# Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Lyapunov Dimension

^{n}. Denote by T

_{x}F the Jacobian matrix of the mapping F at the point x. The continuous differentiability of F implies that:

_{j}(A) the singular numbers of (n×n)-matrix A, such that α

_{1}(A) ≥ α

_{2}(A) ≥ … ≥ α

_{n}(A). Consider the relation:

**Definition 1.**We say that a local Lyapunov dimension of the mapping F at the point x ∈ K is a number:

_{L}(F, x) = 0 if α

_{1}(T

_{x}F) < 1, and dim

_{L}(F, x) = n if α

_{1}(T

_{x}F)α

_{2}(T

_{x}F) … α

_{n}(T

_{x}F) ≥ 1.

**Definition 2.**[20,21] The Lyapunov dimension of the mapping F on the set K is a number:

**Definition 3.**A local Lyapunov dimension of a one-parameter group of mappings F

^{t}at the point x ∈ K is a number:

**Definition 4.**The Lyapunov dimension of mappings F

^{t}on the set K is a number:

_{0}, there exists a solution of Equations (3)x(t, x

_{0}), defined on t ∈ [0, +∞). Here, x(0, x

_{0}) = x

_{0}.

^{t}(x

_{0}) = x(t, x

_{0}) a shift operator along the solutions of Equations (3) and suppose that the set K ⊂ ℝ

^{n}is bounded and invariant: F

^{t}K = K, ∀t ∈ ℝ

^{1}. T

_{x}F

^{t}is the Jacobian matrix of the mapping F

^{t}at the point x.

_{0}, there exists a solution of Equations (4)y(t, y

_{0}), defined on t ∈ [0, +∞). Here, y(0, y

_{0}) = y

_{0}.

^{1}. ${T}_{y}{\tilde{F}}^{t}$ is the Jacobian matrix of the mapping ${\tilde{F}}^{t}$ at the point y.

^{−}

^{1}: y ↦ x that is continuously differentiable in the neighborhood of y.

**Lemma 1.**If for${T}_{x}{F}^{t}\underset{x\in K}{\mathrm{sup}}{\omega}_{{d}_{0}}({T}_{x}{F}^{t})<1$, then for the Lyapunov dimension of mapping F

^{t}of the set K, the following estimate:

_{0}, $s=\underset{x\in K}{\mathrm{sup}}\frac{{\displaystyle {\sum}_{i=1}^{j}{\mu}_{i}}}{|{\mu}_{j+1}|}$, 0 < s ≤ 1, j is the largest integer: j ∈ [1; n]; µ

_{i}= LE

_{i}(x), LE

_{i}(x) are Lyapunov exponents (LEs), ${\sum}_{i=1}^{j}{\mu}_{i}>0$, µ

_{j}

_{+1}< 0.

**Proof.**Consider an arbitrary number ${s}_{0}:0<\underset{x\in K}{\mathrm{sup}}\frac{{\displaystyle {\sum}_{i=1}^{j}{\mu}_{i}}}{|{\mu}_{j+1}|}<{s}_{0}\le 1$.

_{0}, Relation (5) is satisfied. □

**Theorem 1.**The Lyapunov dimension of the mapping F

^{t}of the set K is invariant with respect to the diffeomorphism$Q:K\to \tilde{K}$. Namely

_{0}+ h

_{y}, $\tilde{x}={x}_{0}+{h}_{x}$, Q

^{−}

^{1}(y

_{0}) = x

_{0}, ${Q}^{-}{}^{1}(\tilde{y})=\tilde{x}$,

^{−}

^{1}(property of diffeomorphism) implies the following relation:

_{1}, m

_{2}∈ ℝ

_{+}, are satisfied.

^{−}

^{1}(y), we obtain relations similar to (10):

_{1}, l

_{2}∈ ℝ

_{+}. Suppose that M = max(m

_{1}, m

_{2}, l

_{1}, l

_{2}).

