Homoclinic orbit and a particle in a double well The physical set-up is a classical particle in a parabolic double well:

Physically, a particle with reasonable amount of potential energy would be able to roll down the slope of the well, roll past the equilibrium point, then up the slope on the opposite side. This oscillation continues until the energy of the particle is overcome by friction or some external energy. 
I found the following definitions of a homoclinic orbit:


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*Qualitative: Trajectories that start and end at the same fixed point are called homoclinic orbits.

*Formal: Suppose $\dot{x}=f(x)$. Suppose there is a fixed point $x=x_{0}$, then a solution $\Psi(t)$ is a homoclinic orbit if $\Psi(t) \rightarrow x_{0} \space as \space t\rightarrow \pm \infty$.
The phase portrait for this physical system describes the oscillation of the particle described above – that is, the particle oscillating between extreme ends of a double well – to be a homoclinic orbit.
However, in the physical set-up, the trajectory of the particle – or oscillation – does not begin at an equilibrium point.
Clearly, if the particle would to begin at either equilibrium point at the bottom of a well, then, classically, all other regions are inaccessible to the particle. 
If it begins at the middle equilibrium point, it certainly would not have sufficient energy to move from one well to the another. It would be confined to either potential well that it falls into. 
I’m hoping someone would provide some insight as to how I can merge the physical idea with the phase portrait. As it stands, the formal definition seems to clash with the qualitative definition of homoclinic orbit.
Given the physical set-up above, if the particle begins at any equilibrium point in the double well, it couldn’t oscillate between the extreme ends of the double well. 
The only way for the particle to oscillate between extreme end of the double well is for the particle to begin rolling from either extreme ends of the slope of the double well. 
But the qualitative definition of homoclinic orbit restricts this.
Yet, however, the phase portrait of this physical system describes a homoclinic orbit.
 A: 
The phase potrait for this physical system describes the oscillation of the particle described above to be a Homoclinic orbit-that is, the particle oscillating between extreme ends of a double well.

No, not even close.
I assume that your misunderstanding originates from confusing phase space and geometrical space.
Suppose, we describe the dynamical system by means of position and velocity, i.e., $x=(y,\dot{y})$, with $y$ denoting the position of our particle and further suppose that the position of the fixed point is $y=0$. Both assumptions are without loss of generality. Without friction, the phase space of this system qualitatively looks like this (different colours correspond to different particle energies):



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*What you presumed to be a homoclinic orbit, corresponds to the outer, blue trajectory;

*purple trajectories are trajectories of a particle whose total energy is that of a resting particle just on the central hill;

*green are some trajectories of particles who cannot over overcome the potential hill in the middle;

*the red crosses mark fixed points in phase space:


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*The particle is in standstill at the bottom of left well;

*the particle is in standstill at the local maximum between the two wells;

*the particle is in standstill at the bottom of right well.



The crucial thing that you seem to miss is that a fixed point in phase space requires the particle to stand still. A particle that can pass the middle local maximum never was in standstill at one of the fixed points, even for $t→−∞$.
As only the second fixed point features instability, any possible proper homoclinic orbit in this system must emerge from it. And indeed, the purple orbits comply with the criteria for a homoclinic orbit in both senses:


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*This trajectory starts and ends at the second fixed point.

*$Ψ→(0,0)$ (the second fixed point) as $t→±∞$. For typical shapes of the potential, you actually need the infinity here. For something like Norton’s dome, the fixed point is actually reached in finite time.


If we have friction, then there are no proper homoclinic orbits in this system, since a particle on any proper orbit is bound to lose energy and thus cannot return to its origin in phase space. For the same reason, you can never have homoclinic orbits when you add friction to a classical system.
