## What is/are Positive Lyapunov?

Positive Lyapunov - First, by using a clock-dependent co-positive Lyapunov-Krasovskii function (CDCLKF), sufficient conditions are derived for stability of SPFDSs with dwell time.^{[1]}A co-positive Lyapunov-Krasovskii function is employed to study the stability and the robustness performance analysis.

^{[2]}

## average dwell time

Then, by using co-positive Lyapunov functional method together with average dwell time approach, some sufficient conditions of input–output finite-time stability for the considered system are derived.^{[1]}First, sufficient conditions derived by use of the co-positive Lyapunov function and the average dwell time method are presented to guarantee the proposed system is positive and exponentially stable.

^{[2]}By using the linear co-positive Lyapunov function (LCLF) and average dwell time (ADT) approach, a state feedback controller via asynchronous switching is designed and sufficient conditions are obtained to guarantee the corresponding closed-loop system is IO-FTS.

^{[3]}Based on the time-scheduled multiple co-positive Lyapunov-Krasovskii functional (MCLKF) method combined with fast average dwell time (ADT) techniques, a sufficient condition is obtained to ensure the underlying system is exponentially stable.

^{[4]}

## mode dependent average

Then, by giving the upper bound and the lower bound of the uncertainty and employing the multiple linear copositive Lyapunov function method and mode-dependent average dwell time scheme, some sufficient conditions are derived and applied to build the interval observer, which supply certain information at any instant: an upper bound and a lower bound are provided for each component of the states.^{[1]}Then, by constructing linear copositive Lyapunov functions and using the mode-dependent average dwell time (MDADT) approach, a static output feedback controller is constructed, and sufficient conditions are derived to guarantee that the corresponding closed-loop system is guaranteed cost finite-time stable (GCFTS).

^{[2]}By developing a novel multiple discontinuous co-positive Lyapunov–Krasovskii functional approach, the stability conditions are established for switched positive time delay systems by a linear programming approach under mode-dependent average dwell time switching.

^{[3]}

## finite time stability

By constructing a time-varying copositive Lyapunov function and utilizing the average impulsive interval approach, the finite-time stability criterion is established for the first time by exposing different impulsive effects.^{[1]}Then, explicit conditions for finite-time stability of positive linear time-delay system are proposed in terms of linear inequalities by introducing a time-varying linear copositive Lyapunov function.

^{[2]}

## guaranteed cost finite

Secondly, by using the average dwell time(ADT) approach and multiple linear co-positive Lyapunov function (MLCLF), two guaranteed cost finite-time controller are designed and sufficient conditions are obtained to guarantee the corresponding closed-loop systems are guaranteed cost finite-time stability(GCFTS).^{[1]}

## Two Positive Lyapunov

Specifically, the approach consists of five stages: (1) a newly proposed 5D hyperchaotic system with two positive Lyapunov exponents is applied to generate a pseudorandom sequence; (2) for each pixel in an image, a filtering operation with different templates called dynamic filtering is conducted to diffuse the image; (3) DNA encoding is applied to the diffused image and then the DNA-level image is transformed into several 3D DNA-level cubes; (4) Latin cube is operated on each DNA-level cube; and (5) all the DNA cubes are integrated and decoded to a 2D cipher image.^{[1]}Interestingly, for some regions we have found the so-called hyperchaos, here given by two positive Lyapunov exponents.

^{[2]}The performance analysis shows that the chaotic system has two positive Lyapunov exponents and high complexity.

^{[3]}It is shown that as a result of a secondary Neimark–Sacker bifurcation, a hyperchaos with two positive Lyapunov exponents can occur in the system.

^{[4]}Both the existence of two positive Lyapunov exponents and the Lyapunov dimension value show the hyperchaotic property of the system.

^{[5]}The new four-wing system exhibits two positive Lyapunov characteristic exponents and a large value of Kaplan-Yorke dimension indicating high complexity of the system.

^{[6]}

## Three Positive Lyapunov

The HFWMS with multiline equilibrium and three positive Lyapunov exponents presented very complex dynamic characteristics, such as the existence of chaos, hyperchaos, limit cycles, and periods.^{[1]}Numerical simulations are presented demonstrating that the system has a chaotic behavior with three positive Lyapunov exponents.

