Geometric Theory of Discrete Nonautonomous Dynamical Systems

Geometric Theory of Discrete Nonautonomous Dynamical Systems
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We go through the informational field, which is really causing the deformation of the phase space leading to these connections. Note that we will just consider the strengths which are directly pushing to create a link between nodes, given by the repulsion rates around all the intermediate nodes and the attraction rate to the final node.

This is not the only option we could have been considered, as we could also have taken into account the attraction rates between intermediate nodes.

Nonautonomous Dynamical Systems

This fact would not produce any relevant difference to our treatment. The normalization of these values so that their sum is 1 allows a natural translation into probability distributions associated to each state, one for the past and another for the future. States nodes in the informational structure are represented with parentheses as notation by the list of elements nodes of the mechanism, Fig 5 left with a value greater than 0.

The state with no elements greater than 0 is represented by 0. Inspired by IIT, the level of information of a mechanism in a state is compared both for the past cause information and the future effect information.

Non-autonomous dynamical systems

Note that each node in the IS determines a subset of it. Cause information measures the different probability distributions to the past obtained by considering the knowledge of just the structure unconstrained past, p up and the structure in the current state cause repertoire, p cr.

Cause information is the distance between both probability distributions. Effect information is computed analogously to those presented for cause information. Finally, the cause-effect information of the informational structure IS is the minimum of cause information and effect information ,. Fig 9 shows the distributions involved in the calculation of cause-effect information in state u 1 , u 3 of the informational structure in Fig 5 that is, the state in which only u 1 and u 3 have a value greater than 0.

Cause and effect probabilities of both the structure unconstrained repertoire given by 12 and dynamics cause repertoire from 13 are compared by using EMD. Cause-effect information cei is the minimum of cause information ci and effect information ei. Information is said to be integrated when it cannot be obtained by looking at the parts of the system but only at the system as a whole. We can measure the integration of an informational structure by trying to reconstruct it from all possible partitions.

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The informational structure of the partition contains the set of stable points of 19 together with the transition probability matrices for them, as defined above. Cause and effect repertoires are calculated for all of them, following 12 and The partition with the cause repertoire closer to the cause repertoire of the IS is MIP cause , the minimum information partition with respect to the cause. Fig 10 shows the minimum information partitions for the informational structure of Fig 5.

1 Introduction, definitions and first results

Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to . PDF | On Mar 4, , Robert J. Sacker and others published Geometric theory of discrete non-autonomous dynamical systems.

States of the ISs are highlighted in pink, observe that only u 1 and u 3 have values greater than 0 in the state of each IS, as the partition is for state u 1 , u 3. Both distances are calculated using EMD.

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Cause and effect repertoires of the MIP partitions for past and future are compared to the repertoires of the informational structure of the complete system. Consciousness flows at a particular spatio-temporal grain, and this is the base for the exclusion postulate in IIT 3. First, in a mechanism with many nodes, it may happen that not all of them are simultaneously contributing to the conscious experience. Indeed, it may be the case that several submechanisms of the same given mechanism are simultaneously integrating information at a given state.

Moreover, consciousness flows in time. But the way it evolves is slower than neuronal firing. In our approach, small changes in the parameters do not always imply a change in the informational structure, if parameters move within the same cone see [ 59 , 60 ]. This fact might explain how a conscious experience may persist while neural activity is chaotically changing. However, when change moves the parameters to a different cone the IS suffers a bifurcation on its structure, so changing the level of integrated information. To show this dependence but no determination between mechanisms and associated ISs, we have tried to model the continuous evolution of integrated information for simplified mechanisms. This is probably one of the virtues of our continuous approach to integrated information.

Nonautonomous Dynamical Systems

In particular, we consider the cases of totally disconnected mechanisms, lattice ones, the presence of a hub, and totally connected mechanisms, showing the key functions of the topology and strength of connections with respect to integrated information. The way we define the cause-effect power of a disconnected mechanism always leads to null integration for the level of information. We highlight that integrated information is positive only if we connect the parts in the mechanism, also showing the dependence on the value of integrated information related to the continuous change on the strenght of the connecting parameter.

Fig 12 shows an example of a disconnected mechanism. Connections are only inside each group but not from one group to the other.

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The consequence is that the IS of the mechanism behave like the Cartesian product of the ISs of the partition left and right ISs in the figure. Top : Disconnected mechanism. Left and right : Each of the partitions in the disconnected mechanism has an associated IS. The nodes of the IS of the whole mechanism not represented in the figure correspond to the Cartesian product of both smaller ISs.

For example, the values for nodes u 2 and u 3 in the state u 2 , u 3 of the whole IS are the same that their values at states u 2 and u 3 of the left and right ISs, respectively states highlighted in pink. We can set the connections between u 2 and u 3 , looking at the integration in different states of the mechanism Fig Top : Integration in state u 1 , u 3. Bottom : Integration in state u 2 , u 3 , u 4.

