I may be wrong, but I think the question should be asked in different terms, and with further justification.

Strictly speaking the sufficient conditions do not know about history, namely change, learning, adaptation, evolution and phase space. For it is precisely the sensitivity to the initial conditions that modify - ontologically as well as methodologically - the possibility of speaking about "sufficient conditions".

As a matter of fact, stirctly speaking, the concern about the sufficient conditions was set out by Leibniz in a very specific context. However, the works by Poincaré, Lorenz, and Prigogine, for instance, made clear that such an approcah was not tenable any longer.

Additionally, when a linear system exhibits a caotic behavior such behavior drastically changes the system so much so that it becomes non-linear. Hence a shift ihn the history and state of the system takes place.

It becomes difficult to conceive of a linear system that is autonomous. For linearity means and entails determinism, whence the system is captive by its past, and the past determines both the present state as well as its future states. As a consequence, the system cannot be said to be autonomous.

Unfortunatly, I do not know the answer to your question for the special case of piecewise linear functions, but perhaps it can be traced back to the more general sufficient condition that in 1-D autonomous systems chaos (of the Li Yorke type) is implied by a positive topological entropy, see for instance:

http://www.scholarpedia.org/article/Topological_entropy

In addition to Carlos' answer, I think that a context is required. For example, the answer to your question could be "3.57".

If you look at a simple chaotic example from biology, the Logistic Equation, then chaotic behaviour can be seen in an autonomous system. The logistic difference equation is given by

(x+1) at n+1 = r*x at n *(1 + x at n)

where r is the driving parameter.

Start with a fixed value r, and an initial value of x0.

The equation runs recursively to obtain x1, x2, . . .xn.

Where r is small, xn (as n goes to infinity) eventually converges to a single number.

In biology, this number (xn as n approaches infinity) represents the population of the species.

When r approaches 3, this simple equation produces interesting results.

When r = 3.0, xn no longer converges and there is bifurcation.

With r > 3, xn oscillates between four values.

As r increases, xn goes through bifurcations of period 8, then 16, then chaos.

When the value of the driving parameter r equals 3.57, xn neither converges or oscillates and its value becomes completely random.

For values of r larger than 3.57, the behaviour is largely chaotic. However, there is a particular value of r where the sequence again oscillates with period of three. The bifurcations then begin again with period 6, 12, 24, then back to chaos.

The Wikimedia diagram is attached

https://dl.dropboxusercontent.com/u/463746/Logistic_Bifurcation_map_High_Resolution.png

I think it is a positive Lyapunov exponent.

Dear Hany,

The answer of what you asked depends on some additional information you should provide

a) If you have time-series obtained from a system (map or flow), having one positive Lyapunov exponent is necessary. Being sure about boundedness of the system makes it sufficient.

b) If you have the real system, you can run it and obtain the time-series. Then go to "a".

c) If you want to find out the answer only based on the structure of your system (its equations and ....), I believe you can find only conditions that improves the chance of finding chaos or at best create the potential of having chaos (for example, an specific system can shows lots of different patterns, including chaos, only by changing the initial conditions. Sometimes the basin of attractions for those different patterns are fractals! So having the structure you cannot say it is chaotic for sure. See Professor Sprott's works).

For finding interesting information and papers about chaos topology, see RG pages of Professors Robert Gilmore, Christophe Letellier, Marc Lefranc and ... .

Best regards,

Sajad

Hello All,

Thanks for answering my question via comments and helpful suggestions.

I guess I didn't ask my question clearly enough, so following I try to make the meaning of question more clear.

In basis, the smooth (continuous) autonomous system (Flow dx/dt=f(x), x in R^n, f(x) C^k smooth function ),

According to my knowledge there are two facts:

i)- A positive largest Lyapunov exponent doesn't, in general, indicates chaos.

ii)- For smooth systems also can approach chaos via the period doubling route or the system has highly sensitive to initial conditions..etc:

Please Note that:

1- The "Linear" smooth system cannot exhibit chaotic behavior or period doubling, i.e. only the "nonlinear" systems can exhibit chaotic behavior or period doubling.

2- In "Linear" nonsmooth (piecewise smooth, discontinuous) system (dx/at=A1*x for h(x)>0 & dx/dt=A2*x

I think two issues should be clarified in the discussion:

(1) A Linear system will exhibit chaos (I agree with Hany)

(2) A piecewise linear system is nonlinear; more accurately it is nonsmooth. Being nonsmooth is highly nonlinear and is prone to chaos.

The other thing that should be clarified when askign the questions is: whether you want a sufficient condition a priori, that is, whether you wish to know a system is chaotic or not before numerical computation is carried out. Such a criterion would be very useful. However, I am not aware of it for nonsmooth systems.

If on other hand you wish to know whether the system is chaotic after you have already otained numerical results, that is easy, for example, checking for sensitivity to initial conditions.I feel some replies above imply the second case.

