I am trying to propose a limit cycle model for a movement using "W" method. In this regard, I am going to find the coefficients of the proposed model (i.e. diff. equation) using multiple regression method. Then, I solve the diff. equation using previously found coefficient and plot the results. However, the result is not a limit cycle. Alternatively, when I manipulate the signs of the coefficients to what it should be, the results will be a limit cycle. Why the signs of coefficients are not correct when using regression method? Does it mean the proposed model is not the model that fits the data?
I am by no means an expert on dynamical systems, but I think there might still some question as to whether gait can always be represented effectively as a dynamical system. Gait does change over time due to mental state, fatigue and learning. So perhaps it does not always have a stable cycle. I have seen some results from the application of dynamical systems to gait that are convincing and some that are less convincing... so I wonder if perhaps the use of dynamical systems theory is a bit question-specific?
Dynamical systems theory is still relevant, even where the system is not autonomous: modified by intention, uneven ground, etc. When learning changes the system's fast responses, as in (for instance) Shou-Han Zhou, Denny Oetomo, Ying Tan, Etienne Burdet, Iven M. Y. Mareels: Modeling Individual Human Motor Behavior Through Model Reference Iterative Learning Control. IEEE Trans. Biomed. Engineering 59(7): 1892-1901 (2012), it effectively modifies the system parameters. So does learning to run more effectively, and so on: for instance, learning to have an amplitude of limit cycle that works well for that runner. So, thinking of gait as controlled by a 'given' dynamics is too simple: but this does not make concepts like limit cycle irrelevant.
Tim raises a good point about how the parameters of the model relate to the conditions of the task. Getting back to the original question, I would suggest that if the model does not fit expectations, then there is a mismatch between the model/parameters and the task/conditions being observed. So, it might be helpful to ask an explicit research question, and consider whether the dynamical systems model is the best way to answer it. I agree with Tim that Dynamical systems theory is always relevant to gait, but it may not always be the best tool for the question being asked. Similarly, I would say that Newtonian mechanics are always relevant to gait, but sometimes EMG is a much better tool to answer a given research question.
If your observations give exact limit cycle, this means that your system is deterministic, not stochastic. In such a case you must solve comparatively simple "inverse problem". If you have ODE model, you must solve exactly the system of algebraic equations. The values of ODE variables in arbitrary time moments become coefficients In this algebraic equations and coefficients of ODE become unknown quantities.
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