A state space can be discrete or continuous. For example the coin toss might be modeled by a state space consisting of two states, heads and tails. When the state space is continuous it is often a smooth manifold. In this case it is called the phase space. For example, a simple pendulum is modeled as a rigid rod that is suspended in a vertical gravitational field from a pivot that allows the pendulum to oscillate in a plane.
In addition to the pendulum's state, the model also depends upon two parameters, the pendulum's length and the strength of gravity. A phase space can also be infinite dimensional, e. This is the case for dynamics that is modeled by partial differential equations. Time may also be discrete or continuous or more generally be represented by a topological group.
Dynamical systems with discrete time, like the ideal coin toss, have their states evaluated only after certain discrete intervals. In the case of the coin toss, the smooth tumbling and bouncing of the coin is ignored, and its state is only viewed when it has come to equilibrium. Other systems that are often modeled with discrete time include population dynamics the discreteness referring to subsequent generations and impacting systems like a billiard where only the state at impact is used.
Dynamical systems first appeared when Newton introduced the concept of ordinary differential equations ODEs into Mechanics. He recognized that even differential equations can be viewed as a discrete-time systems by strobing, i.
This, of course, is required in any computational algorithm and also in any experimental measurement since it is only possible to measure finitely many values. The evolution rule provides a prediction of the next state or states that follow from the current state space value.
An evolution rule is deterministic if each state has a unique consequent, and is stochastic or "random" if there is more than one possible consequent for a given state. Deterministic evolution rules are invertible if each state has a unique precedent or preimage. Invertible maps can be continuous with continuous inverses homeomorphism or be smooth and smoothly invertible diffeomorphism.
A simple example is the Logistic map of population dynamics. This is not true generally for ODEs. A semi-flow is a flow defined only for nonnegative values of time. Semi-flows commonly arise for partial differential equations. A stochastic evolution with discrete time but continuous phase space is an iterated function system. Iterated function systems can generate interesting dynamics even when the functions are contraction maps. In this case the orbits are often attracted to some fractal set. The symbol '' 1 '' represents the re-injection of the orbit towards the saddle cycle, while the infinite number of symbols '' 0 '' represent the divergent movement of the orbit around this point.
Such an orbit is represented by the last sequence that appears in a complete and infinite alternating binary tree, so the observation of a homoclinic orbit in a dynamical system implies the existence of an infinite number of unstable periodic orbits, of all possible periods. The occurrence of a homoclinic orbit has been identified as the mechanism responsible for the onset of chaos in many dynamical systems.
The Rossler attractor  and some experimental systems as glow discharge  or lasers with a saturable absorber  are well-studied examples of more complex systems that have their chaotic behavior associated with the presence of a homoclinic orbit.
In such cases the system is said to display homoclinic chaos. The analysis of those dynamical systems has been based on the identification of homoclinic bifurcations  - a sequence of saddle-node bifurcations that alternates with period-doubling bifurcations - that leads to the appearance of a homoclinic orbit. The occurrence of such a sequence of bifurcations, before the observation of a homoclinic orbit, has many times been considered sufficient to assume homoclinicity.
This sequence induces a ramified structure in first-return maps, builded in a special way so they can capture the number of turns that a trajectory performs around a saddle point or an unstable hyperbolic saddle cycle [2,3] before being reinjected in the attractor. Each of the branches of the first-return maps is associated with a specific number of turns that the system gives around the unstable saddle point or cycle. The extrapolation, for smaller values of the control parameter R , of the sequence of unstable periodic orbits found in attractors A, B and C and shown, respectively, in tables I , II and III , allows us to conjecture about the existence of a homoclinic orbit.
This is better seen in table IV , which shows, for attractors A, B and C , the last level n for which all sequences predicted by the alternating binary tree have been found. One can observe that the last sequence in the alternating binary tree is, up to some level n , of the form , with increasing values of n for decreasing values of R. The increase in the number of zeros in the orbits of table IV is associated with an increase in the divergent movement around the saddle cycle, in a typical behavior of systems with a homoclinic orbit.
The analysis of the symbolic planes and fragmentation patterns gives further support to this conjecture. The changes in the dynamics introduced by the development of chaos in the Matsumoto-Chua circuit can be observed both in the first-return maps and in the symbolic planes, for attractors A, B, C , and D. The symbolic plane gives a way of summarizing the ordering of symbolic sequences in a two-dimensional map.
