I have mathematical problem.
I want to know the constraint condition that oscillation occurs (differential equation).
For example,
dA/dt=-v1*A/(k1+A)+k2*B+vin-vout*A
dB/dt=v1*A/(k1+A)-k2*B
v1,k1,k2,vin,vout are free parameters.
I want to know constraint condition (equation) of these free parameters when oscillation occurs.
For a planar system such as the one you describe it is typical to look for parameter values for which a Hopf bifurcation occur.
Any good book on ODEs/dynamical systems should be able to provide more detail but the main idea is to locate the equilibria then linearise the system about these equilibria in the usual way and study the Jacobian.
For planar systems a Hopf bifurcation occurs (generally) when you get 2 purely imaginary eigenvalues. There are some other conditions too in the Hopf bifurcation theorem, such as crossing the imaginary axis with non-zero velocity, which are usually satisfied by "well-behaved" systems.
Finally the Hopf bifurcation theorem can be used in higher dimensional systems with similar conditions, but now you must have a pair of purely imaginary eigenvalues and all the other eigenvalues of the system must be strictly negative.
I can recommend the following paper which proficiently describes the Hopf bifurcation theorem:
Chapter New Numerical Methods for High Dimensional Hopf Bifurcation Problems
With respect to time, the oscillations will be observed when your system's Jacobian matrix have complex pair of eigen values, which is not the case in your defined problem.
For studying the behavior of changes with respect to each variables, you need to go with phase plane diagrams. If there is a loop (i.e. when the phase portrait is ellipse, which is also termed as state of stability ;not asymptotically stable) then there will be oscillations. Physically it means ,increase in one variable make other variable to decrease . After reaching to its minimum, the same variable start increasing to sustain the equilibrium and make the first variable to decrease and the process continues.
When we solve for equilibrium points your system only satisfy when Vin = Vout. and thus does not bring condition of oscillations. Please check your system again.
Just search for solutions of the form A, B = KA,B cos(ωA,Bt) and find under what conditions on the parameters of the system the ωA,B are real.
Hopf bifurcation requires an equilibrium point. Adding the differential equations gives vin = vout at equilibrium, which is a bit of a problem if vin != vout. If vin = vout, then B = (v1/k2).A/(k1+A) gives a 1-D curve of equilibria. This makes oscillation impossible around an equilibrium. Check your equations!
Check the Poincaré-Bendixon theorem for some basic knowledge on the appearance of limit cycles in 2D systems.
Oscillations can occur when system has centers (may be a Hamiltonian system) or limit cycle attractors (a dissipative system). In Hamiltonian systems we find the roots of Jacobian matrix, if these are pure imaginary, there is center and you will have an oscillation (not limit cycle) around the center.
For a dissipative nonlinear system you can determine using Poincare-Bendixon theorem whether periodic orbits (limit cycle) exist or not as mentioned by Dr-Aguirre.
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