Continuation and bifurcation associated to the dynamical spectral sequence

2013 ◽  
Vol 34 (6) ◽  
pp. 1849-1887 ◽  
Author(s):  
R. FRANZOSA ◽  
K. A. DE REZENDE ◽  
M. R. DA SILVEIRA
Keyword(s):  
Spectral Sequence ◽  
Directed Graphs ◽  
Chain Complex ◽  
Morse Decomposition ◽  
Connection Matrix ◽  
Transition Matrices ◽  
Dynamic Interpretation ◽  
Connection Matrices ◽  
Parameterized Family ◽  
Bifurcation Behavior

AbstractIn this paper we consider a filtered chain complex $C$ and its differential given by a connection matrix $\Delta $ which determines an associated spectral sequence $({E}^{r} , {d}^{r} )$. We present an algorithm which sweeps the connection matrix in order to span the modules ${E}^{r} $ in terms of bases of $C$ and gives the differentials ${d}^{r} $. In this process a sequence of similar connection matrices and associated transition matrices are produced. This algebraic procedure can be viewed as a continuation, where the transition matrices give information about the bifurcation behavior. We introduce directed graphs, called flow and bifurcation schematics, that depict bifurcations that could occur if the sequence of connection matrices and transition matrices were realized in a continuation of a Morse decomposition, and we present a dynamic interpretation theorem that provides conditions on a parameterized family of flows under which such a continuation could occur.

A global two-dimensional version of Smale’s cancellation theorem via spectral sequences

2015 ◽  
Vol 36 (6) ◽  
pp. 1795-1838 ◽  
Author(s):  
M. A. BERTOLIM ◽  
D. V. S. LIMA ◽  
M. P. MELLO ◽  
K. A. DE REZENDE ◽  
M. R. DA SILVEIRA
Keyword(s):  
Sequence Analysis ◽  
Spectral Sequence ◽  
Matrix Theory ◽  
Chain Complex ◽  
Connection Matrix ◽  
Local Version ◽  
Two Dimensional ◽  
Dimensional Version ◽  
Cancellation Theorem ◽  
Flows On Surfaces

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

scholarly journals Morse decompositions and connection matrices

1988 ◽  
Vol 8 (8) ◽  
pp. 227-249 ◽  
Keyword(s):  
Metric Spaces ◽  
Morse Decomposition ◽  
Connection Matrix ◽  
Matrix Formalism ◽  
Compact Metric Spaces ◽  
Connection Matrices ◽  
Morse Decompositions ◽  
Morse Sets

AbstractThis paper surveys the work of Charles Conley and his students on Morse decompositions for flows on compact metric spaces, as well as the more recent development of the connection matrix formalism for detecting connections between the Morse sets of a Morse decomposition.

scholarly journals Spectral sequences in Conley’s theory

2009 ◽  
Vol 30 (4) ◽  
pp. 1009-1054 ◽  
Author(s):  
O. CORNEA ◽  
K. A. DE REZENDE ◽  
M. R. DA SILVEIRA
Keyword(s):  
Spectral Sequence ◽  
Flow Line ◽  
Reverse Flow ◽  
Chain Complex ◽  
Connection Matrix ◽  
Spectral Sequences ◽  
Direct Connection ◽  
Zero Differential ◽  
Action Type ◽  
A Chain

AbstractIn this paper, we analyse the dynamics encoded in the spectral sequence (Er,dr) associated with certain Conley theory connection maps in the presence of an ‘action’ type filtration. More specifically, we present an algorithm for finding a chain complex C and its differential; the method uses a connection matrix Δ to provide a system that spans Er in terms of the original basis of C and to identify all of the differentials drp:Erp→Erp−r. In exploring the dynamical implications of a non-zero differential, we prove the existence of a path that joins the singularities generating E0p and E0p−r in the case where a direct connection by a flow line does not exist. This path is made up of juxtaposed orbits of the flow and of the reverse flow, and proves to be important in some applications.

scholarly journals Inverse Problem for Ising Connection Matrix with Long-Range Interaction

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1624
Author(s):  
Leonid Litinskii ◽  
Boris Kryzhanovsky
Keyword(s):  
Inverse Problem ◽  
Long Range ◽  
Random Number ◽  
Connection Matrix ◽  
Range Interaction ◽  
Connection Matrices ◽  
Ising Systems ◽  
Long Range Interaction ◽  
Matrix Spectrum ◽  
Analytical Expressions

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of its eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions.

scholarly journals Connecting orbits in one-parameter families of flows

1988 ◽  
Vol 8 (8) ◽  
pp. 359-374 ◽  
Keyword(s):  
Phase Space ◽  
Parameter Space ◽  
Product Space ◽  
Conley Index ◽  
Morse Decomposition ◽  
Connection Matrix ◽  
Connecting Orbits ◽  
Original Family ◽  
Parameter Interval ◽  
Parameter Values

AbstractGiven a family of flows parametrized by an interval and a Morse decomposition which continues across the interval, a procedure is devised to detect connecting orbits at various parameter values. This is done by putting a small drift on the parameter space and considering the flow on the product of the phase space and the parameter interval. The Conley index and connection matrix are used to analyse the flow on the product space, then the drift is allowed to go to zero to obtain information about the original family of flows. This method can be used to detect connections between rest points of the same index for example.

