraic one can be very specific on what "too large" and "too steep" mean by analyzing Anosov's proof, but in general such an analysis would be more complicated.)
Structural stability says then that even when $\alpha$ and $\beta$ are anonymous (but small enough with small enough derivatives), the dynamical behavior of $g$ will be the same as the dynamical behavior of $f$, up to a topological coordinate change. That is to say, there is a homeomorphism
$$\Phi(x,y)=(\phi^1(x,y),\phi^2(x,y))$$
such that $\Phi\circ f= g\circ \Phi$. I'm leaving it to you to expand this final equation in terms of the coordinatewise formulas and obtain two equations involving $\alpha,\beta,\phi^1,\phi^2$.
A slightly simpler example of a stable dynamical system is given by expanding maps on the circle.
An expanding map on the circle is a $C^1$ (continuous with continuous derivative) map $f\colon \mathbb{S}^1 \rightarrow \mathbb{S}^1$ with $f'(z) > 1$ for every $z$. These maps are, in particular, local diffeomorphisms and indeed covering maps of finite degree. Here $\mathbb{S}^1$ denotes the unit circle $\{z \in \mathbb{C} \colon |z| = 1\}$. The simplest example is $f(z) = z^2$, called the angle-doubling map, since it doubles the argument of a complex number: $f(e^{i\theta}) = e^{i2\theta}$.
However, it is easier to view $\mathbb{S}^1$ as $\mathbb{R}/\mathbb{Z}$, that is, identifying real numbers whose difference is an integer. In this case, if we have a function $F\colon \mathbb{R} \rightarrow \mathbb{R}$ such that:
(1) $F$ is $C^1$,
(2) $F'(x) > 1$ for every $x$,
(3) There exists some $d \in \mathbb{N} \setminus \{0,1\}$ such that $$ F(x+1) = F(x) + d $$ for every $x \in \mathbb{R}$,
then $F$ induces a map $f\colon \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$ that is an expanding map on the circle. Every expanding map on the circle can be constructed in this way. The number $d$ is the topological degree of $f$, that is, $\#f^{-1}(x) = d$ for every $x \in \mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$. In particular expanding maps on the circle are not diffeomorphisms but smooth endomorphisms.
It turns out that $d$ is a complete topological invariant in the class of expanding maps on the circle: if $f$ and $g$ are expanding maps with the same topological degree $d$, then they are topologically conjugate. That means there exists a homeomorphism $h\colon \mathbb{S}^1 \rightarrow \mathbb{S}^1$ such that $$ h \circ f = g \circ h. $$ Note that $h$ is not, in general, of class $C^1$, but that's a different topic...
Since the class of expanding maps on the circle is stable under $C^1$ perturbations (and so is the topological degree), it follows that every $C^1$ map $g\colon \mathbb{S}^1 \rightarrow \mathbb{S}^1$ sufficiently close to an expanding map $f$ in the $C^1$ topology is also an expanding map with the same degree. Hence, $g$ is conjugate to $f$. In fact, the conjugacy $h$ becomes closer to the identity map as the perturbation becomes smaller. That means $f^n(x)$ stays uniformly close to $g^n(h(x))$ for all $n \geq 0$, so their orbits are nearly indistinguishable if $g$ is very close to $f$ in the $C^1$ topology.
One advantage of this example is that it's easier to explain what we mean by "$C^1$ close." If $W\colon \mathbb{R} \rightarrow \mathbb{R}$ is a $C^1$ function such that $W(x+1) = W(x)$ (that is, $W$ is periodic with period $1$), then for every $F$ satisfying (1)-(2)-(3), the function $F+W$ also satisfies (1) and (3). That implies that $F+W$ also induces a $C^1$ map $g\colon \mathbb{S}^1 \rightarrow \mathbb{S}^1$. Moreover, if the $C^1$ norm of $W$ is small, that is, $$ \|W\|_{C^1} = \max_{x \in \mathbb{R}} |W(x)| + \max_{x \in \mathbb{R}} |W'(x)| $$ is small enough, then $F+W$ also satisfies (2), so the induced map $g$ is also an expanding map on the circle. That is what we mean when we say "$g$ is a small $C^1$ perturbation of $f$."
We finish with a concrete example. If we pick $F(x) = 2x$, then the induced map on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$ is the angle-doubling map. Now consider the perturbation $W_\epsilon(x) = \epsilon \cos(2\pi x)$. If $\epsilon$ is small enough, then $F + W_\epsilon$ still satisfies the required conditions and induces a degree $2$ expanding map on the circle, which is topologically conjugate to the angle-doubling map.
Things get more complicated when defining $C^1$ perturbations on manifolds that are not as simple as the circle or the torus (see Whitney topologies on the space of $C^r$ maps on manifolds), but it is fair to say that the idea is quite similar. We can also define expanding maps on more complicated manifolds. The study of expanding maps on manifolds and their structural stability began with the Ph.D. thesis of Michael Shub (1967).
