N-DIMENSIONAL CANONICAL CHUA'S CIRCUIT

1993 ◽  
Vol 03 (01) ◽  
pp. 239-258 ◽  
Author(s):  
LJ. KOCAREV ◽  
LJ. KARADZINOV ◽  
L. O. CHUA
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Measure Zero ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Electrical Circuit ◽  
Continuous Vector ◽  
Linear Vector ◽  
Linear Dynamic ◽  
Canonical Chua’S Circuit

In this paper we present an n-dimensional canonical piecewise-linear electrical circuit. It contains 2n two-terminal elements: n linear dynamic elements (capacitors and inductors), n - 1 linear resistors and one nonlinear (piecewise-linear) resistor. This circuit can realize any prescribed eigenvalue pattern, except for a set of measure zero, associated with (i) any n-dimensional two-region continuous piecewise-linear vector fields and (ii) any n-dimensional three-region symmetric (with respect to the origin) piecewise-linear continuous vector fields. We also proved a theorem that specifies the conditions for a vector field, realized with our canonical circuit, to have two or three equilibrium points.

ON THE GENERALITY OF THE UNFOLDED CHUA'S CIRCUIT

1996 ◽  
Vol 06 (05) ◽  
pp. 801-832 ◽  
Author(s):  
CHAI WAH WU ◽  
LEON O. CHUA
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Topological Conjugacy ◽  
Chua’S Oscillator ◽  
State Observability ◽  
Almost All ◽  
Special Case ◽  
Scalar Nonlinearity

In this paper, we study the generality of Chua's oscillator by deriving a class of vector fields that Chua's oscillator is equivalent to. For the class of vector fields with a scalar nonlinearity, we prove that under certain conditions, two such vector fields are topologically conjugate if the Jacobian matrices at each point have the same eigenvalues and the equilibrium points are matched up. We show how these conditions are related to the complete state observability of a corresponding linear system. These results are used to show that the n-dimensional Chua's oscillator is topologically conjugate to almost every vector field in this class. We comment on the special case when the vector field is piecewise-linear and in particular when the vector field is 2-segment piecewise-linear. These results are illustrated by transforming several systems studied in the literature into equivalent Chua's oscillators. We also extend some of these results to the case of several scalar nonlinearities. As a corollary we prove that almost all piecewise-linear vector fields with parallel boundary planes are topologically conjugate if the boundary planes and equilibrium points are the same and the eigenvalues in corresponding regions are the same. We also give a dual result of topological conjugacy.

scholarly journals Three-Saddle-Foci Chaotic Behavior of a Modified Jerk Circuit with Chua’s Diode

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1803
Author(s):  
Pattrawut Chansangiam
Keyword(s):  
Equilibrium Point ◽  
Chaotic Behavior ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Equilibrium Points ◽  
Initial Point ◽  
Electrical Circuit ◽  
Nonlinear Resistor ◽  
Voltage Current Characteristic ◽  
Time Waveform

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

THE ŁOJASIEWICZ EXPONENT AT AN EQUILIBRIUM POINT OF A STANDARD CNN IS 1/2

2006 ◽  
Vol 16 (08) ◽  
pp. 2191-2205 ◽  
Author(s):  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Vector Field ◽  
Equilibrium Point ◽  
Critical Point ◽  
Vector Fields ◽  
Equilibrium Points ◽  
Łojasiewicz Inequality ◽  
Łojasiewicz Exponent ◽  
Lojasiewicz Inequality ◽  
Output Trajectory ◽  
Lojasiewicz Exponent

