Posted by ""Welch, Cory"" <Cory_Welch@nrel.gov>
I am currently engaged in a debate regarding continuous-time vs.
discrete-time modeling. As a result, I came across two articles that claim a first order difference equation (they use the logistic equation as an example) is capable of a) oscillation, and b) chaotic behavior. As any good system dynamicist knows, this is of course impossible. These articles are referenced countless times in ecology literature.
At first glance, it seems to me that the articles are mistakenly interpreting instability resulting from using discrete time rather than continuous time as ""true"" oscillation.
First, I am wondering whether I am correct in my interpretation.
Second, I wonder anyone on this list knows of a published rebuttal specific to the contentions set forth in these articles (I am aware of the discussion on oscillation in Business Dynamics).
Finally, if anyone can point me toward publications regarding the merits of continuous-time vs. discrete time modeling, it would be greatly appreciated.
Best regards,
Cory Welch
Articles referred to above:
Robert M. May ""Simple mathematical models with very complicated dynamics"" Nature, Vol 261, June 10, 1976, pp. 459-467.
Robert M. May, George F. Oster ""Bifurcations and Dynamic Complexity in Simple Ecological Models"" The American Naturalist, Vol. 110, No. 974 (Jul. - Aug., 1976), pp. 573-599
Cory J. Welch
Senior Energy Analyst
National Renewable Energy Laboratory
Strategic Energy Analysis and Applications Center Posted by ""Welch, Cory"" <Cory_Welch@nrel.gov> posting date Thu, 31 May 2007 10:06:46 -0600 _______________________________________________
QUERY First order oscillation
-
""Welch, Cory"" <Cory_Welch@n
- Junior Member
- Posts: 2
- Joined: Fri Mar 29, 2002 3:39 am
-
John Sterman <jsterman@MIT.ED
- Junior Member
- Posts: 18
- Joined: Fri Mar 29, 2002 3:39 am
QUERY First order oscillation
Posted by John Sterman <jsterman@MIT.EDU>
A continuous time first-order system of the form
1. dx/dt = f(x)
cannot oscillate, whether it is linear or nonlinear. The system has only one eigenvalue, so can never oscillate (even if it is nonlinear). To get oscillation in continuous time requires at least
2 state variables. To get chaos in continuous time requires at least three states and particular nonlinearities.
A discrete-time first-order system (mapping) of the form
2. x(t+1) = F(x(t))
can oscillate and, if it is nonlinear in a particular way, can generate period doublings, chaos, and other interesting dynamics.
One way to think about this is to reformulate the discrete-time map in continuous time. The difference-equation formulation implies that there is a time delay of 1 ""period"" in the feedback loop from x to
its rate of change. Further, that delay is a ""pipeline"" delay:
output(t) = intput(t-L), where L is the length of the delay, in this case, one ""period"". The pipeline delay is the limit of the Erlang delay family as the order of the Erlang delay goes to infinity (it is thus also called an infinite-order delay). Consequently, the continuous time equivalent of the discrete-time mapping eq. 2. is actually an infinite-order system. Such a system can oscillate of course and can generate chaos etc.
Difference equations became popular (in economics, at least) because economic data are typically published at regular intervals such as quarterly or annually, and it was convenient for econometric estimation to model economic dynamics as proceeding in discrete steps. However, while difference equations are often useful, one must be careful because there is an irreducible time delay of at least 1 period in every feedback loop. If the length of this period is long relative to the time constants for the real-world processes being modeled, the result can be the introduction of spurious dynamics (e.g., oscillations caused by the time step). Further, any more complex delays must be integer multiples of the ""period"" between time steps in the difference equations. This is often inappropriate and inconsistent with the data.
It's generally better practice to model the dynamics of a system continuously, explicitly representing relevant time delays as the data suggest, including their mean and distribution, rather than assuming a pipeline delay of 1 period in the updating of every state variable.
John Sterman
Posted by John Sterman <jsterman@MIT.EDU> posting date Fri, 1 Jun 2007 07:39:50 -0400 _______________________________________________
A continuous time first-order system of the form
1. dx/dt = f(x)
cannot oscillate, whether it is linear or nonlinear. The system has only one eigenvalue, so can never oscillate (even if it is nonlinear). To get oscillation in continuous time requires at least
2 state variables. To get chaos in continuous time requires at least three states and particular nonlinearities.
A discrete-time first-order system (mapping) of the form
2. x(t+1) = F(x(t))
can oscillate and, if it is nonlinear in a particular way, can generate period doublings, chaos, and other interesting dynamics.
One way to think about this is to reformulate the discrete-time map in continuous time. The difference-equation formulation implies that there is a time delay of 1 ""period"" in the feedback loop from x to
its rate of change. Further, that delay is a ""pipeline"" delay:
output(t) = intput(t-L), where L is the length of the delay, in this case, one ""period"". The pipeline delay is the limit of the Erlang delay family as the order of the Erlang delay goes to infinity (it is thus also called an infinite-order delay). Consequently, the continuous time equivalent of the discrete-time mapping eq. 2. is actually an infinite-order system. Such a system can oscillate of course and can generate chaos etc.