^{[2]}

## One Positive Lyapunov

This work presents numerical evidence that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent.^{[1]}In addition, there is involvement of chaos in mathematical systems that have one positive Lyapunov exponent.

^{[2]}

## positive lyapunov exponent

Specifically, the approach consists of five stages: (1) a newly proposed 5D hyperchaotic system with two positive Lyapunov exponents is applied to generate a pseudorandom sequence; (2) for each pixel in an image, a filtering operation with different templates called dynamic filtering is conducted to diffuse the image; (3) DNA encoding is applied to the diffused image and then the DNA-level image is transformed into several 3D DNA-level cubes; (4) Latin cube is operated on each DNA-level cube; and (5) all the DNA cubes are integrated and decoded to a 2D cipher image.^{[1]}It is well known that iterated function systems generated by orientation preserving homeomorphisms of the unit interval with positive Lyapunov exponents at its ends admit a unique invariant measure on (0, 1) provided their action is minimal.

^{[2]}Positive Lyapunov exponent (LE) is vital to identify a dynamical system being chaotic or hyperchaotic.

^{[3]}In this manner, since the existence of a positive Lyapunov exponent (LE+) is taken as an indication that chaotic behavior exists, and due to the huge search spaces of the design variables of chaotic oscillators, we show the application of differential evolution (DE) and particle swarm optimization (PSO) algorithms to maximize LE+.

^{[4]}Four different cases of the interaction of systems characterized by different numbers of positive Lyapunov exponents are considered.

^{[5]}The HFWMS with multiline equilibrium and three positive Lyapunov exponents presented very complex dynamic characteristics, such as the existence of chaos, hyperchaos, limit cycles, and periods.

^{[6]}Numerical simulations are presented demonstrating that the system has a chaotic behavior with three positive Lyapunov exponents.

^{[7]}Interestingly, for some regions we have found the so-called hyperchaos, here given by two positive Lyapunov exponents.

^{[8]}The performance analysis shows that the chaotic system has two positive Lyapunov exponents and high complexity.

^{[9]}Under the dual Roth type condition, we associate to a class of functions with \emph{subpolynomial} deviations of ergodic averages (corresponding to relative homology classes) \emph{distributional} limit shapes, which are constructed in a similar way to the \emph{limit shapes} of Birkhoff sums associated in \cite{MMY3} to functions which correspond to positive Lyapunov exponents.

^{[10]}The oscillators with the high positive Lyapunov exponent are implemented into a field-programmable gate array (FPGA), and afterwards they are synchronized in a master-slave topology applying three techniques: the seminal work introduced by Pecora-Carroll, Hamiltonian forms and observer approach, and open-plus-closed-loop (OPCL).

^{[11]}In this paper, we investigate the relationship between the coupling strengths and the extensive behaviour of the sum of the positive Lyapunov exponents of multiplex networks formed by coupled dynamical units.

^{[12]}Extensive homoclinic chaotic motion has been obtained from the bistable cc-beam resonator and validated by means of a positive Lyapunov exponent.

^{[13]}Positive maximal Lyapunov exponents reflected electric field dependence with positive Lyapunov exponents.

^{[14]}Time-delay chaotic systems can have hyperchaotic attractors with large numbers of positive Lyapunov exponents, and can generate highly stochastic and unpredictable time series with simple structures, which is very suitable as a secured chaotic source in chaotic secure communications.

^{[15]}This work presents numerical evidence that for discrete dynamical systems with one positive Lyapunov exponent the decay of the distance autocorrelation is always related to the Lyapunov exponent.

^{[16]}When a new chaotic oscillator is introduced, it must accomplish characteristics like guaranteeing the existence of a positive Lyapunov exponent and a high Kaplan–Yorke dimension.

^{[17]}The dynamics of the systems was shown to be chaotic by their positive Lyapunov exponents and the noninteger fractal dimension of their scattering fractals.

^{[18]}This system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 80 K), a large state amplitude and energy, and power spectral density with a wide bandwidth.

^{[19]}Other indications that the system has entered chaos include multiple frequencies, non-overlapping phase diagram, and positive Lyapunov exponent.

^{[20]}Rössler and Chen systems with time delay are shown to be hyperchaotic, which exhibits a more complex dynamics, including multiple positive Lyapunov exponents and infinite dimension.

