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which is the expected result. Lemma 3 Let [MATH] be a nonlinear TBAN of order [MATH] and [MATH] be the nonlinear term of [MATH] when every nearest neighbour of its central node [MATH] is active. Then: |
[EQUATION] Proof. First, let us show that [MATH] . It suffices to multiply [MATH] by [MATH] [EQUATION] Given [MATH] defined by: [EQUATION] |
we have: [EQUATION] By hypothesis, [MATH] . As a consequence, we have [MATH] . Moreover, given that nonlinear term [MATH] is symmetric: |
[EQUATION] So, we can write: [EQUATION] Expanding left and right members of the equation above leads to: [EQUATION] which is equivalent to: |
[EQUATION] Let us proceed to the following change of variables: let [MATH] (resp. [MATH] ) be the denominator of the left member (resp. of the right member) and [MATH] (resp. [MATH] ) the numerator of the left member (resp. of the right member) of the equation above. We have then: |
[EQUATION] Let [MATH] be such that: [EQUATION] We have: [EQUATION] Thus, we can write: [EQUATION] And, thus, we have: [EQUATION] |
Hence, by hypothesis: [EQUATION] which is the expected result. From Lemmas and , it is easy to derive the following theorem that highlights an empirical sufficient condition of phase transitions in nonlinear TBANs of order [MATH] on [MATH] |
Theorem 1 Let [MATH] be a nonlinear TBAN of order [MATH] . We have: [EQUATION] which means that the symmetry property of the non linear term is an empirical sufficient condition for detM to vanish, allowing consequently phase transitions to occur. |
Proof. From Lemma and because of the parity of the cardinal of [MATH] , we can write: [EQUATION] Then Lemma leads to: [EQUATION] |
which is always true. As a result, since Lemmas and are based on the hypothesis of symmetry of the non linear term, we have from Lemma [EQUATION] which is the expected result. |
# Source: arxiv 1011.5492 # Title: Chimera states in coupled sine-circle map lattices # Sections: all # Downloaded: 2026-03-02T08:58:32.390003+00:00 |
Chimera states in coupled sine-circle map lattices Abstract Systems of coupled oscillators have been seen to exhibit chimera states, i.e. states where the system splits into two groups where one group is phase locked and the other is phase randomized. In this work, we report the existence of chimera states in a system ... |
Introduction Coupled oscillator systems have long been studied as good models of a variety of experimental and natural systems, as well as paradigmatic systems for the observation and analysis of complex spatio-temporal behaviour. The study of synchronised behaviour in systems of identical oscillators, as well as that ... |
In this work, we study the existence and stability of chimera states in two interacting populations of coupled sine circle maps. These systems are the map analogs of coupled oscillator systems. Coupled map systems show many of the varied phenomena observed in the case of continuous extended systems, but are far more nu... |
The model The single sine circle map evolves via the evolution equation [EQUATION] Here, [MATH] is the nonlinearity parameter and [MATH] is the winding number of a single sine circle map in the absence of the nonlinearity. This map shows a tendency to mode lock as the parameter [MATH] is increased and the phenomena of ... |
[MATH] space jensen:pra:84 . The winding numbers of the mode locked tongues show a Devil’s staircase structure. There are several studies on the spatiotemporal dynamics of diffusively coupled sine circle map lattices on regular sites nandini:pre:96 The model that we consider consists of two interacting populations of... |
[EQUATION] where [MATH] and [MATH] [MATH] is the angle at time [MATH] and lies between [MATH] and [MATH] . The parameters [MATH] and [MATH] are taken to be uniform at each site. The values of [MATH] and [MATH] should lie between [MATH] and [MATH] and [MATH] . We considered [MATH] oscillators in each group and the resul... |
The synchronization within the population of each kind can be characterised by the order parameter [MATH] where the angle brackets denotes the average over the oscillators having the same dynamics. Thus for the completely synchronized, and the chimera state, the average is taken over all the oscillators of group 2. The... |
We explore the phase diagrams of the system at [MATH] values which exhibit the variety of possible solutions and also provide clues to the bifurcation sequences that take place. |
The phase diagrams and stability analysis 3.1 The [MATH] phase space We explore the [MATH] phase space with [MATH] , where a good spectrum of dynamical behaviour is seen and map out the observed types of spatial behavior. The value of [MATH] is varied from 0 to 1 and [MATH] from 0 to 0.5. The phase diagram is symmetric... |
the system either is either in a completely synchronized state or a two clustered state depending upon the [MATH] values for the initial conditions mentioned before. The yellow [MATH] -s and the black [MATH] -s represent the completely synchronised and two clustered state respectively and form the most observed states.... |