In the sixties, Łojasiewicz proved a fundamental inequality for vector fields defined by the gradient of an analytic function, which gives a lower bound on the norm of the gradient in a neighborhood of a (possibly) non-isolated critical point. The inequality involves a number belonging to (0, 1), which depends on the critical point, and is known as the Łojasiewicz exponent. In this paper, a class of vector fields which are defined on a hypercube of ℝn, is considered. Each vector field is the gradient of a quadratic function in the interior of the hypercube, however it is discontinuous on the boundary of the hypercube. An extended Łojasiewicz inequality for this class of vector fields is proved, and it is also shown that the Łojasiewicz exponent at each point where a vector field vanishes is equal to 1/2. The considered fields include a class of vector fields which describe the dynamics of the output trajectories of a standard Cellular Neural Network (CNN) with a symmetric neuron interconnection matrix. By applying the extended Łojasiewicz inequality, it is shown that each output trajectory of a symmetric CNN has finite length, and as a consequence it converges to an equilibrium point. Furthermore, since the Łojasiewicz exponent at each equilibrium point of a symmetric CNN is equal to 1/2, it follows that each (state) trajectory, and each output trajectory, is exponentially convergent toward an equilibrium point, and this is true even in the most general case where the CNN possesses infinitely many nonisolated equilibrium points. In essence, the obtained results mean that standard symmetric CNNs enjoy the property of absolute stability of exponential convergence.

Limit Cycles Bifurcating from a Period Annulus in Continuous Piecewise Linear Differential Systems with Three Zones

2017 ◽  
Vol 27 (02) ◽  
pp. 1750022 ◽  
Author(s):  
Maurício Firmino Silva Lima ◽  
Claudio Pessoa ◽  
Weber F. Pereira
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Poincaré Map ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Vector ◽  
Period Annulus ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Two Parameter

We study a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map and some techniques for proving the existence of limit cycles for smooth differential systems, we prove that this class admits at least two limit cycles that appear by perturbations of a period annulus. Moreover, we describe the bifurcation of the limit cycles for this class through two examples of two-parameter families of piecewise linear vector fields with three zones.

scholarly journals ∞-jets of diffeomorphisms preserving orbits of vector fields

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Sergiy Maksymenko
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Hamiltonian Vector ◽  
Homogeneous Polynomials ◽  
Local Flow ◽  
Linear Vector ◽  
Identity Component ◽  
Hamiltonian Vector Fields

AbstractLet F be a C ∞ vector field defined near the origin O ∈ ℝn, F(O) = 0, and (Ft) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝn → ℝn at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney Wr topology. Then contains a subset consisting of maps of the form Fα(x)(x), where α: ℝn → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = .In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O.We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.

scholarly journals LIMIT CYCLES FOR A CLASS OF CONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS WITH THREE ZONES

2012 ◽  
Vol 22 (06) ◽  
pp. 1250138 ◽  
Author(s):  
MAURÍCIO FIRMINO SILVA LIMA ◽  
JAUME LLIBRE
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Vector Fields ◽  
Poincaré Map ◽  
Piecewise Linear ◽  
Differential Systems ◽  
Linear Vector ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
Unique Limit

In this paper, we consider a class of planar continuous piecewise linear vector fields with three zones. Using the Poincaré map, we show that these systems admit always a unique limit cycle, which is hyperbolic.

BIFURCATION SETS OF SYMMETRICAL CONTINUOUS PIECEWISE LINEAR SYSTEMS WITH THREE ZONES

2002 ◽  
Vol 12 (08) ◽  
pp. 1675-1702 ◽  
Author(s):  
EMILIO FREIRE ◽  
ENRIQUE PONCE ◽  
FRANCISCO RODRIGO ◽  
FRANCISCO TORRES
Keyword(s):  
Linear Systems ◽  
Canonical Form ◽  
Vector Fields ◽  
Piecewise Linear ◽  
Oscillatory Behavior ◽  
Van Der Pol ◽  
Linear Vector ◽  
Piecewise Linear Systems

Planar symmetrical continuous piecewise linear vector fields with three zones are thoroughly analyzed. Special emphasis is placed on their oscillatory behavior. The study is made by using a Van der Pol canonical form which captures the most interesting dynamics and minimizes the number of parameters to be dealt with. This work is a continuation of a previous paper [Freire et al., 1998] and uses the same approach and techniques.

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