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Fig 14 shows the changing level of integration of the mechanism in states u 1 , u 2 , u 3 and u 1 , u 3 , u 5 as the strength of the connections grows up. Top : Integration in state u 1 , u 2 , u 3. Bottom : Integration in state u 1 , u 3 , u 5. Note that the level of integrated information is not only a consequence of the topology of the mechanism a lattice one in this case , but also on the strength of the connecting parameters and the particular state. In case of state u 1 , u 2 , u 3 the three nodes with a positive value are consecutive and in u 1 , u 3 , u 5 while u 1 and u 5 are linked in the cycle, u 3 is separated.

Fig 15 shows a mechanism where node u 3 has the role of a hub connecting all other nodes. Bottom : Integration in state u 1 , u 2 , u 3. It is observed that, even for a totally connected mechanism, integrated information is a delicate measure which generically does not hold with positive values. The concept of informational structure is not only related to the understanding of brain processes and their functionality but, more broadly, as an inescapable tool when analyzing dynamics in complex networks [ 19 , 20 ]. The architecture of biodiversity [ 63 , 64 ] is thus referred as the way in which cooperative systems of plants-animals structure their connections in order to get a mechanism complex graph achieving optimal levels in robustness and life abundance.

The nested organization of this kind of complex networks seems to play a key role for higher biodiversity. However, it is clear that the topology of these networks is not determining all the future dynamics [ 65 ], which seems also to be coupled to other inputs as modularity [ 66 ] or the strength of parameters [ 67 ].

This is also a very important task in Neuroscience [ 54 , 68 ]. Informational structures associated to phenomena described by dynamics on complex networks contain all the future options for the evolution of the phenomena, as they possess all the information on the forwards global behaviour of every state of the system and the paths to arrive into them. Moreover, the continuous dependence on the parameters measuring the strength of connections and the characterization of the informational structure allow to understand the evolution of the dynamics of the system as a coherent process, whose information is structured, so giving a comprehension to the appearance of sudden bifurcations [ 69 ].

From a dynamical system point of view [ 44 ], informational structures introduce several conceptual novelties in the standard theory. In particular, for phenomena associated with high dimensional networks or models by partial differential equations [ 15 , 18 , 29 ]. On the one hand, an informational structure has to be understood as a global attractor at each time , i.

Because we allow parameters to move in time, the informational structure also depends on time, so showing a continuous deformation of the phase space by time evolution. It is a real space-time process explaining all the possible forwards and internal backwards dynamics.

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On the other hand, the understanding of attracting networks as informational objects is also new in this framework, allowing a migration of this subject into Information Theory. It is remarkable that dynamical systems cover from trivial dynamics, as global convergence to a stationary point, to the much more richer one usually referred as chaotic dynamics [ 43 , 44 , 57 ] and dynamics of catastrophes [ 70 ].


Abstract We consider the Chua's circuit with a cubic nonlinearity. Ros, H. Baider and J. Serena Matucci , University of Florence, Italy. Joint work with Ricardo Almeida and Agnieszka B.

While the first one can be found with total generality, attractors with chaotic dynamics can be only described in detail in low dimensional systems. Note that some of the classical concepts of dynamical systems have been reinterpreted. In particular, the global attractor it is understood as a set of selected solutions organized in invariant sets and their connections in the past and future which creates the informational structure.

This network, moreover, produces a global transformation of the phase space the informational field enriching every point in the phase space on the crucial information on possible past states and possible future states. It is the continuous change on time of informational structures and fields what allows to talk on an informational flow for which we can analyse the postulates of IIT.

Our approach deals with dynamics on a graph for which a global description of the asymptotic behaviour is posible by means of the existence of a global attractor. That is, dissipative dynamical system for which a global attractor exists. It is important to remark the Fundamental Theorem of Dynamical Systems in [ 33 ], inspired in the work of Conley, because it gives a general characterisation of the phase space by gradient and recurrent points, so leading to a global description of the phase space as invariants and connections among them.

This is the really crucial point which allows for a general framework on the kind of systems to be considered. The Lotka-Volterra cooperative model we consider is for the applications to IIT, because in this case we have a detailed description of the global attractor, which allows to make concrete computations for the level of integrated information on it. Thus, in application to L-V systems, we are considering heteroclinic channels inside the global attractor.

But all the treatment in the paper can be done for every dissipative systems for which a detailed description of the global attractor is available, i. Gradient dynamics [ 15 , 27 , 29 , 31 ] describes the dynamics from heteroclinic connections and associated Lyapunov functionals [ 29 , 32 ], and it naturally suits into higher order systems including infinite dimensional phase spaces [ 27 , 34 , 35 ]. Thus, although our description of informational structures associated to Lotka-Volterra systems is general enough to describe real complex networks, we think the concept may be also well adapted both for different topologies in the networks and also different kind of non-linearities defining the differential equations [ 51 , 52 ].

Actually, for comparison with data and experiments associated for real brain dynamics we certainly need to allow much more complex networks as primary mechanisms as well as different kinds of dynamical rules and nonlinearities. But, in all of them, we expect to find the existence of dynamical informational structures providing the information on the global evolution of the system. The concise geometrical description of these structures and their continuous dependence on sets of parameters is a real open problem in the dynamical systems theory.

We have presented a preliminary approach to the notion of integrated information from informational structures, leading to a continuous framework for a theory inspired by IIT, in particular in order to define integrated information of a mechanism in a state.