Now we study three [MATH] sections at three distinct values of [MATH] . One is the shift map case where [MATH] , another is the [MATH] plane which is known to mode lock and the third is a low value ( [MATH] ), where chimera states are seen and there is rich dynamical behavior. |
3.2 The Shift Map case The simplest case of sine circle maps is the shift map case with [MATH] . For a single shift map given by |
[EQUATION] the system has periodic orbits for rational values of [MATH] The corresponding equation for the coupled system is [EQUATION] |
For the shift map case, the system always evolves to a chimera state for the initial conditions mentioned earlier. The numerically evaluated largest lyapunov exponent shows that the system is chaotic in the entire [MATH] phase space. The order parameter defined by [MATH] remained constant with very small fluctuations w... |
[EQUATION] where [EQUATION] [EQUATION] [EQUATION] [EQUATION] We have [MATH] and [MATH] . The eigen values need to be numerically obtained since the stable state is the chimera state. |
3.3 Linear Stability Analysis for [MATH] Fig. shows that in the [MATH] phase space plane with [MATH] , the system is always having either a complete synchronized state or a two clustered state. We study this plane analytically to obtain the bifurcation boundary. In the completely synchronized state |
[MATH] where [MATH] varies from [MATH] to [MATH] . In this case the Jacobian matrix reduces to [EQUATION] with [EQUATION] and [EQUATION] |
This can be reduced by a similarity transformation to the form [EQUATION] Now the Jacobian matrix has a block diagonalized form with each nonzero block being a circulant matrix such that |
[EQUATION] and [EQUATION] This matrix has eigenvalues [MATH] [MATH] and [MATH] fold degenerate eigen values [MATH] The Lyapunov exponents in terms of the eigen-values of the Jacobian matrix can be written as |
[EQUATION] [EQUATION] and [EQUATION] The numerically obtained value of the largest lyapunov exponent for the synchronised state agrees with the value given by Eq. . Fig. (a) shows the numerical LE for the entire [MATH] and the system is chaotic for most of the values. In the periodic regions, for [MATH] and [MATH] the ... |
3.4 The [MATH] parameter space for [MATH] The third slice along [MATH] exhibits rich spatial dynamics. As noted earlier, chimera states are easily found at the lower values of [MATH] . For [MATH] above [MATH] and for [MATH] near [MATH] the system has clustered chimera states (black [MATH] -s) which bifurcate to cluster... |
Conclusions In this paper, we proposed a couple map model with two interacting species of sine circle maps coupling. Chimera states are easily seen in this system given two sets of initial conditions for the two species, one random and one synchronised. Special choices of initial conditions are not required. The [MATH]... |
Acknowledgements. NG thanks CSIR for partial support in this work. Travel support was provided by the Office of Naval Research Global (ONR Global) to NG and CRN :under the Visiting Scientist Program. |
# Source: arxiv 1012.1221 # Title: Infinite Time Cellular Automata: A Real Computation Model # Sections: all # Downloaded: 2026-03-02T08:58:29.501364+00:00 |
Fabien Givors Gregory Lafitte Nicolas Ollinger Infinite Time Cellular Automata: a Real Computation Model Abstract We define a new transfinite time model of computation, infinite time cellular automata . The model is shown to be as powerful than infinite time Turing machines, both on finite and infinite inputs; thus inh... |
Introduction When the second and third authors of this paper were in their PhD years, their common advisor, Jacques Mazoyer, had encouraged them to define a computation model on real numbers using cellular automata. The second author had defined at the end of the Nineties a cellular automata generalization running into... |
Transfinite time computation models were first considered in 1989 by Hamkins and Kidder. Hamkins and Lewis later developed the model of infinite time Turing machines and its theory in HL00 . Koepke Koe06 defined another transfinite time model based on register machines, that was later refined by Koepke and Miller KM08 ... |
In BSS89 , Blum, Shub and Smale introduced, also in 1989, a model of computation, coined BSS machines , intended to describe computations over the real numbers. It can be viewed as Turing machines with tapes whose cells (or Random Access Machines with registers that) can store arbitrary real numbers and that can comput... |
In this paper, we introduce a transfinite time computation model, the infinite time cellular automata The model is arguably more natural and uniform than other transfinite time models introduced, for the same reasons cellular automata are more natural and uniform than Turing machines. There is no head wandering here an... |
We show that the infinite time cellular automata have the same computing power than infinite time Turing machines, both on finite and infinite inputs. They thus inherit the nice properties of this latter model. We then show how to simulate the BSS machines with infinite time cellular automata in exactly [MATH] steps. W... |
Infinite time cellular automata {defi} An infinite time cellular automaton (ITCA) [MATH] is defined by [MATH] , the finite set of states of [MATH] , linearly ordered by [MATH] and with a least element |
