I have a discrete time dynamical system $$x(t+1)=f(x(t))$$ I would like to write a continuous time system that approximates it well; that is, a continuous system of the form $$\frac{dx}{dt} = g(x)$$, solve the continuous system, and use the solution to approximate the values of the original discrete time system. Are there any methods for doing this? I did some search and found conversions from a continuous system to a discrete one - what about the other way around? Also, how to bound the error of this approximation?
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1Can you list the sources you refer to? This can't be done in general: https://math.stackexchange.com/questions/3235295/equivalence-of-discrete-and-continuous-dynamical-systems – dkv Mar 01 '21 at 15:49
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Try $g(x)=f(x)-x$ or $g(x)=(f(x)-x)-\frac12(f'(x)-1)(f(x)-x)=\frac12(3-f'(x))(f(x)-x)$. See https://math.stackexchange.com/a/3642860/115115 for a deeper explained example. – Lutz Lehmann Mar 01 '21 at 20:13
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Thank you for your solution. Do you have any thoughts on how you can bound the error of such an approximation? – Nithish kumar Mar 11 '21 at 23:53
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You would need a bound on $f(x)-x$ or at least a Lipschitz constant, and that it is small. Then you can apply error estimates for Euler and Taylor-step methods in reverse. Without that there will be very likely a disconnect between the two approaches. – Lutz Lehmann Mar 12 '21 at 08:57