What is the minimum dimension requirement to exhibit chaotic behavior for continuous dynamical systems? I came across that it would be greater than or equal to three. But the Dixon system violates the Poincare-Bendixson theorem. My question is do we have any other systems other than Dixon system which violates Poincare-Bendixson theorem? Or is there any other two dimensional system that can exhibit chaos.
What i also know is that you need at least three equations in a system of autonomous odes to have chaos.
But i think probably the violation of the smoothness condition has something to do with low dimensional chaos.
Dear Suresh,
as far as I know, a minimum of three equations are needed in a system of autonomous odes to exhibit chaos. As pointed out by Jomar, the Dixon system is a particular case since the system is singular at the origin, therefore violating the Poincare-Bendixson condition of smoothness.
There are some variants of the Dixon system that also display low-dimensional chaos, as well as other 2D systems involving singularities in terms of log(abs(x)). You can have a look in the book by Julien C. Sproot "Elegant chaos: algebraically simple chaotic flows":
http://books.google.es/books?id=buILBDre9S4C&pg=PA109&lpg=PA109&dq=Dixon+system+chaos&source=bl&ots=bqWM3MbNI6&sig=5nek5WlvjtdjIWS-UKXgwrNb-oo&hl=es&sa=X&ei=k6jKUqO2L4WJ0AWux4GoDg&ved=0CDUQ6AEwAA#v=onepage&q=Dixon%20system%20chaos&f=false
Apart of this, for 1D or 2D systems exhibiting chaos, I'm only aware of non-autonomous systems (time-dependent, or systems with time delays).
If, as usual, you mean by "chaos" any obstruction to predictability, there is also the intertwining of basins of attraction. If you have a system with two attractors, their basins could accumulate on each other. Here some bibliograhpy, starting from which you can find more.
Yang, "A proof for a theorem on intertwining property of attraction basin boundaries in planar dynamical systems" Chaos, Solitons & Fractals 01/2003; DOI:10.1016/S0960-0779(02)00154-6
LI, Ding, Chen "Intertwined basins of attraction of dynamical systems" Applied Mathematics and Computation 01/2009; 213:272-274. DOI:10.1016/j.amc.2009.03.018
L.Tedeschini-Lalli " Smoothly intertwined basins of attraction in a prey-predator model" Acta Applicandae Mathematicae 01/1995; 38(2):139-147
According to the definition of S. Strogatz [Nonlinear Dynamics and Chaos, Chap. 9.3], chaos requires three ingredients:
- Aperiodic long-term behavior.
- A deterministic system.
- Sensitive dependence on initial conditions.
The last item is equivalent to stating that nearby trajectories diverge exponentially (positive Lyapunov exponent). Continuous systems in a 2-dimensional phase space cannot experience such a divergence, hence chaotic behaviors can only be observed in deterministic continuous systems with a phase space of dimension 3, at least. Note that non-autonomous 2-dimensional systems are equivalent to autonomous 3-dimensional systems. They can therefore exhibit chaotic behavior.
With respect to the answer of Alfonso, I'd add that:
- delay systems are infinite dimensional systems; therefore, chaos can be observed in such systems, even if the state has dimension 1;
- in virtue of my above comment, 1D time-dependent systems are in fact 2D; they should not exhibit chaotic behavior.
Also note the importance of smoothness. Discontinuous systems can exhibit chaotic behaviors even if they are 1D, e.g., logistic map.
Even in one spacial dimension (two dimension in phase space), chaos appears. See Konishi & Kaneko, J. of Phys. A25 (1992) 6283.
As Alexandre stated, continuous systems in a 2-dimensional phase space do not allow the local divergence necessary for sensitive dependence on initial conditions. In fact, Peixoto's Theorem essentially says "no chaos unless at least a 3 dimensional phase space." Yet there are 1 dimensional iterative systems exhibiting chaos as well as 1 dimensional first order differential delay equations that are chaotic (Physics Letters A 366 (2007) 397–402).
The key is not continuity but rather degrees of freedom. A differential delay equation or a discrete map have infinite degrees of freedom -- in the former, there is freedom to choose oo many values of the initial history, and in the latter, there are oo many possible ways of continuously linking iterates of a map to each other.
Thus, a chaotic dynamical system must have at least 3 degrees of freedom.
