Methods of Qualitative Theory in Nonlinear Dynamics

Shilnikov, L. P.; Shilnikov, Andrey; Turaev, Dmitry; Chua, Leon O.

doi:10.1142/4221

Cited by 362 publications

(268 citation statements)

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“…The breakthrough in this direction came in recently with the discovery of two novel transitions occurring at the loss of stability of a tonic spiking periodic orbit though the homoclinic saddlenode bifurcation of periodic orbits. The first transition, reversible and continuous, which occurs in the reduced model of the leech heart interneuron (Shilnikov andCymbalyuk 2004, 2005) and in a modified HindmarshRose model of square-wave burster (Shilnikov and Kolomiets 2008), is based on the blue sky catastrophe (Shilnikov et al 1998(Shilnikov et al , 2001. The feature of the second transition mechanism is the bi-stability of the coexisting tonic spiking and bursting in a neuron model (Shilnikov and Cymbalyuk 2004;.…”

Section: Introductionmentioning

confidence: 99%

Variability of bursting patterns in a neuron model in the presence of noise

et al. 2009

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Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.

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Section: Introductionmentioning

confidence: 99%

Variability of bursting patterns in a neuron model in the presence of noise

et al. 2009

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“…Any point on a periodic trajectory returns to the starting position in times T, 2T, 3T, etc. The minimum of these times is called period [14]. In the neighborhood of the equilibrium state of a saddle-type nonlinear system, there exists an invariant manifold that includes all trajectories until they remain in this neighborhood.…”

Section: Dtmentioning

confidence: 99%

“…In Hamiltonian systems [4,14], the closed solution (3.2) separating periodic motions is termed a separatrix. …”

Section: Curves Separating Periodic Solutions Of a Bistablementioning

confidence: 99%

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Oscillations of conservative systems with complex trajectories

Martynyuk

Nikitina

2008

Int Appl Mech

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The principles of formation of closed and quasiperiodic orbitally stable trajectories of conservative systems are formulated. It is revealed that irregular oscillations are due to the orbital instability of quasiperiodic oscillations. A bistable oscillator with periodic forcing is considered. The existence of random oscillations at low level of energy and the conditions for orbitally stable oscillations are established Keywords: domain of periodic solutions, random oscillations, chaos Introduction. Modern methods of qualitative analysis of motion are based on studies due to Poincaré [11], Birkhoff [3], Lyapunov [7], Andronov [1], etc. (see [4,5,14] and the references therein). In the last 20 years, there have been extensive studies in this area [6,8,13] promoted by applications in mechanics [2,12,15,17,[19][20][21][22][23] and other natural sciences. The concept of complex motions includes chaos, meaning, for example, walk on sections of an orbitally unstable trajectory for an infinite time. An example of regular motion is quasiperiodic motion. The objective of the analysis of conservative systems with complex trajectories is to estimate the domain in which and the energy at which the motions are regular. This can be done with simple (two-frequency) models of mechanical systems with several singular points. Considering classical problems, we attempt to resolve this issue based on the symmetry principle due to 31]. After a preliminary analysis, we will estimate these domain and energy. An analysis of quasiperiodic trajectories shows that they are subject to neutral attraction. By neutral attraction is meant that the integral sum of quasicharacteristic exponents of the system of variational equations is equal to zero. In the periodic case, the domain of existence of a closed trajectory may not be a domain of stable oscillations. By stability of oscillations is meant orbital stability. We will consider a system that has a family of solutions in some domain.1. Preliminaries. Consider a material system with a finite number of degrees of freedom. Its motion is described by the equation

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“…(1.1). The qualitative study of the ω-periodic system (1.2) is based on the investigation of the Poincaré map in the phase plane [5].…”

Section: Introductionmentioning

confidence: 99%

Qualitative Theory and Identification of Dynamic System with One Degree of Freedom

Volkova

Schneider

2005

Int Appl Mech

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Dynamic systems described by nonlinear differential equations of the second order are studied. It is assumed that certain preliminary information on the dissipative or elastic characteristics of systems is known. A new approach is demonstrated to obtaining full information on unknown or partially known characteristics of a system from measurements of not only displacements but also velocities and accelerations Keywords: dynamic system with one degree of freedom, nonlinear differential equations of the second order, identification, experimental observation, external excitation, graphical criteria of nonlinearity 1. Introduction. The identification of nonlinear dynamic systems is often more a subjective art than a direct application of some particular method in systems theory. A nonlinear problem is subjective because although there are many methods from which to choose, there is no general approach to detect and characterize the type of nonlinear systems.Identification problems differ in their purpose and initial data [6, 10, 12]. The following methods and approaches are applied to identify a mechanical system [1,8,11]: spectral analysis and reverse-path formulation, Volterra and Wiener series, nonlinear autoregressive moving average models, the restoring force method, the describing function methods, direct parameter estimation, Hilbert transforms and wavelet transforms, neural networks, and general nonlinear experimental testing techniques. Some of them apply only to particular types of systems [4].Despite the great number of investigations in this field of science, the development of new methods and approaches is currently central.Mathematical models of objects, constructed in one way or another, should meet two main requirements: they must be relatively simple and must make it possible to solve the corresponding problems.We will consider mechanical systems described by the scalar differential equation

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Methods of Qualitative Theory in Nonlinear Dynamics

Cited by 362 publications

References 0 publications

Variability of bursting patterns in a neuron model in the presence of noise

Variability of bursting patterns in a neuron model in the presence of noise

Oscillations of conservative systems with complex trajectories

Qualitative Theory and Identification of Dynamic System with One Degree of Freedom

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