^{[21]}We prove that the set of $C^r$, $0\leq r \leq \infty$ (or analytic) $k+1$-tuples of quasi periodic cocycles taking values in $SL_2(\mathbb{R})$ such that the random product of them has positive Lyapunov exponent contains a $C^0$ open and $C^r$ dense subset which is formed by $C^0$ continuity point of the Lyapunov exponent For $k+1$-tuples of quasi periodic cocycles taking values in $GL_d(\mathbb{R})$ for $d>2$, we prove that if one of them is diagonal, then there exists a $C^r$ dense set of such $k+1$-tuples which has simples Lyapunov spectrum and are $C^0$ continuity point of the Lyapunov exponent.

^{[22]}The presence of positive Lyapunov exponent indicated chaotic behavior of the map.

^{[23]}There have been several recent proofs of one-dimensional Anderson localization based on positive Lyapunov exponent that hold for bounded potentials.

^{[24]}Results show that the short-range interactions of two solitons will evolve into chaotic self-trapped optical beams, which keep the beam width nearly invariant and possess the chaotic properties denoted by the positive Lyapunov exponents and spatial decoherence.

^{[25]}It is shown that as a result of a secondary Neimark–Sacker bifurcation, a hyperchaos with two positive Lyapunov exponents can occur in the system.

^{[26]}The existence of positive Lyapunov exponent showed the chaotic attractor present in the system.

^{[27]}The numerical simulation of the Landau-Lifshitz-Gilbert equation indicates the positive Lyapunov exponent for a certain range of the feedback rate, which identifies the existence of chaos in a nanostructured ferromagnet.

^{[28]}The positive Lyapunov exponent and spatial decoherence denote the chaotic behavior, while the invariance of the beam width during the evolution and the quasi-elastic collision during the interaction demonstrate the soliton-like properties of the self-trapped beams.

^{[29]}Both the existence of two positive Lyapunov exponents and the Lyapunov dimension value show the hyperchaotic property of the system.

^{[30]}Over the last 40 years, the design of n-dimensional hyperchaotic systems with a maximum number (n−2) of positive Lyapunov exponents has been an open problem for research.

^{[31]}We investigate the transition from periodic to chaotic dynamics and show the increase of the number of positive Lyapunov exponents as the number of atoms grows.

^{[32]}Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out of time order correlators.

^{[33]}In addition, there is involvement of chaos in mathematical systems that have one positive Lyapunov exponent.

^{[34]}The predictive performance is examined in terms of the Kolmogorov−Sinai entropy and the Kaplan − Yorke dimension of a chaotic attractor in comparison with those for chaotic flow models having a single positive Lyapunov exponent.

^{[35]}The numerical results show that there exists a positive Lyapunov exponent in the VdPVP.

^{[36]}The chaos has been investigated based on positive Lyapunov exponent value.

^{[37]}This ILM may have been previously overlooked because of its positive Lyapunov exponent, meaning that there might be larger ranges of parameters capable of supporting these energy localizations.

^{[38]}

## positive lyapunov function

Then, by giving the upper bound and the lower bound of the uncertainty and employing the multiple linear copositive Lyapunov function method and mode-dependent average dwell time scheme, some sufficient conditions are derived and applied to build the interval observer, which supply certain information at any instant: an upper bound and a lower bound are provided for each component of the states.^{[1]}The average impulsive interval approach and co-positive Lyapunov function method are employed to derive the sufficient conditions.

^{[2]}By means of the mode-dependent dwell time approach and a class of discretized co-positive Lyapunov functions, some stability conditions of switched positive linear systems with all modes unstable are derived in both the continuous-time and the discrete-time cases, respectively.

^{[3]}ABSTRACT The key result of the paper exploits the duality of arbitrary switching positive linear systems, in order to derive a sufficient condition for the existence and construction of diagonal quadratic copositive Lyapunov functions.

^{[4]}In this paper, a complete procedure for the study of the output regulation problem is established for a class of positive switched systems utilizing a multiple linear copositive Lyapunov functions scheme.

^{[5]}First, by employing the idea of impulse interval partitioning, an impulse-time-dependent discretized copositive Lyapunov function is proposed to analyze the robust stability and L1-gain performance of the considered system without control inputs, and several stability conditions and L1-gain criteria are respectively derived.

^{[6]}Then, by constructing linear copositive Lyapunov functions and using the mode-dependent average dwell time (MDADT) approach, a static output feedback controller is constructed, and sufficient conditions are derived to guarantee that the corresponding closed-loop system is guaranteed cost finite-time stable (GCFTS).

