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 $\alpha,\beta,\phi^1,\phi^2$.