[MATH] and [MATH] , the local rule of [MATH] satisfying [MATH] , so that [MATH] is a quiescent state. configuration is an element of [MATH] The local rule [MATH] induces a global rule [MATH] on configurations such that |
[MATH] for [MATH] and [MATH] Starting from a configuration [MATH] , the evolution of length [MATH] of [MATH] is given by [MATH] [EQUATION] |
{defi} Let [MATH] a particular state we will refer to as the halting state. An evolution of length [MATH] is called a computation if the state [MATH] appears in the last configuration, [MATH] , but not before this stage. |
We settle a convention on the way we code integers or real numbers in our model. For example, we could code integers and real numbers in binary (using [MATH] and another state as symbols for [MATH] and [MATH] ) on the right cells (cells whose indices belong to [MATH] ). |
{defi} Let [MATH] be a space for which a coding has been settled. (For example, [MATH] [MATH] [MATH] or [MATH] .) A (partial) function [MATH] on [MATH] [MATH] , is said to be infinite time computable if there is an infinite time cellular automaton such that for each [MATH] , there is a computation starting with a confi... |
A set [MATH] is infinite time decidable if its characteristic function is infinite time computable, and is infinite time semi-decidable if it is the domain of an infinite time computable function. |
We use the term “semi-decidable” instead of “enumerable”, since contrarily to the classical computability concepts, in the transfinite time context, being semi-decidable is not equivalent to being the range of a computable function. |
Properties of infinite time cellular automata 2.1 Comparisons with other infinite time models Hamkins, Kidder and Lewis HL00 have defined an infinite time Turing machines model and Koepke Koe06 has defined an infinite time register machines model, that was later refined by Koepke and Miller KM08 |
The infinite time Turing machines (ITTM) work as a classical Turing machine with a head reading and writing on a bi-infinite tape, moving left and right in accordance with the instructions of a finite program with finitely many states. At successor stages of computation, the machine operates in exactly the classical ma... |
The infinite time register machines behave like standard register machines at successor stages. At limit times, the register contents are defined using lim inf’s of the previous register contents. The difficulty here is that the lim inf does not necessarily exist in that case, since a register can contain arbitrary lar... |
The infinite time Turing machines are strictly stronger than infinite time register machines: the halting problem for infinite time register machines can be decided by an ITTM. |
Theorem 1 Infinite time cellular automata have the same computing power of infinite time Turing machines. Proof 2.1 The right to left implication goes as follows: |
At successor stages, the simulation of ITTM by ITCA works the same way it does in the non-infinite-time case. At limit stages, we have to put the configuration of the ITCA in the limit configuration simulation of the ITTM simulated. To do this, we need to be able to know that we are at a limit stage. It suffices to hav... |
[MATH] . We also have to use the same trick on the cells visited by the head to make sure that at limit stages, if the ITTM becomes stationary, we can wipe out the stationary state from our simulation tape and enter in the special limit state. The ITCA can then prepare the configuration to continue the ITTM simulation. |
The left to right implication: it takes [MATH] steps with an ITTM machine to simulate an ITCA global step (on an infinite input). It is just then a matter of determining whether the ITTM is at a limit stage or not. This is easily achieved by an ITTM since it enters a special limit state at limit stages. |
2.2 Features of those infinite time models Hamkins, Kidder, Lewis and Welch HL00 Wel99 Wel00b Wel00a have shown many properties of infinite time Turing machines. By Theorem , infinite time cellular automata have many of these same properties. We state in the following the properties inherited by infinite time cellular ... |
Theorem 2 The set of reals coding well-orders is infinite time decidable. The hyperarithmetic sets are those that are decidable in time less than some recursive ordinal. Every [MATH] set is decidable and the class of decidable sets is contained in [MATH] |
{defi} An ordinal [MATH] is clockable if there is an ITCA computation starting from the all-but-one quiescent configuration [MATH] [MATH] and [MATH] ) and that halts after exactly [MATH] steps (meaning that the [MATH] configuration, [MATH] , is the first configuration in which the halting state [MATH] appears). |
A real [MATH] is writable if it is the output of an ITCA computation. An ordinal is writable if it is coded by such a real. There are of course only countably many clockable and writable ordinals, since there are only countably many local rules. |
Theorem 3 Every recursive ordinal is clockable. Even [MATH] is clockable. Beyond that, there are many intervals of non-clockable ordinals. The supremum of clockable ordinals is recursively inaccessible . Moreover, the writable ordinals however form an initial segment of the ordinals. The supremum of the writable ordina... |