^{[7]}By constructing a time-varying copositive Lyapunov function and utilizing the average impulsive interval approach, the finite-time stability criterion is established for the first time by exposing different impulsive effects.

^{[8]}Using linear copositive Lyapunov functions, a control framework for the distributed model predictive controller of constrained positive systems is established.

^{[9]}First, a set of state-feedback controllers for the considered system are designed by using a stochastic co-positive Lyapunov function integrated with linear programming approach.

^{[10]}First, sufficient conditions derived by use of the co-positive Lyapunov function and the average dwell time method are presented to guarantee the proposed system is positive and exponentially stable.

^{[11]}Then, explicit conditions for finite-time stability of positive linear time-delay system are proposed in terms of linear inequalities by introducing a time-varying linear copositive Lyapunov function.

^{[12]}Then, all tractable conditions guaranteeing the solvability of the addressed problem are presented in terms of linear matrix equalities as well as linear vector inequalities by means of the constructed multiple linear copositive Lyapunov functions.

^{[13]}Secondly, by using the average dwell time(ADT) approach and multiple linear co-positive Lyapunov function (MLCLF), two guaranteed cost finite-time controller are designed and sufficient conditions are obtained to guarantee the corresponding closed-loop systems are guaranteed cost finite-time stability(GCFTS).

^{[14]}First, a linear co-positive Lyapunov function is constructed for positive systems.

^{[15]}Unlike the existing results, the obtained condition permits the ascent of the multiple linear copositive Lyapunov functions caused by detection delay and false alarm, which is less conservative.

^{[16]}By using the linear co-positive Lyapunov function (LCLF) and average dwell time (ADT) approach, a state feedback controller via asynchronous switching is designed and sufficient conditions are obtained to guarantee the corresponding closed-loop system is IO-FTS.

^{[17]}Then a time-varying co-positive Lyapunov function for periodic piecewise positive systems is employed and a sufficient condition for the asymptotic stability of the system is established.

^{[18]}Then the interval observer is constructed and we analyze the ultimately uniform boundedness of the error systems by multiple linear copositive Lyapunov function.

^{[19]}Necessary conditions of decay-rate-dependent exponential mean stability are proved by applying the available stochastic stability results of positive MJLSs, and the sufficiency is addressed by a linear stochastic co-positive Lyapunov function method.

^{[20]}Then, through multiple linear co-positive Lyapunov function, sufficient conditions of stochastic stability for semi-Markov positive systems are derived.

^{[21]}Some improved stability conditions are given for the positive LTV systems and switched positive LTV systems by using time-varying copositive Lyapunov functions (CLFs) and switched time-varying CLF, respectively.

^{[22]}First of all, sufficient conditions are obtained for stochastic stability and L 1 performance under the transition rate in a manner of stochastic variation and arbitrary variation by means of choosing a linear co-positive Lyapunov function.

^{[23]}Then, via a switched dwell-time-dependent co-positive Lyapunov functions (SDTLFs) approach, convex sufficient conditions on L1-gain analysis and asynchronous L1-gain control of DSPLSs with interval uncertainties are derived.

^{[24]}Based on the comparison principle, with the help of discretised multiple linear copositive Lyapunov functions, some delay-independent stability results are obtained.

^{[25]}

## positive lyapunov functional

Then, by using co-positive Lyapunov functional method together with average dwell time approach, some sufficient conditions of input–output finite-time stability for the considered system are derived.^{[1]}An improved delay-dependent MADT that takes delay-independent MADT switching as a special case is provided, and a mode-dependent and time-dependent linear copositive Lyapunov functional is presented to establish the exponential stability with the weighted l 1 performance of the filtering error system under the improved MADT.

^{[2]}First, a stochastic co-positive Lyapunov functional is constructed for the systems.

^{[3]}

## positive lyapunov characteristic

Finally, we are about to investigate the invariant properties of one through the transformations such as topological conjugacy, topological equivalence and kinematically similar and then show that topological entropy of one is equal to sum of positive Lyapunov characteristic exponents.^{[1]}The new four-wing system exhibits two positive Lyapunov characteristic exponents and a large value of Kaplan-Yorke dimension indicating high complexity of the system.

^{[2]}