_{1}< D

_{2}and that d is an arbitrary number: D

_{1}< d < D

_{2}. Consider the transformation Q. Then, from (14) and the definition of the Lyapunov dimension, it follows that D

_{2}≤ d, but this contradicts the initial assumption. Consequently, the inequality must be valid:

^{−}

^{1}, D

_{2}< d < D

_{1}. Then, from (15), it follows that D

_{1}≤ d. This is in contrast with the above assumption. Consequently:

^{t}on compact sets.

## 3. Lyapunov Dimension of the Shimizu–Morioka System

**Theorem 2.**Suppose that for the integers j ∈ [1, n] and s ∈ [0, 1], there exist a continuously differentiable function υ(x) and a nonsingular matrix S, such that:

_{L}K ≤ j + s.

**Theorem 3.**Suppose that K is a bounded invariant set of System (20): (0, 0, 0) ∈ K and that the following relations:

**Proof.**Consider a nonsingular matrix:

_{1}≥ λ

_{2}≥ λ

_{3}.

_{1}, µ

_{2}, µ

_{3}, µ

_{4}are running parameters, s ∈ [0, 1).

_{2}if B

_{1}B

_{3}≥ 0. This implies that (27) is equivalent to the system of inequalities:

_{4}> 0 if $2{\mu}_{1}\ge \frac{1}{2\lambda +1}({k}^{2}+{\mu}_{3})$. Then, from (30), we obtain:

_{4}if the discriminant of its left-hand side is nonnegative. This implies that:

_{1}≤ 0, for the last inequality to be valid, it is sufficient that:

_{L}K ≤ 2 + s for all s, satisfying (39). This implies that:

**Remark 1.**An oscillation can generally be easily numerically localized if the initial data from its open neighborhood in the phase space lead to a long-term behavior that approaches the oscillation. Therefore, from a computational perspective, it is natural to suggest the following classification of attractors [29–32], which is based on the simplicity of finding their basins of attraction in the phase space: an attractor is called a self-excited attractor if its basin of attraction intersects with any open neighborhood of an equilibrium; otherwise, it is called a hidden attractor. For a self-excited attractor, its basin of attraction is connected to an unstable equilibrium, and therefore (standard computational procedure), self-excited attractors can be localized numerically by the standard computational procedure: by constructing a solution using initial data from an unstable manifold in a neighborhood of an unstable equilibrium, observing how it is attracted and visualizing the oscillation. In contrast, the basin of attraction for a hidden attractor is not connected to any equilibrium. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of the multi-stability: the coexistence of attractors in multi-stable systems). Well-known examples of the hidden oscillations are nested limit cycles in the 16th Hilbert problem (see, e.g., [32,33]) and counterexamples to the Aizerman and Kalman conjectures on the absolute stability of nonlinear control systems [32,34–36].

## 4. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Self-excited local attractors in: (a) System (18) for α = 0.375, λ = 0.81; (

**b**) System (20) for α = 0.375, λ = 0.81.

**Figure 2.**Self-excited local attractors in: (

**a**) System (18) for α = 0.191450, λ = 0.81, “Burke and Shaw-like”; (

**b**) System (20) for α = 0.191450, λ = 0.81, “Burke and Shaw-like”.

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Leonov, G.A.; Alexeeva, T.A.; Kuznetsov, N.V.
Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System. *Entropy* **2015**, *17*, 5101-5116.
https://doi.org/10.3390/e17075101

**AMA Style**

Leonov GA, Alexeeva TA, Kuznetsov NV.
Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System. *Entropy*. 2015; 17(7):5101-5116.
https://doi.org/10.3390/e17075101

**Chicago/Turabian Style**

Leonov, Gennady A., Tatyana A. Alexeeva, and Nikolay V. Kuznetsov.
2015. "Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System" *Entropy* 17, no. 7: 5101-5116.
https://doi.org/10.3390/e17075101