USING A FEED-FORWARD NETWORK TO INCORPORATE THE RELATION BETWEEN ATTRACTEES AND ATTRACTORS IN A GENERALIZED DISCRETE HOPFIELD NETWORK

1996 ◽  
Vol 07 (03) ◽  
pp. 273-286 ◽  
Author(s):  
ROELOF K. BROUWER
Keyword(s):  
Pattern Recognition ◽  
Cost Function ◽  
Gradient Descent ◽  
Descent Method ◽  
Hopfield Network ◽  
Connection Matrix ◽  
Gradient Descent Method ◽  
Feedforward Network ◽  
Feed Forward Network ◽  
Connection Matrices

This paper demonstrates how a feedforward network with constant connection matrices may be used to train a Hopfield style network for pattern recognition. The connection matrix of the Hopfield style network is asymmetric and its diagonal is non-zero. The Hopfield style network referred to as a GDHN is trained to incorporate a relation between attractees and attractors. The attractees represent class samples and the attractors represent class prototypes. The feedforward network is trained using a gradient descent method. Gradients are fed forward in the network to obtain a gradient for a cost function.

scholarly journals Can we conjointly record direct interactions between neurons in vivo in anatomically-connected brain areas? Probabilistic analyses and further implications

2020 ◽  
Author(s):  
Sean K. Martin ◽  
John P. Aggleton ◽  
Shane M. O’Mara
Keyword(s):  
Large Scale ◽  
Directed Graphs ◽  
Brain Regions ◽  
Anatomical Connectivity ◽  
Brain Areas ◽  
Transformation Rules ◽  
Connection Matrices ◽  
Source Transformation ◽  
Alternative Approaches

AbstractLarge-scale simultaneous in vivo recordings of neurons in multiple brain regions raises the question of the probability of recording direct interactions of neurons within, and between, multiple brain regions. In turn, identifying inter-regional communication rules between neurons during behavioural tasks might be possible, assuming conjoint activity between neurons in connected brain regions can be detected. Using the hypergeometric distribution, and employing anatomically-tractable connection mapping between regions, we derive a method to calculate the probability distribution of ‘recordable’ connections between groups of neurons. This mathematically-derived distribution is validated by Monte Carlo simulations of directed graphs representing the underlying anatomical connectivity structure. We apply this method to simulated graphs with multiple neurons, based on counts in rat brain regions, and to connection matrices from the Blue Brain model of the mouse neocortex connectome. Overall, we find low probabilities of simultaneously-recording directly interacting neurons in vivo in anatomically-connected regions with standard (tetrode-based) approaches. We suggest alternative approaches, including new recording technologies and summing neuronal activity over larger scales, offer promise for testing hypothesised interregional communication and source transformation rules.

COMPUTATION OF CONNECTION MATRICES USING THE SOFTWARE PACKAGE conley

2009 ◽  
Vol 19 (09) ◽  
pp. 3033-3056
Author(s):  
MOHAMED BARAKAT ◽  
STANISLAUS MAIER-PAAPE
Keyword(s):  
Software Package ◽  
Index Theory ◽  
Conley Index ◽  
Bifurcation Parameter ◽  
Transition Matrices ◽  
Hilliard Equation ◽  
Conley Index Theory ◽  
Connection Matrices ◽  
Cahn Hilliard Equation ◽  
Definition Of

In this paper we demonstrate the power of the computer algebra package conley, which enables one to compute connection and transition matrices, two of the main algebraic tools of the CONLEY index theory. In particular, we study the CAHN–HILLIARD equation on the unit square and extend the results obtained in [Maier-Paape et al., 2007] to a bigger range of the bifurcation parameter. Besides providing several explicit computations using conley, the definition of connection matrices is reconsidered, simplified, and presented in a self-contained manner in the language of CONLEY index theory. Furthermore, we introduce so-called energy induced bifurcation intervals, which can be utilized by conley to differential equations with a parameter. These bifurcation intervals are used to automatically path-follow the set of connection matrices at bifurcation points of the underlying set of equilibria.

DIRECTED GRAPHS AND KRONECKER INVARIANTS OF PAIRS OF MATRICES

2008 ◽  
Vol 17 (01) ◽  
pp. 75-132 ◽  
Author(s):  
JACOB TOWBER
Keyword(s):  
Directed Graphs ◽  
Isomorphism Problem ◽  
Connection Matrix ◽  
Combinatorial Properties ◽  
Irreducible Factorization ◽  
Pleasant Surprise ◽  
Complete Set ◽  
Finite Directed Graphs ◽  
Invertible Matrices ◽  
The Given

Call two pairs (M,N) and (M′,N′) of m × n matrices over a field K, simultaneously K-equivalent if there exist square invertible matrices S,T over K, with M′ = SMT and N′ = SNT. Kronecker [2] has given a complete set of invariants for simultaneous equivalence of pairs of matrices. Associate in the natural way to a finite directed graph Γ, with v vertices and e edges, an ordered pair (M,N) of e × v matrices of zeros and ones. It is natural to try to compute the Kronecker invariants of such a pair (M,N), particularly since they clearly furnish isomorphism-invariants of Γ. Let us call two graphs "linearly equivalent" when their two corresponding pairs are simultaneously equivalent. There have existed, since 1890, highly effective algorithms for computing the Kronecker invariants of pairs of matrices of the same size over a given field [1,2,5,6] and in particular for those arising in the manner just described from finite directed graphs. The purpose of the present paper, is to compute directly these Kronecker invariants of finite directed graphs, from elementary combinatorial properties of the graphs. A pleasant surprise is that these new invariants are purely rational — indeed, integral, in the sense that the computation needed to decide if two directed graphs are linearly equivalent only involves counting vertices in various finite graphs constructed from each of the given graphs — and does not involve finding the irreducible factorization of a polynomial over K (in apparent contrast both to the familiar invariant-computations of graphs furnished by the eigenvalues of the connection matrix, and to the isomorphism problem for general pairs of matrices).

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