The gravothermal instablity(see, e.g. Lynden-Bell & Wood(1968)MNRAS,138,495)occurs only if the spatial dimension is greater or equal to three. I don't know the relation between the instability and chaos.
Answering to the question "Do we have chaotic behaviour in two dimensional continuous dynamical systems?" what matters is whether the system is smooth or not. For continuous 2D systems not of class 1, i.e. piecewise smooth continuous 2D ODE equations, chaos is perfectly possible. For smooth systems the dimension must be at least 3.
As I mentioned before, in a system with discrcrete nap(konishi-Kaneko) chaos occuredd.
yes, in a discrete map it is well known that chaos occurs also in one-dimension,
(since a long time, 1960 or before)
my answer refers to flows, i.e. ODE, and what I have written is also well established since a long time.
As far I know even for non-smooth continuous systems should have 3 dimension. The Chua oscillator is a popular example for non-smooth piece-wise continuous systems. Do we have example for piecewise smooth continuous 2D ODE which can generate chaos.
Dimension two is enough. See two-dimensional Filippov Systems
for example in di Bernardo M, Budd CJ, Champneys AR, Kowalczyk P. Bifurcation and chaos in piecewise-smooth dynamical systems: theory and applications. Springer-Verlag 2007, as well as in many papers in the last 15 years.
See also Nosov V, Dominguez H, Ortega Herrera J.A., Meda-Campaña J.A. (2012). Complex dynamics and chaos in commutable pendulum. Revista Méxicana de Fisica v.58, No 1, 6-12.
NO is correct only for smooth systems (in continuous time),
as already remarked, for piecewise smooth systems it may be yes also in dimension two.
Laura,
I finally had a moment to read the Nosov, et al, paper, and it is very interesting. I hope everyone takes the time to check it out! However, in their example, a trajectory can self-intersect without being a cycle. Thus, the phase space of the system is more than 2 dimensions (3 is my guess), and thus, I still believe that my statement that chaos requires more than 2 degrees of freedom is correct. I am pretty sure that phase space for a Filippov system is at least 3 dimensional.
Yes, for the system in that paper I totally agree: it can be considered as a two-dimensional non autonomous system, and thus a three dimensional autonomous one.
The crutial for the existence of chaos in a system is number of degrees of freedom and number of conservation laws.
If the number of degrees of freedom is larger than the number of conservation laws , the system is chaotic independently of its dimensionality.
Chaos can not be in two dimensions, since the phase trajectory can not intersect with itself.
Discrete chaotic systems, such as the logistic map, exhibit chaos whatever their dimensionality.
Discrete system x_(k+1)=F(x_k) is like continuous system with delay x(t)=F[x(t-to)]. And this system have infinity dimension.
Since dimension of a dynamical system is by definition the dimension of the "solution space", i strongly doubt that a nonlinear discrete time system is "like" a delay system. In discrete time case of dimension 1; by only knowing the initial condition of dimension 1 at some time instance k, you can obtain the all future values by iterations, and all will be "embedded" in 1D . On the other hand initial condition to a delay differential equation is a function defined on (-t_0,0) and the dynamics are embedded in an infinite dimensional function space and it can be shown that the solution space is infinite dimensional..
In discrete time case nothing prevents the trajectory from jumping from one invariant subspace(manifold) to another, while in continuous case trajectories must follow the invariant subspaces, if they are initiated on the invariant subspaces. The continuity requirements on solution prevents the trajectory to switch between invariant subspaces if you start on them(I am assuming that the right hand side of the diff. eq. is at least Lipschitz continuous in both cases).. I think this allows one to have chaotic dynamics in 1D discrete time system but prevents it in continuous autonomous systems of 1 or 2 dimensions.. In delay differential eq case, you already have plenty of dimensions., hence very 'simple' delayed dynamical systems can generate chaotic solutions (On the other hand the systems the form dot x(t) = F(x(t-t_0)) is not 1D, it is infinite dimensional.)
If one relaxes the continuity requirements on the right hand side of the diff. eq., (as in above examples) many things are possible (in fact discrete time case relaxes the time continuity by definition ? Just thinking...) However, i think, it will be very hard to prove that these systems are chaotic in rigorous sense, which shadows the soul of the chaotic system paradigm : "The 'simple' systems can generate extremely complicated behavior."
To sum up: the topology of the state and solution spaces determine necessary conditions for chaos. The book "Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer 1991), by Stephen Wiggins has very satisfying coverage of the topic.
