Existence of Black hole radiation (through conservation) and Analysis of classification of black holes in classical dynamical system

Question

I was taking nonlinear dynamical system where I found a confusion with black hole. (The dependency here was with respect to single $t$ and space was "flat", in a sense that the metric does not carry curvature, think it as the simplest Malinowski metric $\mu_{00}=-\mu_{11}=-1$.)

In a conserved dynamical system, there could not be an attracting fixed point(the fixed point that all the trajectories close to the fixed point converge to the fixed point.), thus it made perfect sense that there must be some particles or energy that gets out. (Otherwise it would meant our universe was all in the same energy states.)

However, black hole was a physical "fixed point", then my question was what kind of fixed point was it? I remembered I looked a NOVA video that half of the stars that being sucked around a black hole disk would get out while the other half would get in. It looked to me that black hole looked strangely like a saddle point(In 2D a saddle point was a point with two invariant lines, the two lines was in a X cross through the fixed point where one line was attracting and the other one was repulsing).

1. If the black hole was a saddle point, it naturally arised the question that where was the other path that gets out of the black hole?

2. If not, there was another possibility that the black hole was not a bounded region. In the sense that beyond the event horizon there was a universe like topological structure.

3. However, black hole, in a sense, was an attracting fixed point, such conclusion was also valid if we assume the continuity was broken around black hole, which was valid because the existence of singularities. However, the question would be which conservative quantity no longer hold?

Asking for comments on the above analysis and questions.

*Due to some people's confusing I made the following two points more clear:

1.The whole thing was that we could get the fact that we knew there would be a form of "radiation" or what soever that emits particles away form the black hole. With or without the knowledge of quantum, gravity or even Newtonian.

2.The core argument here was simply the assumption that space was continuous and the universe we knew was a conserved system, nothing else. It doesn't matter which theory you use, as long as the physical theory agree with the above two assumptions which they all were, then you got the black hole radiation. Otherwise there must be something wrong with your theory, in which classical theory does.

Once we established the fact that the black was not an attracting fixed point, the really interesting part came from if we assume case 1 that the black hole was a saddle point.

a. If we assume the symmetry of the space, then there was no reason to believe or presume that the black hole repulse energy much shower than it absorb the energy. Then it came to our attention that there was a huge amount of energy that was missing while counting the emission, because there was supposed to be almost equal amount of energy emitted as that was absorbed by the black hole.

b. A more interesting case was given when the black hole was placed in the vacuum, because being a saddle point with huge amount of energy in the region meant there must be tremendous amount of interaction/commutation between black hole and surrounding environment.

This came up with two possible explanation:

1. was the fact we noticed one thing the black hole keeps "absorbing" was the trajectories of the particles. Meanwhile, by curving the space itself, black hole "created" more volume/length for the particles and trajectories to travel. We knew already that gravity cost energy, in a sense, by creating the curved space or maintain the gravitational files, what the black hole "emits" was actually the space itself, while it constantly absorbing space and particles, where hawking radiation came almost nature in such environment as the space wiggles. Thus, the energy was confined into "space flux" that was kept being absorbed and emitted by the black holes. Further, to keep the conservation of energy, the black hole essentially had to create more space or transmit the energy into gravitational wave or through gravity. Or

2. there could assume that there was some new form of energy that there were not yet able to observe.

Answer 1

I believe that OP's main problem is that he/she is trying to formulate a question about black holes in the language of dynamical systems without really knowing general relativity, so a general advice would be to have a look at some intro course in GR to gain the correct lingo.

As for the more specific questions. If we consider the dynamical system of a point mass in black hole space time, one thing we can notice is that system is not really conservative. On a geometric level this is because spacetime lacks a global timelike Killing vector field (KVF). Such vector field exists locally everywhere outside the horizon, so there is a conservation of energy for particle motion not crossing the horizon (read this answer by Ben Crowell for a simple intro on timelike KVF). In particular, it is possible to have a stable orbital motion around black hole, just like in case of Newtonian gravity. However once a material object enters the horizon there is no longer a conservative dynamical system to be had here so OP's attempt at classification is not applicable.

Another point:

I looked a NOVA video that half of the stars that being sucked around a black hole disk would get out while the other half would get in.

This is not due to the black hole, but due to stars interacting gravitationally with each other. With time some stars gain enough kinetic to escape, while remaining stars have lower average energy and thus are orbiting closer to each other. The black hole becomes relevant once enough energy is carried away by escaping stars so that remaining are close enough to start falling into it.