Difference equations became popular (in economics, at least) because economic data are typically published at regular intervals such as quarterly or annually, and it was convenient for econometric estimation to model economic dynamics as proceeding in discrete steps. However, while difference equations are often useful, one must be careful because there is an irreducible time delay of at least 1 period in every feedback loop. If the length of this period is long relative to the time constants for the real-world processes being modeled, the result can be the introduction of spurious dynamics (e.g., oscillations caused by the time step). Further, any more complex delays must be integer multiples of the ""period"" between time steps in the difference equations. This is often inappropriate and inconsistent with the data.
It's generally better practice to model the dynamics of a system continuously, explicitly representing relevant time delays as the data suggest, including their mean and distribution, rather than assuming a pipeline delay of 1 period in the updating of every state variable.
John Sterman
Posted by John Sterman <jsterman@MIT.EDU> posting date Fri, 1 Jun 2007 07:39:50 -0400 _______________________________________________
-
yaman barlas <ybarlas@boun.ed
- Newbie
- Posts: 1
- Joined: Fri Mar 29, 2002 3:39 am
QUERY First order oscillation
Posted by yaman barlas <ybarlas@boun.edu.tr>
Dear Cory;
It IS possible for a 1st order time-discrete model (i.e difference) equation to oscillate. It does NOT even have to be non-linear. Here is a short proof: take the simplest linear, constant-coefficient, 1st order equation x(k+1) = ax(k).
Now try an a value between -1 and 0. You will get damped oscillations. For a<-1, you will get growing oscillations. You may want to play with different a's.
As for the interpretation: there is NO numerical stability/error problem here at all. When you iterate this first order model, what you obtain is actually its EXACT mathematical behavior. So, the true behavior of a first order time-discrete system can be oscillatory.
What we are used to hear in SD community ('first order systems can not
oscillate') assumes the model is continuous.
It is great that you brought this up. The implication of this important difference between continuous and discrete systems is that we must be careful and EXPLICIT in our assumption in this respect. When we build a continuous model, we must mean it and must never use arbitrary/careless dt values. (Such dt values would imply some 'arbitrary' discrete-time models, with potentially very different dynamics). Similarly, when we build time-discrete models and use dt=1, we must again mean it and know the fact that would be delaing with quite different dynamics. Time continuous and discrete models are quite different creatures.
About your final question on chaos in the first order discrete logistic model, yes, it is well known with this property. (The articles you refer to are very important classics). Since this particular model is non-linear, it CAN exhibit strange attractors and chaos, even though it is first order. Again, here is an
example: The simplest discrete logistic equation is: x(k+1) = ax(k)(1-x(k)).
It is known to start exhibiting near chaotic behavior, after a=3.8 and becoming
wilder and wilder as it approaches 4. Perhaps you may want to play with it?
best wishes,
Yaman Barlas
---------------------------------------------------------------------------
Professor, Industrial Engineering Dept.
Bogazici University,
34342 Bebek, Istanbul, TURKEY
Posted by yaman barlas <ybarlas@boun.edu.tr> posting date Fri, 1 Jun 2007 19:44:18 +0300 _______________________________________________
Dear Cory;
It IS possible for a 1st order time-discrete model (i.e difference) equation to oscillate. It does NOT even have to be non-linear. Here is a short proof: take the simplest linear, constant-coefficient, 1st order equation x(k+1) = ax(k).
Now try an a value between -1 and 0. You will get damped oscillations. For a<-1, you will get growing oscillations. You may want to play with different a's.
As for the interpretation: there is NO numerical stability/error problem here at all. When you iterate this first order model, what you obtain is actually its EXACT mathematical behavior. So, the true behavior of a first order time-discrete system can be oscillatory.
What we are used to hear in SD community ('first order systems can not
oscillate') assumes the model is continuous.
It is great that you brought this up. The implication of this important difference between continuous and discrete systems is that we must be careful and EXPLICIT in our assumption in this respect. When we build a continuous model, we must mean it and must never use arbitrary/careless dt values. (Such dt values would imply some 'arbitrary' discrete-time models, with potentially very different dynamics). Similarly, when we build time-discrete models and use dt=1, we must again mean it and know the fact that would be delaing with quite different dynamics. Time continuous and discrete models are quite different creatures.
About your final question on chaos in the first order discrete logistic model, yes, it is well known with this property. (The articles you refer to are very important classics). Since this particular model is non-linear, it CAN exhibit strange attractors and chaos, even though it is first order. Again, here is an
example: The simplest discrete logistic equation is: x(k+1) = ax(k)(1-x(k)).
It is known to start exhibiting near chaotic behavior, after a=3.8 and becoming
wilder and wilder as it approaches 4. Perhaps you may want to play with it?
best wishes,
Yaman Barlas
---------------------------------------------------------------------------
Professor, Industrial Engineering Dept.
Bogazici University,
34342 Bebek, Istanbul, TURKEY
Posted by yaman barlas <ybarlas@boun.edu.tr> posting date Fri, 1 Jun 2007 19:44:18 +0300 _______________________________________________
-
Scott Rockart <srockart@duke.