I would argue that Marek's comment is correct, and in fact x_(k+1)=F(x_k) is equivalent to x(t) = F(x(t-t_0)).
I invite to the following articles:
http://www.sciencedirect.com/science/article/pii/S096007799900171X
http://www.sciencedirect.com/science/article/pii/S0960077904006320
Discrete dynamics of the form x_{k+1}=F(x_k) is *not* equivalent to delay differential equations of the form \dot{x}(t)=F(x(t-\tau)). The former is 1-D (precise knowledge of x_k is sufficient to predict the future) whereas the latter is infinite dimensional (you need knowledge of the entire trajectory x(t) over an interval of length \tau.
Sampling the continuous system with sampling time \tau gets you a discrete time system - but this is neither necessarily a 1-D map nor is it necessarily sufficient information to reconstruct (under all circumstances) the continuous delay differential system (and certainly not in 1-D).
You can't have continuous trajectories displaying chaotic dynamics in 2-D as the definition (bounded together with locally divergent states) imply that you will run out of space and trajectories will be forces to cross (which they cannot). 3D gives you enough "space" to have bounded and locally separating trajectories that don't cross (and as a consequence this will give you the chaotic mixing).
It has been well known in control theory that the behaviour of delay differential equations of the form \dot{x}(t)=F(x(t-\tau)). is similar to partial differential equations in some sense. I knew it around 1970 when I was in Department of Electrical Engineering.
x(t) = F(x(t-t_0)) is qualitatively equivalent to x_(k+1)=F(x_k), but is not equivalent to delay differential equations.
Ivan - No worries. I think the thread got rather confused - at one point we were talking cross-purposes about different things. As Marek said - without the derivative the correspondence between x(t) = F(x(t-t_0)) and x_(k+1)=F(x_k) is fine: this is exactly what we do all the time with discrete sampling of continuous dynamical systems and the difference between the time index and the k-indexing is one of labelling. I assumed the question to be about something slightly different - the delay differential equation.
As we know 3 (or more) ODEs:
dx/dt=f1(x,y,z)
dy/dt=f2(x,y,z)
dz/dt=f3(x,y,z)
can give chaos.
Above we can write as:
x(k+1)=x(k)+f1[x(k),y(k),z(k)]*delta t
y(k+1)=y(k)+f2[x(k),y(k),z(k)]*delta t
z(k+1)=z(k)+f3[x(k),y(k),z(k)]*delta t.
We can reduced it, for example, to:
x(n+1)=x(n)+f1[x(n),y(n),z(n)]*(delta t)*M1 for {y(n+1), z(n+1)}=0 only.
This means that we can reduced 3 ODEs to one discrete equation:
x(n+1)=F[x(n)].
And because 3 ODEs can give chaos, chaos can give one discrete equation too.
As far as I could see the Dixon system Mr. Kumarasamy referring to is a 2D 'impulsive' dynamical system of the form
d^2 x(t) / dt^2 = - w^2 x(t)+A f(x, t) \sum_(n=0)^\infty \delta(t - nT)
which models a kicked harmonic oscillator.(Please correct me if this is not the correct system)
If this is the case, the r.h.s. of this diff. eq. is not continuous in any sense. As far as I know, there is no mathematical result that forbids existence of chaos in such a system. I think, it will not be correct to say that this system violates Poincare-Bendixon theorem since the continuity assumption of the theorem is not satisfied already.
This system can also viewed as a non-autonomous dynamical system, where the input impulse train is coupled to the states in multiplicative form. In this case, it is possible to find a transformation to re-write the model in 3D autonomous form. Nothing mathematically to avoid chaos again.
I think the answer would be: There can be 2D dynamical systems with discontinuous right hand sides with chaotic solutions (in mathematically rigorous sense). The result that 2D continuous time system can not have chaos assumes that, the r.h.s. is at least Lipschitz continuous. If this assumption is relaxed you can have chaos, but it would be really hard to prove that the solution is chaotic rigorously.
On the other hand any physical system (that can actually be realized) will always have a continuous right hand side. I believe, this fact made the statement "There can be no continuous time dynamical system whose dimension less than 3" true, if you don't take the possibilities of systems modelled by functional differential equations (the family that includes delayed systems) or partial diff. eq. as pointed out by Prof. Inagaki in to account. In such cases, the system is infinite dimensional and you have plenty of 'space' to have tangled trajectories without violating the uniqueness of the solutions, characteristic to chaos.