This method has been widely employed to identify topologically similar systems see examples in references ,  and . Each point a , b in a Cartesian plane is associated with a possible sequence in a way described below. If all possible orbits are present, the plane is completely filled. Empty points, represent orbits that have been pruned and are forbidden. From the symbolic plane it is possible to identify the pruning fronts and the kneading sequences. The coordinates a , b of a specific orbit or trajectory is built in a unique way from the symbolic sequence of zeros and ones that represents the orbit, through the following procedure.
Let the symbolic sequence. The symbol s 0 representing the current position of the system splits the sequence into two parts, a forward sequence given by s 0 s 1 s The symbolic coordinates a , b for the point that will represent this orbit are then given, in binary notation, by. This formalism can be extended to represent a dynamics described by 3 or more symbols see, for example, and . In Figs. The first return maps of attractors A, B and C show that those attractors have a unimodal behavior, and can essentially be described by a symbolic dynamics made of two symbols.
Comparing with Figs.
Now the first-return map is bimodal. The symbolic plane, now built with a dynamics of three symbols, is completely different, with many forbidden regions. In unimodal maps with a unidimensional structure, the pruning front, in the symbolic plane, is a continuous line, while in two-dimensional maps the pruning front is discontinuous . We see, however, that those pruning fronts get closer and closer to each other as we go from attractor A to C 1. In order to measure how good is the one-dimensional approximation, we can define an index d , equal to the integer part of log 2 , where a 1 and a 2 define the smaller and larger pruning fronts, respectively.
The index d gives the level, in the binary tree, up to which the behavior of the system can be considered one-dimensional. Those results show that the one-dimensional approximation becomes better and better as the chaotic regime evolves. We are aware that, if we have a 2D map, the existence of a 10 n orbit does not force the existence of all previous orbits of the natural sequence, as it happens in 1D maps. However, we think that we have strong indications that, at least in these case, as the chaotic regime evolves and the dynamics of the system becomes more and more one-dimensional, it approaches the dynamical behavior of a typical unimodal map, and our observations may in fact indicate the existence of homoclinic chaos.
It would be interesting to observe if, in other systems, for which it is well established that the hyperbolic regime is reached, the same scenario is found. The fragmentation patterns see Fig. Through those patterns one can build a pictorial representations of the symbolic sequences that represent the trajectories. We associate a black block with symbol 1 and a white block with symbol 0.
For instance, the sequence would be represented by the sequence of blocks ''black-white-white''. The blocks are placed in sequence, side by side, from left to right, up to 50 blocks; after that, a new row of symbols is added on top of the previous one. As the chaotic regime evolves, the allowed trajectories will present an increasing number of symbols 0 , associated with the divergent movement around the saddle cycle, as compared to the number of symbols ''1'' , associated with the reinjection movement.
By visual inspection, it is possible to see that the orbits of attractor A Fig. In conclusion, we have extracted the unstable periodic orbits from attractors A, B, C and D of the Matsumoto-Chua circuit. We have used symbolic dynamics in order to show that as the chaotic regime evolves the Matsumoto-Chua circuit has a dynamics that is increasing one-dimensional. We have presented numerical evidence that a homoclinic orbit may be present in the Matsumoto-Chua circuit.
We found that there is a pattern in the way new unstable periodic orbits are created as the control parameter R is continuously decreased. From this pattern, if the system had a truly 1D dynamics, it would be possible to infer the existence of a homoclinic orbit. If this behavior comes to be checked in similar systems, it will be a new approach to the problem of identifying the onset of homoclinicity that could be used in many other problems, either in experimental situations or numerical simulations.
For each of the studied attractors attractors A, B and C , we have extracted the unstable periodic orbits, built first-return maps, codified the dynamics and ordered the orbits according to the natural order in an alternating binary tree. Every level of the binary tree is then filled up as in a one-dimensional map, until the hyperbolic regime is attained.
We conjecture that, because the behavior of the system becomes close to a one-dimensional map, maybe a homoclinic orbit exists in this circuit. We also built symbolic planes and fragmentation patterns for all the studied sequences and trajectories, which gives further support to our conjecture. We acknowledge the financial support of the Brazilian agency CNPq. Gaspard, R.
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