- Junior Member
- Posts: 3
- Joined: Fri Mar 29, 2002 3:39 am
QUERY First order oscillation
Posted by Scott Rockart <srockart@duke.edu>
Hi Cory,
Tu (Dynamical Systems, 2nd edition 1994, Springer-Verlag) has a nice concise discussion of limit cycles and chaotic behavior in difference and differential equations which makes specific use of the logistic equation as an example (see section 10.3). His discussion supports rather than refutes the conclusions that both oscillation and chaos are possible in the first order difference equations, but notes that continuous functions must be at least second order for limit cycles and third order for chaos.
Sterman makes similar points (omitting proofs) on page 290 of Business Dynamics (see footnote). In light of that, the debate seems to be whether and when the behavior of the difference equations reflect the behavior of the systems of interest rather than merely reflecting integration errors.
Best,
Scott
Posted by Scott Rockart <srockart@duke.edu> posting date Fri, 01 Jun 2007 16:17:34 -0400 _______________________________________________
Hi Cory,
Tu (Dynamical Systems, 2nd edition 1994, Springer-Verlag) has a nice concise discussion of limit cycles and chaotic behavior in difference and differential equations which makes specific use of the logistic equation as an example (see section 10.3). His discussion supports rather than refutes the conclusions that both oscillation and chaos are possible in the first order difference equations, but notes that continuous functions must be at least second order for limit cycles and third order for chaos.
Sterman makes similar points (omitting proofs) on page 290 of Business Dynamics (see footnote). In light of that, the debate seems to be whether and when the behavior of the difference equations reflect the behavior of the systems of interest rather than merely reflecting integration errors.
Best,
Scott
Posted by Scott Rockart <srockart@duke.edu> posting date Fri, 01 Jun 2007 16:17:34 -0400 _______________________________________________
-
""Welch, Cory"" <Cory_Welch@n
- Junior Member
- Posts: 2
- Joined: Fri Mar 29, 2002 3:39 am
QUERY First order oscillation
Posted by ""Welch, Cory"" <Cory_Welch@nrel.gov>
All,
Thank you all for your responses. Sterman's response cleared it up for me entirely. My original posting should have read that oscillation is impossible in a first-order ""differential"" rather than ""difference""
equation.
I'd still question, however, whether a ""difference"" formulation should be used to model ecological systems. It seems there are many papers out there pointing to difference formulations as an explanation for oscillation and/or chaotic behavior in natural systems. Rather, it seems to me that a continuous time formulation would be more appropriate, modeling any non-continuous type behavior (e.g., seasonal mating, etc.) with an appropriate order delay to study its affect on the dynamics.
All my best,
Cory
Cory J. Welch
Senior Energy Analyst
National Renewable Energy Laboratory
Strategic Energy Analysis and Applications Center Posted by ""Welch, Cory"" <Cory_Welch@nrel.gov> posting date Sat, 2 Jun 2007 14:47:26 -0600 _______________________________________________
All,
Thank you all for your responses. Sterman's response cleared it up for me entirely. My original posting should have read that oscillation is impossible in a first-order ""differential"" rather than ""difference""
equation.
I'd still question, however, whether a ""difference"" formulation should be used to model ecological systems. It seems there are many papers out there pointing to difference formulations as an explanation for oscillation and/or chaotic behavior in natural systems. Rather, it seems to me that a continuous time formulation would be more appropriate, modeling any non-continuous type behavior (e.g., seasonal mating, etc.) with an appropriate order delay to study its affect on the dynamics.
All my best,
Cory
Cory J. Welch
Senior Energy Analyst
National Renewable Energy Laboratory
Strategic Energy Analysis and Applications Center Posted by ""Welch, Cory"" <Cory_Welch@nrel.gov> posting date Sat, 2 Jun 2007 14:47:26 -0600 _______________________________________________
-
""Douglas Franco"" <dfranco@c
- Junior Member
- Posts: 8
- Joined: Fri Mar 29, 2002 3:39 am
QUERY First order oscillation
Posted by ""Douglas Franco"" <dfranco@cantv.net>
The continuous model of the discrete formulation
output(t) = intput(t-L), where L is the length of the delay
does need infinite levels. Therefore, this continuous model can oscillate, as John pointed out.
But it is a model.
Sometimes, there is more in our models than it is in the real world.
Sometimes, there is more in real world than there is in our models.
Douglas Franco
Posted by ""Douglas Franco"" <dfranco@cantv.net> posting date Sat, 2 Jun 2007 08:51:14 -0400 _______________________________________________
The continuous model of the discrete formulation
output(t) = intput(t-L), where L is the length of the delay
does need infinite levels. Therefore, this continuous model can oscillate, as John pointed out.
But it is a model.
Sometimes, there is more in our models than it is in the real world.
Sometimes, there is more in real world than there is in our models.
Douglas Franco
Posted by ""Douglas Franco"" <dfranco@cantv.net> posting date Sat, 2 Jun 2007 08:51:14 -0400 _______________________________________________