All the mathematical results about the necessary dimension of the dynamical system to have chaos is for continuous systems. As I tried to explain earlier, in the discrete time systems the trajectories have much more freedom in comparison, allowing even 1D discrete time systems be chaotic.
I insist, that the discrete time systems are not qualitatively equivalent to delay systems. Two systems that generate outputs which are exactly the same does not imply that the underlying dynamics are qualitatively equivalent. To discuss the equivalence, one needs the dimension of the state, input and output spaces be well defined in the first place.
Michael Small -- Thanks. I think I lost the thread, too. It would be nice if we could post mathematical symbols right in this thread ...
You cannot have chaos in autonomous continuous two-dimensional systems. The exceptions are if there is a delay, or if the system is non-autonomous.
In piecewise smooth vector fields we proved exactly what you asked. See at arxiv:
Chaotic Planar Piecewise Smooth Vector Fields With Non Trivial Minimal Sets
and
On Poincar\'e-Bendixson Theorem and Non-Trivial Minimal Sets in Planar Nonsmooth Vector Fields
Suresh Kumarasamy, please see the papers that I said before. Maybe we can write some work about this subject.
The first one is a paper accepted at Ergodic Theory of D.S. and the other is in reviewing process.
as regrads chaos in two dimensions it is mathematically not feasible. However in delayed two dimensional system, chaotic picture may be evolved.
It is possible in a two-dim system in continuous time when it is
piecewise smooth, see also the answer of T. de Carvalho
In my book, Elegant Chaos: Algebraically Simple Chaotic Flows (World Scientific, 2010), there is a short chapter on two-dimensional autonomous systems of ODEs that exhibit S-chaos including several variants and simplifications of the Dixon system. All these systems have a local singularity in the flow and a mechanism that forces the orbit to cross the singularity in finite time. Thus there is no violation of the Poincare-Bendixson theorem since the flow is discontinuous at that singularity. Here's an example that is particularly elegant:
x' = log|x| - y
y' = log|x| + x
Replace log|x| with 0.5*log(x^2) if you prefer.
Dear Suresh,
although it is often claimed otherwise, there is no chaos in Dixon's system. Because of the singularity at the origin in the definition of the vector field, it is non-trivial to study the applicability of Poincare-Bendixson for it (the arguments given by Dixon et al in their paper are erronous, one must work quite a bit harder to prove that Poincare-Bendixson does not directly apply to their system). However, it is not difficult to derive a system which has exactly the same trajectories as Dixon's system and which satisfies all assumptions of Poincare-Bendixson. This proves rigorously that there is no chaos in Dixon's system. Dixon and his collaborators conjectured a chaotic behaviour solely on the basis of numerical simulations. If one analyses more closely the parameter range where they claim to see chaos, then one notices that for these parameter values the system has two elliptic sectors at the origin and thus infinitely many homoclinic orbits. Probably, the numerics of Dixon et al was not able to cope with these homoclinic orbits and thus created an impression of chaos which does not exist.
Regards Werner
If the two dimensional system is for ced by an external periodic force, it can show chaotic behavior within certain parameter range, such as the forced Duffing oscillator model.
adding delay to equations can be mentioned as another way that two dimension continuous differential system can generate chaotic behavior. like that:
x'=y(t-T);
y'=y(1-x^2)-x(t-T);
T: delay;
but for normal two dimension system that have no discontinuation or other stuff, it is proved that it can't generate the chaotic behavior;
If we add delay or diffusion in the continuous dynamical system then the system can exhibit chaos.
Yet, technically speaking adding external forcing, time-delay or diffusion makes the system dimensionality to increase anyway. If the external forcing is stochastic then it becomes infinite-dimensional. It also becomes infinite-dimensional due to addition of time-delay. Due to the change in initial conditions, that are now functions, the space of the solutions changes too and the same goes for adding diffusion since the addition of spatio-temporal dependence makes it a PDE system.
Absent a precise definition of what chaos is assumed to be, no conclusive answer can be given. However it's the result of the absence of a sufficient number of conservation laws.