One of the beautiful theorems of ITTMs that carry through to ITCAs is the Lost Melody Theorem. The real constructed in this theorem is like a lost melody that you can recognize when someones hums it to you, but which you cannot sing on your own. |
Theorem 4 (Lost Melody Theorem) There is a real [MATH] which is recognizable ( [MATH] is decidable), but not writable. There are different ways to construct such lost melody reals. One way is to consider the supremum [MATH] of the ordinal stages by which an ITTM computation, on an empty input, either halts or repeats. ... |
[MATH] (and thus [MATH] ) are somehow a generalization of the busy beaver problem to transfinite time computations. It is then not surprising that [MATH] cannot be computable, since that would render the infinite time halting problem decidable. It is recognizable because it is possible to reconstruct the [MATH] hierarc... |
Computations on the reals 3.1 Blum-Shub-Smale model Blum, Shub and Smale BSS89 introduced the BSS model A simplified presentation of the BSS model goes through defining “Turing machines with real numbers”. We follow Hainry’s presentation in his PhD thesis Hai06 |
{defi} simplified BSS machine (or shortly, BSS machine) is composed of an infinite tape and a program. The infinite tape is made of cells, each containing a real number. We denote the tape by [MATH] The program is a numbered (finite) sequence of instructions. The number of each instruction in the program sequence is se... |
go right : changes the tape to [MATH] go left : changes the tape to [MATH] branch if greater than [MATH] : if the current cell ( [MATH] ) is greater than [MATH] , then it branches to a specified location in the program; |
branch if equal to [MATH] : if the current cell ( [MATH] ) is equal to [MATH] , then it branches to a specified location in the program; |
make a computation : the current cell ( [MATH] ) is changed to be equal to the result of a computation from [MATH] [MATH] and possible constants ( [MATH] ). A computation is one of the following: |
[MATH] [MATH] [MATH] [MATH] [MATH] The machine starts by executing the first instruction (with the least number) of the program and continues by executing the next instruction and so on until it branches. It halts when it has no more instructions to execute. |
It is possible to give a definition for a more general BSS machine on rather arbitrary structures. In this paper, we will stick with simplified BSS machines on [MATH] |
For a lot more on the BSS model and computation on the real numbers, the reader is referred to the book by Blum, Cucker, Shub and Smale BCSS98 |
3.2 BSS by [MATH] -ITCA We show how to simulate a simplified BSS machine with an ITCA. Theorem 5 A simplified BSS machine can be simulated by an ITCA in [MATH] steps. |
Proof 3.1 Consider a BSS machine. At each time step [MATH] of a computation, only a finite number of non-zero real numbers are defined: [MATH] constants inside the program and [MATH] cells of the tape. For the sake of clarity, suppose that the BSS machine works on reals in the interval [MATH] . Imagine that we encode t... |
Each computation instruction of the BSS machine has the nice property that it can be computed, in time [MATH] , by a Turing machine working synchronously on the representation of its operands. For each finite initial portion of the tape, it is left untouched by such a machine after some finite time. Moreover, at the co... |
Each branch instruction of the BSS machine can be achieved in a similar way: one can choose a initial hypothesis on the branch and start a computation by a Turing machine working synchronously on the representation of the operands ; either the machine eventually halts contradicting the hypothesis, or its head moves inf... |
Packing it all together, we can simulate a BSS machine if we can launch as many Turing computation threads as needed. Encode the [MATH] reals encodings as a single infinite word and put some finite control at the beginning of it. The finite control plays the role of the head of the BSS machine, keeping the current stat... |
Notice that a key point of the construction is that there is always enough room for bookkeeping: if a thread needs more space, it just can wait for more space to be available before continuing its computation, either it will eventually happen, or a backtrack will occur that can be handled. |
The result of a computation is obtained easily: at time [MATH] , if the control eventually converged to an accepting state, the control part encodes an accepting state and the encoded reals contain the result of the computation ; if the BSS machine did not converge, the control part does not encode an accepting state. |
Let us now explain how the described simulation can be carried on a ITCA. Given a BSS machine, the ITCA is constructed as follows. Only a semi-infinite part of the configuration is used. Cell [MATH] encodes control and the cells on its right encode the reals, the computation area. The computation area is constructed in... |
