Of course! This is an excellent question that sits at the intersection of chemical kinetics and dynamical systems theory. You are correct that many well-known chemical oscillators, when modeled, fall into the 1-fast/2-slow category. However, the 2-fast/1-slow structure you're looking for is a classic paradigm for generating bursting oscillations in many fields, including chemistry and neurobiology.

Here are some examples and the general framework you're looking for.

The General Mathematical Structure

The system you are describing is a canonical example of a fast-slow dynamical system. In three dimensions with two fast variables () and one slow variable (), the general form is:

where is a small, positive parameter () that explicitly separates the time scales.

The "bursting" behavior arises from the interaction between the fast subsystem (the dynamics) and the slow drift of the variable.

1. Fast Subsystem: If we treat as a fixed parameter, the first two equations, and , describe a 2D system. The behavior of this system (e.g., whether it has a stable steady state or a stable limit cycle) depends on the value of .
2. Slow Variable as a Bifurcation Parameter: The slow variable evolves according to its own dynamics, which are influenced by the state of the fast variables. As slowly changes, it effectively sweeps the fast subsystem through different dynamical regimes.
3. Bursting Mechanism: Bursting typically occurs when the slow drift of pushes the fast subsystem across a Hopf bifurcation.
   • Quiescent Phase: For a certain range of values, the fast subsystem has a stable fixed point. The full system will be near this point, and will drift slowly.
   • Oscillatory Phase: As drifts, it crosses a bifurcation point where the fixed point loses stability and a stable limit cycle is born (a supercritical Hopf bifurcation is the simplest case). The variables will rapidly spiral out to this limit cycle, producing a train of fast oscillations.
   • Termination: During the oscillatory phase, continues to drift. Eventually, it may cross another bifurcation point (e.g., a saddle-node of limit cycles or a homoclinic bifurcation) that destroys the limit cycle, causing the system to return to a stable fixed point and re-enter the quiescent phase. The slow dynamics of then reverse its course, starting the cycle over again.

Example 1: The Koper Model

A concrete chemical model that fits your description perfectly was proposed by Marc Koper to study oscillations in electrochemical systems, but it's presented as a general kinetic model. It exhibits precisely the kind of bursting you're interested in.

The model describes the concentrations of two fast intermediates, and , and one slow reactant, .

The reaction scheme can be abstractly written as:

1. (Slow supply of reactant A)
2. (Production of fast intermediate X)
3. (Conversion of X to Y)
4. (Autocatalysis involving Y)
5. (Removal of Y)

The corresponding ODEs are:

Note: I've used a slightly different but common formulation of the model. The original paper has slightly different scaling.

Here:

• are the concentrations of the fast intermediates.
• is the concentration of the slow reactant.
• The subsystem is a variant of the "Brusselator" or "autocatalator" type oscillator.
• The variable acts as the control parameter for the fast subsystem. As slowly increases and decreases due to its own supply/consumption dynamics, it pushes the system in and out of an oscillatory regime via Hopf bifurcations.

Reference: Koper, M. T. M. (1995). Bifurcations of mixed-mode oscillations in a three-variable autonomous model. The Journal of Chemical Physics, 102(1), 419-428.

Example 2: The "Slow-Inflow" Brusselator

A more textbook example can be constructed by taking a famous 2D chemical oscillator and adding a slow dynamic to one of its parameters. The Brusselator is a perfect candidate.

The standard Brusselator model is:

Here, and are concentrations of reactants that are assumed to be constant (i.e., held in large excess). This system has a Hopf bifurcation as is varied, leading to a limit cycle.

To create a 2-fast/1-slow burster, we can assume that the reactant is not constant but is supplied to the reactor slowly. Let's rename its concentration to .

The modified system becomes:

Here:

• are the fast variables.
• (formerly ) is the slow variable.
• The dynamics for represent a slow inflow from a reservoir (concentration ) and slow consumption by the first reaction step (rate proportional to the rate of change of from that step).
• The slow variable now plays the role of the bifurcation parameter, and the system can exhibit bursting as oscillates slowly, turning the fast oscillations on and off.

This "modified Brusselator" is a very clear example of the design principle and is often used in textbooks on nonlinear dynamics to illustrate bursting.

Summary for the Mathematician

You are looking for a 3D fast-slow system with a 2D fast subsystem. These are prime candidates for modeling bursting and mixed-mode oscillations (MMOs). The key mechanism is a slow passage through a Hopf bifurcation.

While prominent models like the Oregonator for the BZ reaction have a 1-fast/2-slow structure, the 2-fast/1-slow structure is also chemically plausible and has been explicitly modeled. The Koper model is an excellent, published chemical example. Furthermore, you can systematically construct such models by taking any 2D chemical oscillator (like the Brusselator) and adding slow dynamics to one of its key control parameters, which is a common technique in theoretical studies.

I hope these examples provide the concrete starting point you were looking for