Furthermore, the ``three ingredients'' mentioned confuse cause and effect. Deterministic chaos is defined by the existence of an attractor in phase space, which means that the dynamics must be mixing and the space of states must be compact. The point of interest is that the compactness is a ``dynamical'' property, not imposed. And so is the mixing property. Absent mixing, perturbations escape and their effects ``die out''.
In particular, the exponential divergence of nearby initial conditions only holds until the attractor has been reached, whereupon the divergence ceases, since the-finite-size of the attractor implies the distance can't grow any more. So it's the leveling off that's as significant as the early time exponential divergence. (Not to mention the fact that the greatest Lyapunov exponent is an average; and it can very well be that this average isn't the typical value of the probability distribution. That's why focusing on the greatest Lyapunov exponent isn't a reliable way for characterizing the dynamics; the probability distribution is much more complex.)
The claim that the Dixon system is a counterexample to the Poincaré-Bendixon theorem is incorrect, since the system, by design, doesn't satisfy the assumptions of the theorem; so it's hardly surprising that it doesn't satisfy its consequences.
Indeed, by construction, all deterministically chaotic systems evade the assumptions of the Poincaré-Bendixon theorem; that's how they evade its consequences.
It should be stressed that the ``formal'' dimensionality is a bit misleading, since the attractor of a deterministically chaotic system doesn't, necessarily, have the dimensionality of the space of states one starts with. There do exist three dimensional systems that aren't chaotic and systems with infinite dimensional spaces of states that aren't, either-they're integrable. Whether a system is chaotic or integrable is a quite non-trivial issue, that involves counting the number of conserved quantities and trying to show that they aren't sufficient to constrain the dynamics.
Dynamical chaos, or rather its stochastic generalization, can be viewed as the spontaneous breakdown of topological supersymmetry which is hidden in all stochastic differential equations and which represents preservation of proximity of points in the phase space during continuous-time flow so that its breakdown essentially indicates (stochastic generalization of) the butterfly effect. Due to certain general properties of stochastic evolution operators, this phenomenon cannot happen in 2D not only in deterministic but also in stochastic models.
Dear Suresh,
we have finally finished a paper where we study Dixon's system in details and where we provide a complete bifurcation analysis of it including the stationary points at infinity. This paper also contains a rigorous proof that there is no chaos in Dixon's system, but only two elliptic sectors with infinitely many homoclinic orbits. At the end, we show that also in Sprott's logarithmic system (mentioned in an answer above) no chaos appears; again there are only regions in phase space where it is fairly difficult to resolve the dynamics numerically. Numerical errors then easily lead to the erroneous impression of chaotic behaviour. The paper is available on the arXiv (arXiv:2002.06593) but also here in ResearchGate in my profile.
Regards Werner
@Nicolis: "In particular, the exponential divergence of nearby initial conditions only holds until the attractor has been reached, whereupon the divergence ceases, since the-finite-size of the attractor implies the distance can't grow any more."
I don't think this is right. First, the highest Lyapunov exponent remains positive on the attractor, in fact, its value is determined by the attractor. Second, the finite size of the attractor never constrains exponential divergence of nearby initial conditions, because nearby can always be taken small enough to allow exponential divergence for a short time without the finite size of the attractor being reached. Third, of course, if you start far from the attractor, there may be no longer exponential divergence of the trajectories, once the attractor has been "reached" (in fact, it can be reached at points that have almost maximal distance, so there is no room for further growth anymore) but that is not what the statement refers to. If you want to explore exponential divergence near or on the attractor, you must take two trajectories that are nearby and already near/on the attractor. You will then find that they will initially diverge exponentially (even though staying on the attractor) and even that a numerical evaluation of the largest Lyapunov exponent converges faster there than with initial conditions far from the attractor.
The exponential divergence of nearby trajectories is permanent throughout the dynamic evolution for chaotic trajectories. What is not permanent ist the condition of closeness of any given pair of trajectories. So to check the divergence numerically, you may keep one of the trajectories, but have to choose a new closer one each time the second one has drifted too far away. That is what people actually do when they calculate the Lyapunov exponent of a trajectory.
Yes, the Duffing oscillator is a nice example of 2D nonlinear systems that exhibit chaos for certain types of inputs.
No sir, we cannot have chaotic behaviour in 2D differential systems (continuous time) according to the Poincare-Bendixson theorem.
Yes, @Parsa Sarosh is right, at least three system of continuous dynamical system is required for getting chaos. However, a single equation of discrete system can exhibit chaotic dynamics.