Each instruction of the BSS machine is simulated as explained before. A move thread simply moves the circle to the previous or next bit, adding a new real if necessary, using its thread stack. A branch thread does not modify reals but checks if the hypothesis was true or false, as depicted on figure . A computation thr... |
It is important to notice that the monotonicity of the forward movement of [MATH] symbols ensures that there is no concurrency problem due to neighbor threads backtracking and coming back forward in a same area: when a thread wants to access an area already explored by its follower, the follower is asked to undo its co... |
The details of the construction use rather classical but tedious CA encoding tricks. The key argument of the proof is that an ITCA can simulate in time [MATH] the work of an unbounded number of Turing heads, thus achieving the same quantity of work than a ITTM in time [MATH] |
By diagonalization, it is easy to see that there are functions on the real numbers, computable by ITCAs in [MATH] steps but that are not computable by BSS machines. |
Concluding remarks We finish this paper by pointing in a direction that we believe to be promising. Koepke Koe05 has defined a transfinite time computation model based on Turing machines that has transfinite space. It can thus compute on arbitrary ordinals and sets of ordinals. Koepke and Siders KS08 have also extended... |
We propose the following definition for ordinal computations over cellular automata. {defi} An ordinal cellular automaton [MATH] is defined by [MATH] , the finite set of states of [MATH] , linearly ordered by [MATH] and with a least element |
[MATH] and [MATH] , the local rule of [MATH] satisfying [MATH] , so that [MATH] is a quiescent state. configuration of length [MATH] is an element of [MATH] . The local rule [MATH] induces on configurations of length [MATH] a global rule |
[MATH] such that [EQUATION] Starting from a configuration [MATH] , the evolution of length [MATH] of [MATH] is given by [MATH] [EQUATION] |
We think that it would be interesting to try to carry the results of the other ordinal machines models over to this ordinal cellular automata model. In this model, there is the limitation due to the fixed length of configurations, which is imposed by the cylindrical nature of our model. There are certainly other ways t... |
# Source: arxiv 1104.5538 # Title: Complex Networks # Sections: all # Downloaded: 2026-03-03T01:56:19.337732+00:00 Complex Networks |
Graph theory was initiated by Euler in the eighteenth century. In mathematics, graph theory consolidated itself in the following decades and centuries. However, it was only until a little more than a decade ago that an explosion of research and applications ocurred, in what is now referred to as network science |
. In particular, the networks have become a central tool in the study of complex systems . The language of “nodes and edges” provided by networks has proven to be very illustrative to model elements of a system (nodes) and their interactions (edges). Having a language that describes interactions is essential for a non-... |
The relevance of the study complex networks resides in the fact that so many real networks do not have a trivial topology. It follows that their properties are also not trivial, opening many research avenues. Examples of these properties include the small world effect, scale-free topologies, modularity, robustness, evo... |
Within ALife, almost all topics benefit from the study of complex networks, since the connectivity of systems is strongly related to their function. For example, cortical networks, genetic regulatory networks, metabolic pathways, artificial chemistries, and ecological webs describe phenomena in terms of nodes and links... |
For this reason, we decided to organize a special session on complex networks at the ALife XII conference in Odense, Denmark, which was held on August [MATH] and [MATH] , 2010. The intention of the session was to foster cross-fertilization between the ALife and complex networks communities. Following the success of the... |
We received fifteen submissions, out of which eight papers were selected with the valuable aid of multiple thorough reviews. A generic unifying framework for diverse complex real-world networks has not yet been developed, and in part this is due to a limited number of available examples of these networks. As pointed ou... |
this can be addressed by development of re-wiring algorithms capable of generating networks with specific characteristics. Such characteristics may, for example, combine scale-free properties and community structures encountered in the real-world. The re-wiring algorithm presented in this work is inspired by observatio... |
Brede presents another model of network generation , where the rates of random addition of nodes and optimal rewiring are explored to generate complex networks with power law tails in degree distributions, hierarchies, non-trivial clustering and degree mixing patterns. |
Another step towards a generic framework for networks science is made by Lizier et al. , who investigate computational capabilities of small-world networks in terms of information-theoretic measures. The analysis includes topological and dynamical phase transitions, and associates specific modes of computation (such as... |
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