Probably it's not what you exactly want, but it can give you a view for the matter from a different angle. Have you tried linear switched systems literature? They are often concerned with a reachable set which is the result of the chaotic like evolution. E.g.: liberzon.csl.illinois.edu or books.google "Switching in Systems and Control"

for a piecewise linear system in discrete time (a map) a sufficient condition is the existence of an homoclinic orbit, for a flow the same condition can be applied to some Poincaré section

I do not think there is a general answer but sufficient conditions known for continuous maps may be applied if you think of the pw linear map (or autonomous system) as a finite number of linear maps on non-invariant and disjoint (but non-separated) regions of the state space. The combined interactions of these two or more partial maps may define a continuous map associated with an "ejector cycle". My book "Nonlinear difference equations: theory with applications to social science models" Section 3.4 has some examples - I can email copies of some pages if needed.

It is widely most theories for smooth dynamic systems are not applicable to nonsmooth dyanmci systems (including piecewise lienar systems).

The following book may be useful:

Mario di Bernardo Laurea; PhD, Alan R. Champneys BSc; DPhil, Christopher J. Budd MA; DPhil, Piotr Kowalczyk MSc; PhD

Piecewise-smooth Dynamical Systems

Applied Mathematical Sciences Volume 163, 2008, pp 47-119 (Springer)

There are properties which work well both for smooth system and for piecewise smooth systems (both continuous and discontinuous). In one-dimensional maps these are properties associated with the appearance/existence of homoclinic orbits, which are the same also for expanding cycles in R^n (L. Gardini, I. Sushko, V. Avrutin, M. Schanz “Critical homoclinic orbits lead to Snap-Back Repellers”. Chaos Solitons & Fractals 44 (2011) 433–449, http://www.mdef.it/uploads/tx_sibibtex/GSAS_CSF_2011.pdf).

These properties are often the skeleton of chaos in piecewise smooth systems in continuous time.

For sure a sufficient condition is the existence of a noncritical homoclinic orbit of an expanding cycle, in maps or in a Poincaré section for flows (of any kind): when it exists then an invariant chaotic sets exists. Besides the paper cited above, for piecewise smooth system the material of our Summer School may be of interest: http://tutorial-sicc.mdef.it/index.php?id=200

I should emphasize that a non-smooth even discontinuous (discrete) system can indeed be stitched together in some cases to generate a continuous, even smooth map on a region of a the state space that is invariant under the composition of the partial maps (linear pieces in this dialog). Unfortunately this fact is not as widely known as the standard, basic facts which are simply well-known sufficient conditions.

Dear Hassan:

A way of dealing with nonsmoothness is to replace it with a consitnuous function that fits it closely. This has been done for bilinear systems, systems with clearance and systems with friction.

Does your "stitched together in some cases to generate a continuous" mean the above?

Dear Huajiang, continuous (usually smooth) approximations to non-smooth maps are intuitive first thoughts and usually very useful but not what I meant here. In the discrete context (I am not sure if this works for continuous time) one could check the following: for simplicity consider two linear (or smooth) pieces A and B that are defined over disjoint, non-invariant and non-separated (common boundary) regions that we can also call sets A and B. Suppose that k iterations (with k>=2) bring points within region A that is governed by one of the linear pieces back to A and that all points within a subset M of A come back to M. Then M is invariant and it may happen that the k-times composite map is continuous (even smooth) on M. This situation makes available widely known theory (stability, Li-Yorke, etc). A piecewise smooth model of combat turns out to have this property (a 2002 SIAM Review paper in my R-Gate pubs list).

It is clear that in piecewise smooth systems, invariant sets having points belonging to “smooth partitions” follow the theory of smooth systems. Things change when such invariant sets (due to some parameter variation) reach some “border”. In applied sciences (as in Engineering, Physics, Social Sciences) many models are described by piecewise smooth systems (continuous or discontinuous). Their dynamic properties cannot be studied by “smoothing” the system. What is to be investigated is exactly what happens when there is a contact with some border. Then border collision bifurcations are to be studied.

I should add that in the example that I mentioned there are no invariant sets for any of the smooth pieces (partitions). The invariant set M belongs to an associated map obtained from the iteration of the piecewise continuous map and the associated map is continuous on M. An analogy (possibly not a good one) is that a cycle with period 2 or more is seen as a fixed point of some iteration of a map.

It is exactly that the meaning: a 2-cycle is an invariant set !

In that case, we imagine that each point of the cycle belongs to a different piece of the discontinuous or non-smooth map so the continuous or smooth theory does not apply to the cycle. In my example, the same idea extends to chaotic orbits that are distributed across two or more partitions.

dear Israel Tankam

in maps (discrete time systems, continuous or discontinuous, smooth or piecewise smooth) the existence of chaotic dynamics is rigorously proved whenever there exists a noncritical homoclinic orbit (noncritical means without critical points); a cycle of period 3 is not necessary

The system should be 3 dimensional and the equilibrium point should be unstable and the lyapunov exponent should be positive.