Narrow Results

Using the Routh–Hurwitz stability criterion and a systematic computer search, 23 simple chaotic flows with quadratic nonlinearities were found that have the unusual feature of having a coexisting stable equilibrium point. Such systems belong to a newly introduced category of chaotic systems with hidden attractors that are important and potentially problematic in engineering applications.

The Cardiac Purkinje Fiber (CPF) is the last branch of the heart conduction system, which is meshed with the normal ventricular myocyte. Purkinje fiber plays a key role in the occurrence of ventricular arrhythmia and maintenance. Does the heart Purkinje fiber cells have the same memory function as the cerebral nerve? In this paper, the cardiac Hodgkin–Huxley equation is taken as the object of study. In particular, we find that the potassium ion-channel $K$ and the sodium ion-channel $Na$ are memristors. We also derive the small-signal equivalent circuits about the equilibrium points of the CPF Hodgkin–Huxley model. According to the principle of local activity, the regions of Locally-Active domain, Edge of Chaos domain and Locally-Passive domain are partitioned under parameters $(a,b)$, and the domain exhibiting the normal human heartbeat frequency range (Goldilocks Zone) is identified. Meanwhile, the Super-Critical Hopf bifurcation of the CPF Hodgkin–Huxley model is identified. Finally, the migration changes between different state domains under external current $I_{\mathrm{\text{ext}}}$ excitation are analyzed in detail.

All of the above complex nonlinear dynamics are distilled and mapped geometrically into a surreal union of intersecting two-dimensional manifolds, dubbed the Hodgkin–Huxley’smagic roof.

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

This paper investigates the Li–Yorke chaos in linear systems with weak topology on Hilbert spaces. A weak topology induced by bounded linear functionals is first constructed. Under this weak topology, it is shown that the weak Li–Yorke chaos can be equivalently measured by an irregular or a semi-irregular vector, which are utilized to establish criteria for the weak Li–Yorke chaos of diagonalizable operators, Jordan blocks, and upper triangular operators. In particular, for a linear operator that can be decomposed into a direct sum of finite-dimensional Jordan blocks, it is Li–Yorke chaotic in weak topology if its point spectrum contains a pair of real opposite eigenvalues with absolute values not less than 1, or a pair of complex conjugate eigenvalues with moduli not less than 1. Interestingly, as a specific example of upper triangular operator, the existence of Li–Yorke chaos in weak topology can be derived for a class of linear operators expressed as the direct sum of finite-dimensional Jordan blocks and a strongly irreducible operator.

In this paper, we compute the Hausdorff dimension of a graph-directed set when the underlying multigraph is a Cartesian product or a tensor product of several multigraphs. We give explicit formulas in terms of the eigenvalues of the graph and the similarity ratios used with each graph.

The eigenvalues of the transition matrix of a weighted network provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to biased walks. Although various dynamical processes have been investigated in weighted networks, analytical research about eigentime identity on such networks is much less. In this paper, we study analytically the scaling of eigentime identity for weight-dependent walk on small-world networks. Firstly, we map the classical Koch fractal to a network, called Koch network. According to the proposed mapping, we present an iterative algorithm for generating the weighted Koch network. Then, we study the eigenvalues for the transition matrix of the weighted Koch networks for weight-dependent walk. We derive explicit expressions for all eigenvalues and their multiplicities. Afterwards, we apply the obtained eigenvalues to determine the eigentime identity, i.e. the sum of reciprocals of each nonzero eigenvalues of normalized Laplacian matrix for the weighted Koch networks. The highlights of this paper are computational methods as follows. Firstly, we obtain two factors from factorization of the characteristic equation of symmetric transition matrix by means of the operation of the block matrix. From the first factor, we can see that the symmetric transition matrix has at least $3\cdot 4^{g-1}$ eigenvalues of $-\frac{1}{2}$. Then we use the definition of eigenvalues and eigenvectors to calculate the other eigenvalues.

In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates $q$ connected branches in each iteration process and assigns the corresponding weight. To reflect the global characteristics of such networks, we study the eigentime identity determined by the reciprocal sum of non-zero eigenvalues of normalized Laplacian matrix. By the recursive relationship of eigenvalues at two successive generations, we find the eigenvalues and their corresponding multiplicities for two cases when $q$ is even or odd. Finally, we obtain the analytical expression of the eigentime identity and the scalings with network size of the weighted scale-free networks.